Combating Bilateral Edge Noise for Robust Link Prediction
Authors: Zhanke Zhou
Jiangchao Yao
Jiaxu Liu
Xiawei Guo
quanming yao
LI He
Liang Wang
Bo Zheng
Bo Han
Venue: NeurIPS 2023 poster (2023)
Keywords: Robust link prediction
Edge noise
TL;DR: We provide an information-theory-guided principle and its two instantiations for robust link prediction under bilateral edge noise.

Abstract: Although link prediction on graphs has achieved great success with the development of graph neural networks (GNNs), the potential robustness under the edge noise is still less investigated. To close this gap, we first conduct an empirical study to disclose that the edge noise bilaterally perturbs both input topology and target label, yielding severe performance degradation and representation collapse. To address this dilemma, we propose an information-theory-guided principle, Robust Graph Information Bottleneck (RGIB), to extract reliable supervision signals and avoid representation collapse. Different from the basic information bottleneck, RGIB further decouples and balances the mutual dependence among graph topology, target labels, and representation, building new learning objectives for robust representation against the bilateral noise. Two instantiations, RGIB-SSL and RGIB-REP, are explored to leverage the merits of different methodologies, i.e., self-supervised learning and data reparameterization, for implicit and explicit data denoising, respectively. Extensive experiments on six datasets and three GNNs with diverse noisy scenarios verify the effectiveness of our RGIB instantiations. The code is publicly available at: https://github.com/tmlr-group/RGIB.

Combating Bilateral Edge Noise for Robust Link Prediction
Zhanke Zhou'
Jiangchao Yao
Jiaxu Liu
Xiawei Guo
Quanming Yao
Li He
Liang Wang*
Bo Zheng*
Bo Han
Honn Kong Baptist University 2CMIC, Shanghai Jiao Tong University
Shanghai AI Laboratory "Alibaba Group Tsinghua Unversity
(eszkzhou, bhanml) (liujiaxu.ijx, guoxiawei, heli, wangliang) taobaa.cco
uunarker@sttu.ddue qyaoaa waaaae@isnninnnnnnnee
bozheng @ allbbaba-inc.com
Abstract
Although link prediction on graphs has achieved great success with the develop- ment of graph neural networks (GNNs), the potential robustness under the edge noise is still less investigated. To close this gap. we first conduct an empirical study to disclose that the edge noise bilaterally perturbs both input topology and target label, yielding severe performance degradation and representation collapse. To address this dilemma, we propose an wwwrmmaiioonttoor-uuuuiiee bust Graph Information Bottleneck (RGIB), to extract reliable supervision signals and avoid representation collapse. Different from the basic information bottle- neck, RGIB further decouples and balances the mutual dependence among graph topology. target labels, and representation, building new learning objectives for robust representation against the bilateral noise. Two instantiations, RGIB-SSL and RGIB-REP, are explored to leverage the merits of different methodologies, iee. self-supervised learning and data reparameterization, for implicit and explicit data denoising, respectively. Extensive experiments on six datasets and three GNNs with diverse noisy scenarios verify the effectivvness of our RGIB instantiations. The code is publicly available at: https:// //g/ thub. com/tmlr- group/RGIS
principle, Ro-
1 Introduction
Link prediction [25] the process of determining whether two nodes in graph are likely to be connected. As fundamental task in graph learning, it has attracted growing interest in real-world applications, e.g., drug discovery [18], knowledge graph completion [3], and question answering [17].
While recent advances in graph neural networks (GNNs) [21, 12, 22, 48, 55] have achieved superior performances, the poor robustness under the edge noise still a practical bottleneck the current deep graph models [11, 10, 36, 8]. Early works explore improving the robustness of GNNs under the node label noise [7, 24] by means of the smoothing effect of neighboring nodes. Other methods achieve similar goal via randomly removing edges [30] actively selecting the informative nodes or edges and pruning the task-irreles inre-irrelevant ones [50, 27]. However, when applying these noise-robust methods to the link prediction with noise, only marginal improvements are achieved (Sec. 5). The attribution is that different from the label noise, the edge noise here is bilateral: can naturally deteriorate both the inpul graph topology and the targel edge labels, as illustrated in Fig. 1. Such bilateral noise is practical in woaldw graph data [1 36). Nonetheless, previous works that only consider unilateral noise in input space or label space cannot effectively deal with such coupled scenario. This raises new challenge tackling the bilateral edge noise for robust link prediction.
"Corespondence to Bo Han Shhanm@@comp.keee cdu.hkk and Jiangehao Yao (sunarker @ Satnarker@sjtu.ddu.co edu.cdd..cc..
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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noisy inputs
node repre. U
edge repre. H
noisy labels Y
GNN encoding
GNN decoding
Predict unseen edges
clean edges
noisy edges
Figure I:
Link prediction with bilateral edge noise. The GNN takes the graph G (A, X) as inputs,
generates the edge representation H, and then predicts the existence of unseen edges with labels Y.
In this paper, we first disclose that the bilateral edge noise leads to the severe edge representation collapse behind the performance degradation (Sec. 3.1), Note the collapsed representation reflected by much lower alignment and poorer uniformity (Sec. 3.2), which are used for quantitive and qualitative analysis of edge noise from representation perspective [35]. To solve this, we propose an cerimaatiionthoor-guuuiiie principle named Robust Graph Information Bottleneck (RGIB) (Sec. 4.1).
Conceptually, the RGIB principle is with new learning objectives that decouple the mutual in- formation (MI) among noisy inputs, noisy la- bels, and representation. As illustrated in Fig. 2, RGIB generalizes the basic GIB [39] learn robust representation that is resistant to the bilat- eral edge noise. Technically, we provide two in- stantiations of RGIB based on different method- ologies: (1) the RGIB-SSI. utilizes contrastive pairs with automatically augmented views to a self- supervised learning manner, which benefits from its intrinsic robustness to label noise (Sec. 4.2); (2) the RGIB-REP explicitly purifies the graph topol- ogy and supervision targets with the reparameterization mechanism, which enables further modeling of the mutual patterns of noise from both input and label spaces (Sec. 4.3). Both instantiations are equipped with adaptive designs, aiming effectively estimate and balance the informative terms in a tractable manner, i.e., the hybrid augmentation algorithm and self-adversarial alignment loss for RGIB-SSI; the relaxed information constraints on topology space and label space for RGIB-REP. Empirically, the two instantiations of RGIB work effectively under extensive noisy scenarios and can be seamlessly integrated with various GNNs (Sec. 5). Our contributions are summarized as follows.
RGIB
A
Y
Y
Clean
Clean
Nolsy
Notty
① I(A: 1(A: P) 3 I(:HIA)
optimal
A [[A (2 3) = ((:H)
Figure 2: The basic GIB and the proposed RGIB.
Note RGIB balances signals O, 3 4 to capture
form the informative regularization
the optimal H and discard noisy signals in red.
To our best knowledge, we are the first to study the robustness problem of link prediction under bilateral edge noise. We reveal that the bilateral noise can bring severe representation collapse and performance degradation, and such negative impacts are general to common datasets and GNNs.
We propose a general learning framework, RGIB, with representation learning objectives promote GNNs' robustness. Two instantiations, RGIB-SSI. and RGIB-REP, are proposed upon different methodologies that are equipped with adaptive designs and extensive theoretical analysis.
Without modifying the GNN architectures, the RGIB achieves state-of-the-art results on 3 GNNs and 6 datasets under various noisy scenarios, obtaining up to 12.9% promotion in AUC. The distribution of leamned representations is notably recovered and more robust the bilateral noise.
2 Preliminaries
Notations. We denote V= (w: as the set of nodes and E [er as the edges. With adjacent matrix A and node features X, a graph is denoted as G (A, where A, = means there is an edge esj between v; and vy. X E D the D-dimension node feature of ú; The link prediction task is to indicate the existence of query edges that are not observed in A. The binary labels of these query edges are denoted as Y, where for negative edges and for positive edges. The I (:::)) indicates the mutual information between two variables. Detailed notations are in Tab. 9.

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GNNs for Link Prediction. Here, we follow the encode-decode framework (Fig. 1) for link predic- tion [22], where the GNN architecture can be GCN [21], GAT [33], or SAGE [15]. Given a L-layer x for each node es E V are
GNN fw( (-) with learnable weights w, the node representation obtained by a L-layer message propagation as the encoding process. For decoding, the logits Φe., of each query edge Eij are computed by projecting the edge representation h. =u; O u probability. Note that the representation of all query edges is denoted as H and namely fw (A, X) H. Finally, the optimization objective is to minimize the binary classification loss with logits &e., and labels Y.j-
Topological denoising approaches. A natural way to tackle the input edge noise is to directly clean the noisy graph. Sampling-based methods, such as DropEdge [30], NeuralSparse [50], and PTDNet [27], are proposed to remove the task-irrelevant edges. Besides, as GNNs can be easily fooled by adversarial networks with only few perturbed edges [4, 52, 9, 6]. defending methods GCN-jaccard [38] and GIB [39] are designed for pruning adversarial edges from the poisoned graphs.
techniques. To tackle the general problem of noisy labels, sample-selection- based methods such as Co-teaching [16] let two neural networks teach each other with small-loss samples based on the memorization effect [2]. Besides, peer loss function [26] pairs independent peer examples for supervision and works within the standard empirical risk minimization framework without noise priors. For tackling label noise on graphs, label propagation techniques [7, 24] are designed to propagate the reliable signals from clean nodes to noisy ones, which are nonetheless entangled with the node annotations and node classification task that cannot be directly applied here.
Label-noise-resittee
3 An Empirical Study of Bilateral Edge Noise
In this section, we attempt to figure out how GNNs behave when learning with the edge noise and what are the latent mechanisms behind it. We first present an empirical study in Sec. 3.1, and then investigate the negative impact of noise through the lens of representation distribution in Sec. 3.2.
3.1 How GNNs perform under bilateral edge noise?
To quantitatively study the impact of edge noise, we simulate different levels of perturbations properly on a range of GNNs benchmark datasets, as elaborated Def. 3. Note the data split manner adopt by most relevant works [22, 48, 55] randomly divides partial edges observations and the others as prediction targets. Hence, the noisy edges will be distributed to input Á and labels Y. With the bilateral noise defined in Def. 3.1, we then conduct a comprehensive empirical study. A further discussion on the edge noise is in Appendix B.I, and full evaluations can be found in Appendix D.I.
Definition 3.1 (Bilateral edge noise). Given a clean training data, i.e., observed graph G = (A,
and labels Y E (0. 1} of query edges, the noisy adjacence Á is generated by directly adding edge noise to the original adjacent matrix A while keeping the node features X unchanged. The noisy labels Y are similarly generated by adding edge noise the labels Y. Specifically, given noise ratio Eas the noisy edges A' (Å = A A') are generated by flipping the zero element in A as one with the probabil- ity Ea- It satisfies that A' O A = O and Ea = [nonzero(A)] [omnareeAA]lInmazeeo((]]]]] nonzero(()). Similarly, noisy la- bels are generated and added to the original labels, where Ey [nonsero(Y ] nonzero(( (Y)) |//nossero(Y))
Observation 3.1. As illustrated in Fig. 3, the bilateral edge noise causes a significant drop in performance, and a larger noise ratio generally leads to greater degradation. It means that the standardly trained GNNs are vulnerable to the bilateral edge noise, yielding a severe robustness problem. As will be shown in Sec. 5, the performance drop brought by the bilateral noise is much greater than that of the unilateral input noise or label noise. However, none of the existing methods can effectively tackle such bilateral noise, since they only consider unilateral noise either in input space or label space. Thus, it is necessary to devise a robust method accordingly. To this end, we further investigate the underneath noise effect follows.
0.90
GCN-Cara GCN CIt--Cieeeerr
Gora
GAT Citeseer
0.05
SAGE-Com
SAGE Citeseer
Test AUC
0.00
0.70
clean
--20%
c=40%
2-60%
Figure 3: Link prediction performance in AUC with the bilateral edge noise.
The bilateral noise ratio 40% means
c = 40% with both noisy A and Y.
3

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Table 1: Mean values of alignment, which are calculated as the L2 dis- lance of representations of randomly per- turbed graphs Ai, A2, i.e.,
positive samples
poshiive samples
possiive samples
positive samples
negative samples
negative samples
negative samples
negattve samples
Align N Ei=1 [| H 2- Representation H. = (Ai. and
H = fw X).
dataset
Cora
Citeseer
(a) Clean data
(b) 20%
(c)
40%
(d) 60%
clean
.616
.455
Figure 4: Uniformity distribution on Cora dataset. Representations
8 20%
.687
.586
of query edges in the test set are mapped to unit circle of R? with normalization followed by the Gaussian kernel density estimation as [35]. Both positive and negative edges are expected be uniformly distributed.
40%
.695
.689
8 60%
.732
.696
3.2
Understanding the noise impact via inspecting the representation distribution
Recall that the edge representation is extracted by forward pass as H = fw (A, and the model optimized by stochastic gradient descent as within Å and Y, the edge representation H can be directly influenced, since the training neglects the adverse effects of data corruption. Besides, the GNN readouts the edge logit &e,, based on node representations h; and hj. which are possibly collapsed under the bilateral edge noise [14, 28).
V Y). When encountering noise
To analyze the edge noise from the perspective of representation H, we use two recent concepts [35] in representation learning: (1) alignment quantifies the stability of GNN when encountering edge noise in the testing phase. It is computed as the distance of representations between two randomly augmented graphs. A higher alignment means being more resistant input perturbations; (2) uniformity qualitatively measures the denoising effects of GNN when learning with edge noise in the training phase. Overall, a greater uniformity implies that the representation of query edges are more uniformly distributed on the unit hypersphere, preserving more information about the original data.
Observation 3.2. As be seen from Tab. and Fig. 4, poorer alignment and worse uniformity are brought by a severer edge noise. As the noise ratio e increases, the learned GNN fw)(-) is more sensitive input perturbations the alignment values sharply increased, and the learned edge representations tend to be less uniformly distributed and gradually collapse into individual subregions. That is, the learned edge representation H is severely collapsed under the bilateral edge noise, which is strongly correlated with the observation in Sec. 3.1 and reflects an undesirable property of GNNs.
Robust Graph Information Bottleneck
Without prior knowledge like the noise ratio, without the assistance of auxiliary datasets, and even without modifying GNN architectures. how can representations be resistant to the bilateral noise? Here, we formally build method to address the above problems from the perspective of robust graph information bottleneck (Sec. 4.1) and design its two practical instantiations (Sec. 4.2 and Sec. 4.3).
4.1 The principle RGIB
Recall in Sec. 3.2, the edge representation H is degraded. To robustify H. one can naturally utilize the information constraint based on the graph information bottleneck (GIB) [39, 46], i.e., solving
min GIB & I(H; (H;YY Y), s.t. I(H: <>>
(1)
where the Spper-parameter constrains the MI I [(H; A) to avoid H from capturing excess task- irrelevant information from A. The basic GIB can effectively defend the input perturbation [39]. However, is intrinsically vulnerable to label noise since it entirely preserves the label supervision and maximizes the noisy supervision T(H; Y). Our empirical results Sec. 5 show that optimization only with Eq. is ineffective when learning with the bilateral edge noise defined in Def. 3.

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unformity (.H())
H(H)
A
H
topology
label constraine Zy 1(P:2g)
AIP)
(H:PIA))
Y
Ä
Y
Å ZA [(A:Aa)
H
Y
Å
H
Sawwer algnmene ICM;; Hq)
(a) RGIB (Eq. 2)
(b) RGIB-SSI. (Eq. 3)
(c) RGIB-REP (Eq. 4)
Figure 5: Digrams of the RGIB principle (a) and its two instantiations (b,c). RGIB-SSL utilizes the automatically augmented views A1, A2 in a contrastive manner to be resistant to input noise. RGIB-SSI is intrinsically robust to label noise due to its self- self-supervised nature. Besides, (GIB-REP explicitly purifies the input graph's 's topology and target labels with the jointly reparameterized Z and Zy. It enables to modeling the mutual patterns of edge noise from both input and label spaces.
Deriving the RGIB principle. The basic GIB of Eq. 1 decreases 1 [H:; A(Y) ) by directly constraining I(H;A) A) to handle the input noise. Symmetrically, the label noise can be hidden in I [H:; Y|)), but trivially constraining T(H;Y) Y) to regularize T(H; ÝIA) is not ideal, since it conflicts with Eq. and also cannot tackle the noise within I(A;)) Y). Thus, it is crucial further decouple the dependence among A, Y, and H, while the noise can exist in areas of T T(H; YIA), [(H; Aly), and (A; Y /1H). Analytically, / (A; Y ||H) (A;; +HH, Y| ( /1 (H [AA, Y). where I (A:Y) a constant and redundancy H (H [A, Y) can be minimized. Thus. the [(A; Y H) can be approximated by H (H), T( H: YIA) and T (H: Al)). Since the two later terms are also noisy, a balance of these three informative terms can be a solution to the problem of bilateral edge noise. Definition 4.1 (Robust Graph Information Bottleneck). Based on the above analysis, we propose a new learning objective to balance informative signals regarding H, as illustrated in Fig. 5(a), i.e,
min RGIB = T(H;Y), S1. 7H < H(H)- T TH;; (H; < vy AM) < <AA- (2)
Specifically, constraints on (H) encourage a diverse H to prevent representation collapse (>TH) and also limit its capacity (<YH) to avoid over-fitting Another two MI terms, [(H; Y(,) and I(H; AlL), mutually regularize posteriors to mitigate the negative impact of bilateral noise on H. The complete derivation of RGIB and a further comparison of RGIB and GIB are in Appendix B.2.
Note that MI terms, e-g., I(H; A'Y)) are usually intractable. Therefore, we introduce two practical implementations of RGIB, i.e., RGIB-SSL and RGIB-REP, based on different methodologies. RGIB- SSL explicitly optimizes the representation H with the sele-super regularization, while RGIB- REP implicitly optimizes H by purifying the noisy Á and Y with the reparameterization mechanism.
4.2 Instantiating RGIB with self-supervised learning
Recall that the edge representation has deteriorated with the supervised learning paradigm. Naturally, we modify it into supervuuee counterpart by explicitly regularizing the representation H (see Fig. 5(b)) to avoid collapse and to implicitly capture reliable relations among noisy edges as
+I II Y)) A, (H ((H)) min RGIB-SSL A YY)
IH H2).
uniformity
slignment
where margins Às Au. d balance one supervised and two self-supervised regularization terms. When A, = A,, = O, the RGIB-SSL can be degenerated to the basic GIB. Note H, and H2 are the representations of two augmented views Å, and A2, namely, H, far ( (A1, and Hu= (Āg, X).
Instantiation. To achieve tractable approximation of the MI terms in Eq. 2, we adopt the contrastive learning technique [5], and contrast pair of samples, i.e., perturbed A₁ Ag that are sampled from the augmentation distribution P(A). Based on which we approximate the expected supervision term by E[[(H; Y [A)] [I ();;H) = (H, Y)] 1/2(C (H; Y) Loie (H2; Y)). Note the approximation also supports multiple samples. Similarly, we approximate the entropy term E [ H)) by 1/2(H (H,)+ H (H2),, where higher uniformity leads to a high entropy H (H) as proved in Prop. 4.2. Lastly, the I (H; ) is estimated by the alignment term [HH; H2) (refer to Prop. 4.3).
5

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Proposition 4.2. A higher information entropy H of edge representation H indicates higher uniformity [35] of the representation's distribution on the unit hypersphere. Proof. See Appendix A.3.
Proposition 4.3. A lower alignment (HH;H)) indicates a lower [H;; A(Y). Since I ; A\Y) S A( Σ (H (2)) +2)++(((1))) A2)) = ; by [HH; H2) can bound a lower I [H;; A and A). Proof. See Appendix A.4.
a constrained alignment estimated
However, directly applying existing contrastive methods like [5, 20] can be casily suboptimal, since they are not originally designed for graph data and neglect the internal correlation between topology A and target Y. Here, propose the following two designs to further improve robust learning.
Hybrid graph augmentation. To encourage more diverse views with lower [(4 A2) and avoid manual selection of augmentation operations, we propose hybrid augmentation method with four augmentation operations as predefined candidates and ranges of their corresponding In each training iteration, two augmentation operators and their sampled from the search space. Then, two augmented graphs are obtained by applying the two operators on the original graph adjacency A. The detailed algorithm is elaborated in Appendix C.
are automatically
Self-adversarial loss terms. With edge representations H1 and H2 from two augmented views, we build the alignment objective by minimizing the representation similarity of the positive pairs (hly. h2y) and maximizing that of the randomly sampled negative pairs (hlj: hmmn), City f Cynn The proposed self self-adversarial alignment loss is Ratign = Σn= R.uss + RReey Importantly, softmax functions pros() (-) and paea(( (-) aim to mitigate the inefficiency problem [32] that aligned pairs are not informative. Besides, the uniformity loss is Raniy - Lij. - || +e 1|0,],, [|n with the Gaussian potential kernel, where edges Eij and emn are respectively sampled from Epos and gneg.
Optimization. Regarding Eq. 3, the objective of RGIB-SSL is L Xahas Da + Xu Runif. Remark 4.1. The collapsed representation comes from trivially minimizing the noisy supervision Y) The alignment and uniformity terms in Eq. 3 can alleviate such noise effects (see Sec. 5).
4.3 Instantiating RGIB with data reparameterization
Another realization is by reparameterizing the graph data on both topology space and label space jointly to preserve clean information and discard noise (as Fig. 5(c)). We propose RGIB-RE REP that explicitly models the reliability of Á and y via latent variables to learn a noise-resistant H, i.e.,
min RGIB-REP
H.I(H: Zy) + XAI(ZA:A) A) + Iy
(4)
supervision
topology constraint
label constraint
where latent variables Zy and Z are clean signals extracted from noisy Y and A. Their complemen- tary parts Zy, and A are considered as noise, satisfying y= + Zy and Ā= A + Z When Zr = y and Z = Å, the RGIB-REP can be degenerated to the basic GIB. Here, the 7(H; Zy) measures the supervised signals with selected samples Zy, where the classifier takes Za (i.e., subgraph of Å) input instead of the original A, ie., H fw (Z A X). Constraints AZ A) and [(zr; Y) aim to select the cleanest and most task-relevant information from Á and Y
Instantiation. For deriving a tractable objective regarding Z and Zy, a parameterized sampler fo(.) sharing the same architecture and weights as (.)()) is adopted here. fo(.) generates the probabilistic distribution of edges that include both À and Y by P (HH ) E (0, 1)1M) where representation H f A, X). Bernoulli sampling is then used to obtain high-confidence edges, i.e., ZA= SA=MAMP( [P] (A) and Zy SAMP SAMP(PlÝ), where | and Zy |
Proposition 4.4. Given the edge number n of A, the marginal distribution of Z is Q ((A))= = F(n) Pij. ZA satisfies A) < (IKL(Pp/ZZ/4))QQ(ZZ = EesE Pij log Pu
(1- (1-P;; -Pi)) log I-Pu bounded by RA. and the label constraint is similarly bounded by Ry. Proof. See Appendix A.5.
T- = RA, where T is a constant. The topology constraint I(Z A: A) in Eq. 4 is
"RRo0 = (hig, h2,)) ||t - n2t 11: where ppsa (hi, h2)) exp[| n3 1B exp( |ntt 12 -k2, 112)- R,000 neg (h mnn (7- ||Abl 1 11 neg (1att, 12 exp(<= | 1//22 exp(a-1|Ably-aa -b2m 11.
6

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Table 2: Method comparison with GCN under bilateral noise, i..,, both the input and label noise exist. The boldface mean the best results in AUC, while the underlines indicate the second- best results.
Cora
Ciesseer
Pubiedd
Facehook
Chameleon
Squirrel 40%
method
20%
40%
60%
20%
40%
60%
20%
40%
60%
20%
40%
60%
20%
40%
60%
20%
60%
Standard
8111 .2419 .6900
7864
.7380
.7085
8870
8748
864]
9829
.9520
.4388
.9166
9499
.9274
.9422
.406
.9386
DropEdge
8017
.7423 .7303
.7635
.7933
.7094
.8711
.8482
.354
.9811
.9682
-9473
.9568
.9488
.9407
.9439
9977
.9365
NeuralSparse
8190
.7318 .7293
7765
.7997
.7148
8908
.8733
8633
98255
.9638
.9456
.9599
.9497
.9402
.9494
.9309
.9297
PTDNet
8047
.7559
.7388
7795
7423
.2833
8872
.833
8623
9725
9674
.9855
9607
9514
.9424
.985
.9266
.9304
teachhing
.7479
.7030
.7238
.7131
8943
.766
9820
.9526
9595
.9483
.9611
.9774
Peer loss
8185
.7468
.7018
.423
.7345
.7104
.961
.88
8566
9807
.9536
9440
9543
.9333
.9267
.9457
.9345 .2866
Jaccard
8143
.7498
.7024
7473
.7324
.7107
8872
8803
8512
9794
.9579
9428
9503 9538
.9344
.943
.927 9244
GIB
8198
.7485
.7148
.7509
.7388
.7121
8899
8729
8544
9773
96088
.417
.9554
.9561
.9321
.9472
.9329
.9302
VIB
8208
.7810 .7218
.7701
.8120
.7185
8927
.8225
8501
9697
.9637
9500
.9529
.9561 .9487
.9431 .9399
.9288
PRI
.7776
.7330
.6981
.7567
.7452
70188
8898
.8001
8487
9601
.6199
.5077
.9513
.9499
.490
.9382
.9133
.9301
SupCon
8240
.7819 .7490
.554
.4558
.229
8853
8718
8525
9588
9508
9297
9561
9531
.4667
9473
.9448
.9301
GRACE
6940 .6929
7633
.7422
.6844
8922
.749
8899
8315
8978
8949
.380
.9663
RGIB-REP
8113
7966 .7591
7875
7519
7312
9017
8834
8652
9832
9770
9519
.9233
9621
.9119
9509
.9555
.9334
RGIB-SSL
.8300
.8554
.8339
8694
.8477
.8337
9225
.8918
8697
9829
9711
9643
9655
9592
9500
9499
9426
9425
Proposition 4.5. The supervision term IH: Zy) in Eq. 4 can be empirically reduced the classifi-
cation loss, i.e,, I(H; Zy) 2 Ezy.zz [log Pu
~ -Lss(Soo (Z Zy), where Lets is the
standard cross-entropy loss. Proof. See Appendix A.6.
Optimization. A relaxation is then conducted on the three MI terms in Eq. 4. With derived bounds in Prop. 4.4 (i.e., regularization RA and Ry) and Prop. 4.5 (i.e., Lots), the final optimization objective is formed as L = XsLats + AARA + Ry and the corresponding analysis Thm. 4.6 is as follows.
Theorem 4.6. Assume the noisy training data Dirain = =A, X, Y) contains a potentially clean subset D (Z X, Z The Zi- and 2 are the optimal solutions of Eq. 4 that Zi =Y, based on which trained GNN predictor f.)()) satisfies fw( Z X)= = Zy +c. The random error is independent of Dauh and c c-> 0. Then, for arbitrary X., A dy E [0. Z = Z', and Zy Zy minimizes the RGIB-REP of Eq. Proof. See Appendix A.7.
Remark 4.2. Note that the methodologies take by RGIB-SSL BSLB-SSLLdde and RGIB-RER i.e., self-supervised learning and data reparameterization are not considered in the original GIB [39]. In Sec. 5, we justify that they are effective in instantiating the RGIB to handle the bilateral edge noise. A detailed comparison of the two instantiations is conducted in Appendix B.3. More importantly, it is possible that new instantiations based on other methodologies are inspired by the general RGIB principle.
5
Experiments
Setup. popular datasets and 3 types of GNNs are taken in the experiments. The edge noise is generated based Def. 3.1 after the commonly used data split where 85% edges are randomly selected for training, 5% as the validation set, and 10% for testing. The AUC is used as the evaluation metric in [48, 55J. The software framework is the Pytorch [29], while the hardware is one NVIDIA RTX 3090 GPU. We repeat all experiments five times, and full results can be found in Appendix E.
Baselines. As existing robust methods separately deal with input noise or label noise, both kinds of methods can be considered as baselines. For tackling input noise, three campling-based approaches used for comparison, i.e., DropEdge [30], NeuralSparse [50], and PTDNet [27]. Besides, we also include Jaceard [38], GIB [39]. VIB [31], and PRI [47], which are designed for pruning adversarial edges. Two generic methods are selected for label noise, i.e., Co-teaching [16] and Peer loss [26]. Besides, two contrastive learning methods are taken into comparison, i.e., SupCon [20] and GRACE [54]. The implementation details of the above baselines summarized in Appendix C.
5.1 Main results
Performance comparison. As shown in Tab. 2, the RGIB achieves the best results in all 6 datasets under the bilateral edge noise with various noise ratios, especially on challenging datasets, egg., Cora and Citeseer, where a 12.9% AUC promotion can be gained compared with the second- best methods. As for the unilateral noise settings shown Tab. 3, RGIB still consistently surpasses all the baselines ad hoc for unilateral input noise or label noise by a large margin. We show that the bilateral edge noise can be formulated and solved by a unified learning framework, while the previous denoising methods can only work for one specific noise pattern, e.g., the unilateral input noise label noise.
7

---

Table 3: Method comparison with GCN under unilateral input noise (upper) label noise (lower).
The boldface numbers mean the best results, while the underlines indicate the second -best results.
Cora
Citeseer 40%
Pubmod
Facebook 40%
Chameloon 40%
Squirrel 40%
input noise
20%
40%
60%
20%
60%
20%
40%
60%
20%
60%
20%
60%
20%
60%
Standard
.8227
.7856
.7490
8054
.7708
.7583
8854
8759
8651
.9819
.9668
.9622
.9008
.933 .9368
.9416
.9955
.9411
Dropkige
8338
.7826
.7454
8025
.730
.473
.8682
88456
8376
99803
96885
.9531
9567
.9335 .9452
.9426
.9576
.9358
SearalSparse
8834
.7794
.7637
8093
.7809
.7468
8931
.8720
8649
9712
.9691
.983
9609
.9400 .9348
.9669
.4033
.9417
PTDNet
.833
.8214
.7770
.119
.7811
.7638
.8903
.8776
.8679
.9725
.966
.9493
.910 .9457
.9360
.9469
.9400
.9379
Jaccard
.8000
.7838
.7617
.8776
.7776
.7725
8987
.8764
86.39
9784
.9022
96338
9507
.9866
.9364
99388
9345 .9400
GIB
8002
.8099
.7741
.0700
-7717
.7988
.9122
.8808
8618
9796
.647
.6650
.9605
.9521
.9116
.9900
9406
.9997
VIB
8603
.8590
8008
8499
8399
.703
8910
8829
8519
9800
.910
9536
9499
9558
.9312
.416
.9022 .297
PRI
.7990 .7736
.7652
.7558
8801
.790
9691
9969
.9529
9528
.901
.9435
.9997
.9419 .9318
SupCon
8349
.8301
8025
8076
.7677
7655
8867
87.39
8558
9647
.9517
.9401
9606
9536
.9468
.9722 .9343 .9305
GRACE
.7777
.7107 .6975
7615
.151
6830
8810
8795
8593
9015
.8833
8395
8994
9007
.864
.9922
.9788 .9363
RGIB-RET
8624
.8313
.8158
8299
7996
.771
9008
.822
8687
.9833
.9723
.9622
9705
.604
9480
.9495
.9432 .9405
RGIB-SSI.
.0224
8577
.8421
8747
.8461
_8245
9126
.8889
8693
9821
.9707
9668
9658
.5500
.986
.4499
.4299 .9429
lahel noise
Cora
Citeseer
Pubmed
Facebook
Chameleon
Squirrel
20%
40%
60%
20%
40%
60%
20%
40%
60%
20%
40%
60%
20%
60% 40%
20%
40%
60%
Standard
8281
8054
.060
7965
.7850
.7659
90.30
.9039
9070
9882
.9880
.9886
.968
.9500
.9362
.9200
.720
.9110
Co-teaching
8446
.8209
81557
.9774
.7877
.7913
.9315
.9291
.9319
9762
.9797
.9638
.9642
.9650
.9533
.9675
.9641
.9655
Peer loss
8325
.8036
.8069
.7991
77900
.7751
91.26
.9101
.210
9769
.9750
.9334
9921
.9501
.9569
.936
.694
.9696
Jaccard
.2899
.8064
.8148
.061
.7887
.6889
9098
.9135
9096
9702
.9725
.9758
.9033
.9659
.9557
.9529
.9122
.9501
GIB
8357
.8137
.8157
79186
.752
7649
9037
.9114
9064
9742
9703
.9711
.651
9582
.9189
9641
.9288
.9601
VIB
.8296
.8036
.068
8034
.7739
9088
.9022
9705
.9790
.9113
9610
9629
.9118
.9222
.9690
PRI
8298
.81.9 .7960
7902
7881
7725
8939
8917
8812
9733
9802
9620
9414
9593
.9599
9658
9698
.9533
SupCon
8491
.8275 8256
8024
7983
.7077
9131
91088
9162
9647
9567
9553
9584
9580
9477
.516
9595
.9511
GRACE
8831
.8237 8193
7909
7630
.7737
9234
9252
9255
8913
8972
8887
9053
9074
9075
.171
.974
.9166
RGIB-REP
8554
8318
.8297
8083
7846
7745
9357
9343
9332
9884
9883
9889
9785
9797
9785
9735
9933
.9377
RGIB-SSI.
.9114
99224
.9411
9204
9218
9250
9594
.9004
9613
9857
9881
9857
9730
9752
9744
9727
9729
.726
Table 4: Combating with adversarial attacks on Cora and Citeseer datasets.
adversarial
Cora
Citeseer
attacks
clean
Fade = 20%
Fads = 10%
Eade = 60%
clean
Eadu = 20%
Fadv = 10%
Sads = 60%
Standard
8686
.7911
.7671
.7014
.8317
.8139
.7736
.7481
RGIB-SSL
.2600
8299
.8095
.8052
.9148
.8656
.8347
.8022
RGIB-RFF
.758
.8408
.7918
.7611
.8415
.8382
.8107
.7893
Remark 5.1. The two instantiations of RGIB can be generalized to different scenarios with their own priority according the intrinsic graph properties. Basically, the RGIB-SSL is more adaptive to sparser graphs, e.g., Cora and Citeseer, where the edge noise results in greater performance degradation. The RGIB-REY can be more suitable for denser graphs, e.g., Facebook and Chameleon.
Combating with ad adversarial attacks. Adversarial attacks on graphs can be generally divided into poisoning attacks that perturb the graph in training time and evasion attacks that perturb the graph in testing time. Here, we conduct the poisoning attacks based on Nettack [57] that only perturbs the graph structure. Notes that Nettack generates perturbations by modifying graph structure or node attributes such that perturbations maximally destroy downstream GNN's predictions. Here, we apply Nettack on Cora and Citeseer datasets as representatives. As shown in Tab. 4, the adversarial attack that adds noisy edges to the input graph also significantly degenerates the GNN's performance. And comparably, the brought damage is more severe than randomly added edges [57, 53, 42]. Crucially, empirically justify that RGIB-SS and RGIB-REP can also promote the robustness of GNN against adversarial attacks on the graph structure, remaining the adversarial robustness of the GIB.
The distribution of learned representation. We justify that the proposed methods can effectively alleviate the representation collapse. Compared with the standard training, both RGIB-RED and RGIB-SSL bring significant improvements to the alignment with much lower values, in Tab. 5. At the same time, the uniformity of learned representation is also enhanced: it can be seen from Fig. 6 that the various query edges tend to be more uniformly distributed on the unit circle, especially for the negative edges. It shows that the distribution of edge representation is effectively recovered. Besides, RGIB-SSL explicitty constrains the representation, its recovery power on representation distribution is naturally stronger than RGIB-REP, resulting in much better alignment and uniformity.
Learning with clean data. Here, we study how the robust methods behave when learning with clean data, ie., no edge noise exists. As shown Tab. 7, the proposed two instantiations of RGIB can also boost the performance when learning on clean graphs, and outperforms other baselines in most cases.
8

---

positive samples
positive samples
poseiiv samples
Table 5: Comparison of alignment. Here, std. is short for standard training, and SSL/REP short for RGIB-SSL/RGIB. REP. respectively.
dataset
Cora
Citeseer std.
negative samples
negetive samples
negative samples
method
std.
REP SSI.
REP SSL.
clean
.616
.524 .475 445 .439 .418
= 20%
.687
.642 .543
5866
.533 .505
5
40%
.695
.679
.578
689
.623 .533
60%
732 .704 .615 ,696 .647 .542
(a) Standard
(b) RGIB-REP
(c) RGIB SSL
Figure Uniformity distribution on Citeseer with == 40%.
Table 6: Comparison on different schedulers. SSL/REP are short for BEII--SLLRRGIB-ER Experiments are performed with 4-layer GAT and e 40% mixed edge noise.
dataset
Cora
Citeseer
Pubmed
10
822
method
SSL
REP
SSL
REP
SSL
REP
100
1.0
10
180
0.01 0.1
19
constant
8398 .7927
.8277
.7742
8596
8416
linear(-)
8427 .7653
8167
.7559
8645
8239
Figure 7: Grid search of hyper-parameter with RGIB-
sin(-)
8436 .7924
8132
76800
8637
8275
SSL (left) and RGIB-RED (right) on Cora dataset with bilateral noise e = 40%. As can seen, neither too
cos(.)
8334
.7833
.7647
8579
8372
cap(.)
8381 .7815
8085
.7669
8617
8177
large nor too small value bring good solution.
Table 7: Method comparison with a 4-yyer GCN trained on the clean data.
method
Cora
Citeseer
Pubmed
Facebook
Chameleon
Squirrel
Standard
8686
.3177
9178
.9800
.9788
.9725
DropEdge
8684
.344
.9444
9869
.7700
9629
Neu NeuralSparse
8715
.405
9366
.9803
.9635
FTDNet
8577
8398
9315
9868
9696
.6400
Co-teaching
8684
.8387
9192
9771
9698
.9626
Peer loss
8313
.7742
.0855
.855
.974
.9422
Jaccard
.8413
.8055
88811
9792
.9003
.9610
GIB
8582
8327
9019
.9911
9628
9635
SupCon
8529
.0033
9131
9692
9717
9619
GRACE
.3299
8236
9358
.953
.8999
.9165
AGIB-REP
8758
8415
9408
.9875
9792
.6600
RGIB-SSL
.9260
.9148
.9593
.9845
.9740
.9646
Optimization schedulers. To reduce the search cost of coefficients in the objectives of RGIB-SSL and RGIB-REP, we set a unified optimization framework that can be formed as L = aLCes + (1 a)R, + (1 R22 Here, we attempt 5 different schedulers tune the only hyper-parameter k t, where t is the normalized time step; (3) sine, 0ç = sin(t a/2): ; (4) cosine, a; = cos(t a/2); and (5) exponential, 0r2 = e As empirical results summarized in Tab. the selection of optimization schedulers greatly affects the final results. Although there is no golden scheduler to always perform the best, the constant and sine schedulers generally better than others among the above candidates. A further hyper-parameter study with grid search of the multiple À in Eq. 3 of RGIB-SSL and Eq. of RGIB-REP are illustrated in Fig. 7.
a [0, 1], including (1) constant, a = (2) linear.
5.2 Ablation study
Hybrid augmentation. As shown Tab. 8, RGIB-SSL benefits from the hybrid augmentation algorithm that automatically generates graphs of high diversity for contrastive learning. e.g., compared with the fixed augmentation, the hybrid manner brings a 3.0% average AUC promotion on Cora.
Self-adversarial alignment loss. Randomly-sanpled pairs with hierarchical information to be contrasted and learned from. The proposed re- weighting technique further enhances high-quality pairs and decreases low-quality counterparts. It refines the signal and brings up 2.1% promotion.

---

1.
mecapery and loss
1.5
moppehoy constrant
08-
0.-
1505
.iggmeet reg artiation
1208
0.-
smaal gresent at
4.
044
aricatin
044
Jubel construie
00
0.0
0.0
Eooch
Eooch
Eooch
(a) RGIB-SSL on Cora (b) RGIB-SSL on Citeseer (c) RGIB-REP on Cora (d) RGIB-REP on Citeseer
Figure 8: Learning curves of RGIB-SSL and RGIB-REP with E 40% bilateral noise.
Table 8: Ablation study for RGIB-SSL and RGIB-REP with 4-layer SAGE. Here, e 60% indicates the 60% bilateral noise, while co and ey represent ratios of unilateral input and label noise.
Cora
Chameleon
variant
60%
60%
60%
60%
60%
60%
RGIB-SSI. (full)
.8966
.8300
.9944
.663
9758
9762
w/o hybrid augmentation
8150 (8.1%))
8604 (1.4% 1)
8757 (2.6%)
.9528 (1.3% 1)
9746 (0.1%))
9695 (0.6% L)
w/o sell-sadverarrall
.8410 (2.1%%))
.8755 (0.25%
8927 (0.7% ()
.9655 (0.1 J)
.9722 (0.2%%
.946 (0.1 (0.1%)L)
wio supervision (A,
.7480 (12.9%.)
78810 (10.5%.))
.7820 (13.09.L)
8626 (10.79.))
8628 (11.5%))
.8122 (12.8%.)
wwo alignment (A., 0)
8194 (4.6%)L
.5.10 (2.5%))
8866 (5.9%L)
.9133 (0.5%) 3)
.749 (0.19)L
.722 (0.4%)-)
w/o uniformity (A= 0)
8355 (2.8%%)
8621 (1.2% ()
.878 (1.3%)
9652 (0.1%% ()
.910 (0.4%)
.9511 (0. (0.1%)))
KGGI-REP (full)
.7611
.8487
.8955
.9577
.9766
.9666
wlo edge selection (Z A)
.7515 (1.2%-))
.8199 (3.3%)
7890 (2.5%LL
9654(0.1 ()
9704 1%%1)
9661 (0. 19k-)
w/o label selection (Zy Y)
.7533 (1.0%.)
8373 (1.3%) D)
.7877 (3.0%)
9484(08%% 1)
9666 (0.4%
9594 (0.8%)
w/o topology constraint A = 0)
.7355 (3.3%).
.7699 (9.2%) 1)
7969 (1.5%) 1)
9503(0.698
9658 (0.4% ()
9635 (0.4%%
wfo label constraint (xy
.7381 (3.0%.)
.8106 (4.4%%.) L)
.032 (0.79.4)
.443(1.2%
.965 (0.4%%))
9669 (0.1%)))
Information constraints. Label supervision contributes the most among the three informative terms, even though it contains label noise. As can be seen from Tab. 8, degenerating RGIB-SSL. to a pure self-supervised manner without supervision (i.e., o 0) leads an average 11.9% drop in AUC.
Edge label selection. The two selection methods adopted in RGIB-REP are nearly equally important. Wherein, the edge selection is more important for tackling the input noise, greater drop will come when removed; while the label selection plays a dominant role in handling the label noise.
Topological/ / label constraint. Tab. 8 also shows that the selection mechanisms should be regularized by the related constraints, otherwise sub-optimal cases with degenerated effects of denoising will be achieved. Besides, the topological constraint is comparably more sensitive than the label constraint.
Learning curves. We draw the learning curves with constant schedulers in Fig. 8, where the learning processes are generally stable across datasets. A detailed analysis with more cases is in Appendix 1.
6 Conclusion
In this work, we study the problem of link prediction with the bilateral edge noise and reveal that the edge representation is severely collapsed under such a bilateral noise. Based on the observation, we introduce the Robust Graph Information Bottleneck (RGIB) principle, aiming to extract reliable signals via decoupling and balancing the mutual information among inputs, labels, and representation to enhance the robustness and avoid collapse. Regarding the instantiations of RGIB, the self- supervised learning technique and data reparameterization mechanism are utilized to establish the RGIB-SSL and RGIB-REP, respectively. Extensive studies on 6 common datasets and 3 popular GNNs verify the denoising effect of the proposed RGIB methods under different noisy scenarios.
Acknowledgements
ZKZ and BH were supported by the NSFC Young Scientists Fund No. 62006202, NSFC Gen- eral Program No. 62376235, Guangdong Basic and Applied Basic Research Foundation No. PRAA1115011661
Alibaba Innovative Research Program, HKBU Faculty Niche Research Ar- eas No. RC-FNNRA-1G2--22/CCCO and HKBU CSD Departmental Incentive Scheme. JCY was supported by the National Key R&D Program of China (No. 20222D0160703). 111 plan (No. BP0719010, and the National Natural Science Foundation of China (No. 62306178). QMY was in part sponsored by the National Natural Science Foundation of China (No. 92270106).
10

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Appendix
Theoretical justification
15
A.1 Notations
15
A.2 Preliminaries for mutual information
15
A.3 Proof for Proposition 4.2
16
A.4 Proof for Proposition 4.3
16
A.5 Proof for Proposition 4.4
17
A.6 Proof Proposition 4.5
18
A.7 Proof for Theorem 4.6
18
Further discussions
19
B.1
Definition of the bilateral edge noise
19
B.2
The difference with GIB and RGIB
20
B.3
The relationship between RGIB-SSL and RGIB-REP
21
B.4
Assumption
B.5
Broader impact
22
B.6
General robustness of GNNs
22
Implementation details
23
C.1 GNNs for link prediction
23
C.2 Details of RGIB-SSL and RGIB-REP
23
further comparison RGIB-SSL and other contrastive learning methods
C,4 Baseline implementations...
25
A further empirical study of noise effects
26
D.I Full evaluation results
26
D.2 Further investigation on the unilateral noise
26
D.3
Decoupling input noise and label noise
27
D.4
Large-scale datasets
28
D.5 Node classification task
29
D.6
Combating feature noise
29
Full empirical results
31
E.1
Further ablation studies on the trade-off parameters À
31
E.2
Combination of RGIB-REP and RGIB-SSL
32
E.3 Further case studies
32
E,4 Full baseline comparison with bilateral and unilateral noise
37
14

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A Theoretical justification
A.1 Notations
As an expansion of the notations Sec. 2, we summarize the frequently used notations in Tab. 9.
Table 9: The most frequently used notations in this paper.
notations
meanings
V (we)
the set of nodes
8 ==(eishis
the set of edges
Ae {0, 13N
the adjacent matrix with binary elements
X E RND
the node features
9 (A,A,
graph (the observation input of GNN)
Equery {eig) K
the query edges to be predicted the labels of the query edges representation of a query edge Eij representation of all the query edges
Y
hig
H
u;
representation of node v; representation of nodes the mutual information of X and Y
I(X;Y)
I(X;Y|Z)
the conditional mutual information of X and Y when observing Z
A.2 Preliminaries for mutual information
Definition A.1 (Informational Divergence). The informational divergence (also called relative entropy or Kullback-Leible distance) between two probability distributions p and q on finite space X (i.e., a common alphabet) is defined as
p(x) ((x)
p(X) (X)
D(pllg) Ep(()) log 3EX
(5)
Remark A.1. D(pllg) measures the distance between and q. However, D(-II-) is not true metric, and it does not satisfy the triangular inequality. D(pll)) non-negative and zero if and only if Definition A.2 (Mutual Information). Given two discrete random variables X and Y. the mutual information (MI) I(X: Y) is the relative entropy between the joins distribution p(x. y) and the product of the marginal distributions p(z) ply)) namely,
I(X; Y) =D(((a, ((((a,))ll((a)p(u))
p(a, v) p(x)p(u)
Σ p(x, v) log
=EX.EY
(6)
p(xly))
Σ p(z, v) log
p(x)
=EXWEY
Remark A.2. I(X:Y) is symmetrical in X and i.e. I(X:Y) H(X) - H(XIY) H(Y)
H(Y[X) X).
Proposition A.3 (Chain Rule for Entropy).
H2 .X))
Ex X1, (e,...,X;,,1,
Proposition A.4 (Chain Rule for Conditional Entropy). X2,,,X2,.'. ,,,YY
Σ H(X,)X X1, Xr. X,-1.Y)
15

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Proposition A.5 (Chain Rule for Mutual Information).
X2,
[[;;; X2: ,I-)).
Corollary A.6. VA, Z, 2,. [(A; 2,) 2 max
Proof. As [())) 2 0. [AA Z,, 2j) [AA + I(A 2,1,, z [(A;
Sim-
ilarly, 2,, 2
I(A; 2z) can be obtained.
Thus, we have [A 2n)
max (I(A; I(A; 2,))
Proposition
for Conditional I(X,,-.. Xn; = - [(X;; Y,,,, (,,,
A.7
(Chain Rule
Mutual
Information).
A.3 Proof for Proposition 4.2
Proof. As introduced in Sec. 4.2, the uniformity loss on two augmented graphs is formed as Runif Eij.mn e -l - N/ + -||,, -A=..ll ll2, For simplicity, we can reduce to one graph, i.e., Runif = Eij.mn-pe [e-1ku, h Al1]. Next, we obtain its upper bound by the following derivation.
log Runis = log Bij.mnnpp (e- |h4 Amnnnni [B]
=log Eig.mmnpe 2t 2t2h hi Amn 21] <Bij-pe log Emnnpe [e 222 Nemn))
N
Σ
log > 22.h3
(7)
log P.ME- (hig)) + log ZrMP
N
(H)) log ZuMP-
Here, (DE) is von Mises-Fisher (vMF) kernel density estimation based on N training samples, and ZvMF the vMF normalization constant. Hi() is the resubstitution entropy estimator of = (-) [1]. As the uniformity loss Lunif can be approximated by entropy (HH) higher entropy H(H) indicates lower uniformity loss, i.e., higher uniformity.
A.4 Proof for Proposition 4.3
Proof. Recall the hybrid graph augmentation technique is adopted by RGIB-SSL for contrasting, we can approximate the expectation [E[(H:; Ā[\) by 1/N Σ T(H,; Á,l with N augmented graphs. Based on this, drive the upper bound of I(H; A(Y) as follows.
N
(H;A & &I(HAA) = I/N Σ
(I(HAA))
1=1
(HHH(((H(AA) A.) + ((H2;
+ H(H, An ((H2,A2) =1/2(I(H,; (((AAA)))) =1/((((H;;H)) H2) +c).
(HI)H(H))++H()))) HH))) - H(H, H2) - -H(ÁI,
(8)
According to the definition of interaction information w.r. three variables, we know that
T(X;Y; 2) Y(X;Y)- -700; 7(X-Y 2 under the condition of Markov chain Z - Y. Thus,
can obtain I(X;Y) 2 Z))
16

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Specifically, the first inequality, ie.. N - proximate the expectation of N views by 2 representative views practice. As these two views are generated by random and unbias perturbation in each training iteration, the approximation error induced here is expected to be reduced.
+ ((H;; is to ap-
As for the second inequality, i.e.,
H (H))) + + H A.) HH2,, A2))
(9)
1 (H), + H (A,) H HHH, H₂) - H (A), A2)), 2
due to HHH,, A1) H [H A2) 2 HHH,, H2) + (A A2). Besides. 1HH; H2) 2 I H A and (Ai 2 HH A1).
That is, the informative variables located in the same representation space are naturally more corre- lated.
Then, we have
=H(H)) + + H()) - I(H;; - 1(H;;
(10)
2 H(HH) + H(H2) ( T(H;; H2) T(A1; A2)
-H H2) A2).
Thus, we can obtain H(H,, A (H2, A,) H(H,, H2) + H(A,, Az). And finally, lower I(H;; H2) can upper bounded as lower I (H; A) and
A.5 Proof for Proposition 4.4
Proof. From the perspective of data processing, the collected noisy edges in the training set can be randomly split into the observed graph (i.e., the input of GNN) or the predictive graph (i.e., the query edges for the GNN to predict). Considering the noisy edges might come from similar sources, e.g., biases from human annotation, the corresponding noise patterns can also be similar between these two kinds of noise that are naturally coupled together to some extent. We treat the label constraint and topology constraint in the same way here, without injecting any additional heuristics or assumptions.
With the marginal distribution Q(ZA)= = Cn P PP n) PP()) 11 Psj. we drive the
upper bound of MI I(ZA; A) as:
P(ZalA)
I I(Z A) -E EZA,A [log(
)]
AP(ZA)
(o(AAl)) (L(P(ZZ))I((ZN))
E (log(
((ZA)
(11)
SE|KI A (0]][((ZA))
P (1 P.j)) log 1- Pij R.r
Σ Pi log
eyEd
where the KI. divergence on two given distribution P(x) and Q(x) defined as (I()))))
Σ. Px() log F(s)/Q(s). Thus, we obtain the upper bound of / A(Z A) as Eq. 11. Similarly, the label constraint is bounded as T(Zy; 8) < Eeyer Pej log Ps + P.j) log 1
I-r = Ry.
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---

A.6 Proof for Proposition 4.5
Proof. We derive the lower bound of I(H: Zy) as follows.
T(H; Zy) = =I w AA): Zy)
Pm Zy 2a)
ZRzy 2 flog
P(Zy)
-Ezy 2, (log Pw [y [A) -log(P(Zy)))
-Rzy, ZA 1log IP,, [Zy Za) + H[Zr)
(12)
ZE Z2,zz.z. ZA (log P. Zy [a)]
1
Lato AA), Zv).
Zy
E2yEZr
Thus, the MI T(:; Zy) can be lower bounded by the classification loss, and min - A(H Zy) in RGIB-REP (Eq. 4) upper bounded by min >./1Zy| Eegezr Lids ww (ZA), Zy) as Eq. 12.
A.7 Proof for Theorem 4.6
Proof. From the theoretical perspective, RGIB-SSL explicitly optimizes the representation H with sule--upe regularizations, i.e., alignment I(H; H2) and uniformity H (H)). H ((2)). By con- trast, RGIB-REP implicitly optimizes H by purifying the noisy Å and Ý with the reparameterizatioo mechanism to extract clean signals in the forms of latent variables Zy and Z The information constraints AZ: Å), (Zy; Y) are directly acting on Zy and Z A and indirectly regularizing the representation H. The Zy and Z are the optimal solutions of RGIB-RED in Eq. 4, and that is the reason of clean labels are expected to be recovered, i.e., Zi = YY
Then, with the relaxation via parametrization in A.6, we first relax the standard RGIB-REP to its upper bound as follows.
min RGIB-REI A X,I(H: 2y) + (ZA +
(13)
- >.IA + Aa/ + Ar
As Z = Y, Eq. 13 can be reduced to min -X,1 (Z,, + AAT(ZA;A) + HY), where I(Y:Y) (Y) Y C Y. Removing the final term with constant HYY), can be further reduced to min (Z) AA; 4; A), where À Aa/A,- Since minimize the -I(ZA; Y) A A) is equal maximize the T(ZA: AT(ZA; A), next, conduct the following derivation:
max Y) AI
max ( - - -
maxx (11T:::4))--1(1112=)-1111131 -(
-max - -1
(14)
.maxx (1--::)))----///Y1)))))-=-
- - -A(
=max(1- A)c - 10o--11-00 (1-))(á:Y]Zz)) A(1))
=(1 - -X)c.
Since MI I(Ā;Y|ZA)) > 0 and 1(ZA: 2 0 are always true, the optimal Z should reach the optimal case. Thus, should makes -(1- (1=1))((Ay\\zA) XI A|Y) simultaneously. Therefore, Za Z maxi- and A(Z ĀlY) mizes (Zx;))-AI(Z)) -
satisfy [(A:Y\ZA) A))(Z,;A), which equal to minimize -I(ZA; Y) + AT(Z;; Å), i.e. the
RGIB-REP. Symmetrically, when ZA = A. Zy Zi maximizes 1(A; Zy) >M(Zy; Y) as [(A; Zv) )) (); A) - (1 A)T(Ý; A| - - YIA).
18

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B Further discussions
B.1 Definition of the bilateral edge noise
Basically, the edge noise can exist in the form of false positive edges or false negative noise. Specifically, the false positive edges are treated as existing edges with label I, but in fact, such edges should not exist. This is the standard noisy label learning problem. Conversely, the false negative edges are about the labels of the non-observed edges, which is usually considered as hard negative mining in GNNs. The predictive probabilities of such edges will be minimized. In terms of dropping edges, it actually will not inject noise into the graph data but only decrease the positive observations, which is beyond the research focus.
Here, we would highlight that our work focuses on the false positive edges as it is more practical and common in real-world scenarios since the data annotating procedure can produce such a kind of noise [37]. Thus, if the bilateral edge noise exists, it is more likely be false positive samples, while the false negative samples are often intractable to be collected and annotated in practice. Actually, investigating the influences of the false negative samples is another line of research, such as [41, 19], which is orthogonal and complementary to our work. Learning from a clean graph can also encounter the problem of false negative samples, which is usually due to the random sampling of negative nodes/edges. Note that tackling the false negative samples is a well-known problem in the area of link prediction while handling the false positive samples is also valuable but still remains exploee
The input noise and label noise do look quite similar and unilateral. From the perspective of data processing, the collected noisy edges in the training set can be randomly split into the observed graph (i.e., the input of GNN) or the predictive graph (i.e., the query edges for the GNN to predict). Considering the noisy edges might come from similar sources, e.g., biases from human annotation, the corresponding noise patterns can also be similar between these two kinds of noise that are naturally coupled together some extent. However, we would claim and justify that the two kinds of noise are fundamentally different.
Although both noises can bring severe degradation to empirical performances, they actually act in different places from the model perspective. As the learnable weights are updated as w := w - V C. H. Y), the label noise acting on the backward propagation can directly influence the model. By contrast, the input noise indirectly acts on the learned weights it appears in the front end of the forward inference, ie.,, H = fw (A, X). Empirically, as results shown Tab. and Tab. 3, the standard GNN (without any defenses) performs quite differently under the same proportion of input noise and label noise. Such a phenomenon can inspire one to understand the intrinsic denoising mechanism or memorization effects of the GNN, and we would leave that further work.
More importantly, from the perspective of defending, it could be easy and trivial to defend if these two kinds of noise the same. However, none of the existing robust methods can effectively defend such a bilateral noise. As can be seen from Table 2. only marginal improvements are achieved when applying the existing robust methods to the bilateral noise. While in Tab. 2 and Tab. 3, these robust methods work effectively in handling the unilateral noise, i.e. only the structure noise or label noise exists. The reason is that the properties of bilateral noise much more complex than unilateral noise. Both sides of the information sources, i.e., A, and Y, should be considered noisy, based on which the defending mechanism could be devised.
As pointed out by recent works [11, 37], graph noise can be produced in the data generation process that requires manually annotating the data. As in [37], in general, graph noise here refers to irredueible noises in graph structures, attributes, and labels. Thus, the studied bilateral edge noise in our work is in line with the generic concept of graph noise in the literature, which is recognized as a common problem in practice but also underexplored challenge in both academic and industrial scenes.
In this work, we focus on the edge noise on adjacency A but not the node feature X. Specifically, we focus on the investigation of the false positive edges as the bilateral noise and design the simulation the optimization
approach in Definition 3. as it is closer to real-world scenarios. Therefore, objective of the proposed RGIB principle, we assume that X is generally clean and attaches more importance to the adjacency A. Besides, the presentation H is encoded from the node feature X. Thus, regularizing H in RGIB objectives can indirectly balance its dependence on X.
19

---

Input noise
Observed graph
Predictive graph
Label noise
GNN
À = A A+AA A'
= -
Derain (A,X,Y)
Figure 9: Illustration of input noise and label noise decomposed from the edge noise. The GNN predictor takes the observed graph as input, where the black edges are known, then predicts the latent edges (blue ones). Under the bilateral edge noise, the original training data D (A, X, Y) can be (i) added with noisy edges as input (A > A) and (ii) perturbed by noisy supervision (Y Y).
B.2 The difference with GIB and RGIB
In short, RGIB generalizes the GIB with improvements both theories and methodologies to learn a robust representation more resistant to the bilateral edge noise.
Conceptually, we would clarify that GIB intrinsically susceptive to label noise since it entirely preserves the label supervision with maximizing. Thus, GIB cannor provide an ideal solution to the inherent edge noise investigated in this work. Different from the GIB, the RGIB takes both noisy inputs and noisy labels into the design of its information bottleneck. RGIB decouples and balances the mutual dependence among graph topology, target labels, and representation, building new learning objectives toward robust representation.
Analytically, the GIB is intrinsically not robust to label noise since it entirely preserves the label supervision with maximizing I(H; Y), in Eq. As illustrated in Fig. the GIB decreases [[H:A|Y] by directly constraining [H; A) handle the input noise. Symmetrically, the label noise can be hidden in the area of I(H; ()),, but trivially constraining I(H; (H) to regularize T(H; A)) is not ideal, since it will conflict with Eq. 1.
Besides, it cannot tackle the noise within I I(Ā: Y), where the two kinds of noise can share similar patterns as the random split manner does not change their distributions in expectation. GIB only indirectly regularizes the MI term I [H: A))), as we introduce Sec. 4.1, which can only solve partial noisy information. By contrast, RGIB takes all the related MI terms, i.e., H(H), I(H; ÍIA) and Á(H)) based which to provide solution that balances these MI terms as more strict information bottleneck.
Thus, it is crucial to further decouple the mutual dependence among A, Y, and H. Regarding the representation H, noise can exist in areas of I(H; Y A), IH A\Y), and [(A; H )). as shown in Fig. 2. As T(Ā: y H) = T(A; F) T(H: ÍIĀ) + T(H; Al H(H) H(H) Á, Y), where ((; Y) is a constant and redundancy H (HI A, can be regularized simultaneously. Therefore, the I(Á; (H) containing the shared noise patterns of Á and Y can be approximated by the other three terms, Le., H (H), T(H; (A)) and 7 [HH; Al)). ). Since the two later terms are also with noise, balance of these three informative terms can be a solution to the problem.
Based the above analysis, we derive the RGIB principle in Def. 4.1 that balances the three important information terms H (H), TH;; Y Á) and T(H; A()). It works as an information bottleneck to filter out the noisy signals in both Á and Y, utilizing the supervision / (H; Y) at the same time.
From the view of methodology, we also provide two instantiations, RGIB-SSI and RGIB-REI to approximate the aforementioned MI terms. These two instantiations benefit from different methodolo- gies, i.e., self-supervised learning and data reparameterization, for implicit and explicit data denoising, respectively. Note that these two methodologies are not considered in the original GIB. However, we show that they can be utilized to implement the specific RGIB instantiation and build effective information bottlenecks for handling the inherent noise.
20

---

Empirically, we justify the effectiveness of the two instantiations of RGIB that outperform GIB in several noisy scenarios. Besides, the GIB is highly coupled with the GAT. By contrast, RGIB does not require any modifications to GNN architecture. It can be seamlessly integrated with various existing GNNs and promote their robustness against inherent noise.
B.3
The relationship between RGIB-SSL and RGIB-REP
Based on the empirical study of several datasets and GNNs, it found that the two instantiations can be generalized to different scenarios. Generally, RGIB-SSL is more adaptive sparser graphs, while RGIB-REP can be more suitable for denser graphs. More importantly, they can be complementary to each other with flexible options in practical applications, summarized in Remark 5.1.
From the theoretical perspective, RGIB-SSL explicitly optimizes the representation H with self- supervised regularizations, i.e., alignment IHH;; H2) and uniformity H (H)), H (H2). By contrast, RGIB-REE REP implicitly optimizes H by purifying the noisy À and Y with the reparameterization mechanism to extract clean signals in the forms of latent variables Zy and Za. The information constraints / AZ A), [(Zy; Y) are directly acting on Zy and A and indirectly regularizing H.
Besides, from the perspective of methodology, both instantiations are equipped with adaptive designs for obtaining an effective information bottleneck. We also conduct further comparison and analysis of the two instantiations that are summarized as follows.
Table 10: Comparison of RGIB-SSI and RGIB-RED
Instantiation
methodology
advantages automated graph
disadvantages
with expensive calculation for the contrastive objectives, especially the uniformity; requires extra graph augmentation operations.
self- supervised learning
augmentation; good effectiveness; can be applied in entirely self-supervised settings without labels. no need to do data augmentation; good efficiency; the input/output constraints do not require extra annotations for supervision and can be casily controlled.
RGIB-SSI.
sensitive the hyper-parameters A; less effective in extremely noisy cases; only applicable fully supervised settings.
data reparame- terization
RGIB-REP
More importantly, it is also possible that new instantiations based on other kinds of methodology inspired by the RGIB principle. We believe that such bidirectional IB that strictly treats the information signals on both the input and label side is helpful to noisy scenarios in practice. For example, can generalize RGIB to the reasoning task knowledge graphs [49, 56), where the multi-relational edges characterize more diverse and complex patterns.
B.4 Assumption
The primary assumption of our work is that the edges of collected graph data can be potentially noisy. As in the submission, we present two ways of realizing the RGIB principle, which actually corresponds to some assumptions in building the objectives with the proper approximation.
RGIB-SSL assumees that the learned representation alignment. As RGIB-SSI. directly acts on the graph representation, it is more suitable for recovering the distribution of representation, especially when encountering representation collapse due the severe edge noise.
be improved with higher uniformity and
RGIB-REP explicitly purifies the input graph's 's topology and target labels with the jointly reparame- terized ZA and Zy. Here, the latent variables ZA.Zy are expected to more clean and informative than the noisy Á.Y. RGIB-REP assumes that the GNN model can identify these latent variables and further benefit its learning procedure against noise.
21

---

In addition, any clean labels and adjacency are not used in training. On the contrary, the model directly learns from the noisy adjacency A and label Y. which is practical as the collected data is potentially noisy in real-world applications. Besides, RGIB and all the baselines are run without any noise priors, e.g., noise type or noise ratio.
Empirically, RGIB-SSL is more adaptive to sparser graphs, e.g. Cora and Citeseer, where the edge noise results in severer representation collapse. RGIB-REP can be more suitable for denser graphs, e.g., Facebook and Chameleon, where the latent variables of edge data extracted by RGIB-REP. More importantly, the two RGIB instantiations be complementary to each other with flexible options in practical applications.
B.5 Broader impact
In this work, we propose the RGIB framework for denoising the bilateral edge noise in graphs and for improving the robustness of link prediction. We conceptually derive the RGIB and empirically justify that it enables the various GNNs to learn the input-invarian and label-invariant edge representations, preventing representation collapse and obtaining superior performances against edge noise. Our objective is to protect and enhance the current graph models, and we do not think our work would have negative societal impacts.
Besides, A similar problem has been noticed in an early work [11], where robots tend to build connections with normal users to spread misinformation on social networks, yielding the degeneration of GNNs for robot detection. In addition, as pointed out by recent survey [37], the bilateral noise can be produced in the data generation process that requires manually annotating the data, As introduced in [37], the bilateral noise here refers to irreducible noises in graph structures, attributes, and labels. In short, the studied edge noise in our work is in line with the generic concept of graph bilateral noise common problem in practice but also an under-explored
in the literature, which is recognized challenge in both academic and industrial scenes.
However, several existing benchmarks, e.g., Cora and Citeseer, are generally clean and without annotated noisy edges. Inevitably, there is a gap when one would like to study the influence of bilateral noise on these common benchmarks. Thus, to fill this gap, it is necessary to simulate the bilateral edge noise properly to investigate the impact of noise, the edge noise exists, it should be false positive samples, while the false negative samples are often intractable to be collected. Thus, we focus on the investigation of the false positive edges as the noise and design the simulation approach in Def. 3.1 it is closer to the real-world scenarios.
Besides, our knowledge, we are the first study the robustness problem of link prediction under the bilateral edge noise. One of our major contributions to this research problem is that we reveal that the bilateral noise can bring severe representation collapse and performance degradation, and such negative impacts are general to common datasets and GNNs. What's 's more, it is also possible that new instantiations based on other kinds of methodology are inspired by the robust GIB principle. We believe that such a bidirectional information bottleneck strictly treating the information source on both the input side and label side is helpful in practice, especially for extremely noisy scenarios.
B.6 General robustness of GNNs
Here, we provide discussion of the robustness based our understanding, which is in two folds.
(1) Training-phass robustness. Briefly, the training data is "hard" due to it can be perturbed with input or label noise or with imbalance problems, e.g., long-tail sample distributions. By contrast, the testing data is comparably "casy" because it can be clean and well-balanced. Here, the robustness is usually evaluated by how the model trained on the "hard tasks" performs "casy tasks", i.e. the performance on testing data. Considering the "hard tasks" can be with different difficulties, e.g., low high noise ratios, the change in model performances is also natural metric to quantify the robustness. A more robust model shall have lower performance changes while maintaining good performance simultaneously [13, 34].
In addition to test accuracy, the loss visualization can utilized to establish quantitative metrics to evaluate the intrinsic robustness of the model, e.g., the KI. divergence between two predictions on an original input and perturbed input [45]. can provide more insights into the model compared with the aforementioned accuracy- based robustness estimation.
22

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(2) Testing-phar phase robustness. Here, the training data is "easy" and the testing data can be "hard", e.g. with perturbation generated by adversarial attacks, or encountering the out-o--distribution samples in testing time. The robustness is evaluated here for a trained model in the testing phase, where the property that if testing sample "similar" to a training sample, i.e., the in-distribution samples, then the testing error is close to the training error [51].
On the other hand, how this model performs on the out-of-distribution samples is usually hard to estimate, whether these samples are naturally collected or manually perturbed. Thus, measuring the performance of these "hard" testing samples, and be used to estimate the robustness of the model. An early work [40] derives generalization bounds for learning algorithms based on their robustness. And recently, oat-of-distribution generalization on graphs has made great progress and attracted more attention from the research community [23]. Besides, adversarial training has been demonstrated to improve model robustness against adversarial attacks and out-of -ntto---iistiittionn aenerruutonngerrliinnnn generalization ability.
C
Implementation details
C.1 GNNs for link prediction
We provide detailed introduction forward propagation and backward update of GNNs in this part.
Formally, let f 1..... L denote the layer index, hf is the representation of the node i, MESS(.) is a learnable mapping function to transform the input feature, AGGREGATE(() captures the 1-hop information from neighborhood (v)) in the graph, and COMBINE(.) is the final combination between neighbor features and the node itself. Then, the I-layer operation of GNNs can be formulated as m. AGGREGATE' ((MESS(A ) euo): (W))))) where the representation of node e is h = COMBINE' (h--1 m.). After L-layer propagation, the final node representations h of each EEV V are obtained. Then, for each query edge eij E Etrain unseen from the input graph, the logit Desy is computed with the node representations hf and h with the readout function, i.e., Desj READOUT hef) + R. Finally, the optimization objective can be defined as minimizing the binary cross-entropy loss, i.e,, min Lets = Ee e., Egtrain lot (de.,) (aap)))--- (1- (1-19 Wky (ogg)))o((1 -oglloo(de (oee)) where (.) is the sigmoid function, and Vij = for positive edges while yia = 0 for negative ones. In addition, we summarize the detailed architectures of different GNNs in the following Tab. 11.
Table 11: Detailed architectures of different GNNs.
GNN
(ESS(.) & AGGREGATE(()
COMBINE(.)
READOUT())
GCN
m² w EjEN()) Va.d, h2
h! am, + w'
De,, h h
GAT
m = =EjEN(O asy W 1-1
h = amm; + W'anh
De., = h h
SAGE
m/ = w' [all Ejevio h3-1
h! = a + w'
DEs = t2T h2
C.2 Details of RGIB-SSL and RGIB-REP
As introduced in Sec. 4.2, the graph augmentation technique [43, 44] adopted here to generate the perpetuated graphs of various views.
To avoid manually selecting and tuning the augmentation operations, we propose the hybrid graph augmentation method with the four augmentation operations as predefined candidates and the ranges of their corresponding candidate's 's operations cover most augmentation approaches except for those operations modifying the number of nodes that unsuitable for the link prediction task.
The search space is summarized in Tab. 12, where the
In each training iteration, two augmentation operators (-)- and T²(-) and their hyper-parameters 0, and 02 randomly sampled from the search space claborated in Algorithm. 1. The two operators will be performed on the observed graph G, obtaining the two augmented graphs, namely,
To avoid abusing notations, we the h; to stand for the final representation hf in later contents.
23

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g' -' and and G²= The edge representations are gained by h - f(g', e:ylw) h O h where the node representations h and h are generated by the GNN model f f((lw) with learnalle weights w, and so is for hig with G².
Then, we learn the self-supervised edge representations by maximizing the edge-level agreement between the same query edge of different augmented graphs (positive pairs) and minimizing the agreement among different edges (negative pairs) with their representations. Note the haj here is the edge representation. Specifically, minimize the representation similarity of the positive pairs (hy hey and maximize the representation similarity of the andomly-sampled negative pairs ( h where Gig + emn.
Table 12: Search space Hr of the hybrid graph augmentation.
Operator Ti(:)
ppermmarrmettr Or;
description
edge removing
Our E (0.0, 0.5)
Randomly remove the observed edges with prob. Per-
feature masking
ff (0.0, 0.3)
Masking the columns of node features with prob. fmm.
feature dropping
Ofa E (0.0, 0.3) NA.
Dropping the elements of node features with prob. Uza.
identity
Do not modify anything.
Algorithm 1 Hybrid graph augmentation.
Require: number of augmentation operators n;
1: initialize @ + (.)
2: i 1i.1.... do
3: randomly selects one augmentation operator from the operator space Ti(.) E Hr:
4: uniformly sample the corresponding hlpeer-parameter 0, (fr,));
5: stores the newly sampled operator T()) (TT(110.)) + T());
6: end for
7: return the hybrid augmentation operators T(-)
C,3 A further comparison of RGIB-SSL and other contrastive learning methods
Regarding the three information terms of RGIB-SSL in Eq. 3, we elaborate a detailed clarification by comparing it with other contrastive learning methods in the following three folds.
8). Compared with the common manner of self celf-supervised contrastive learning, RGIB-SSL also considers utilizing noisy labels Y. We show that, although being potentially noisy, the labels Y can also benefit learning. Besides, our experiments show that both the supervised contrastive learning method (e.g., SupCon) and the self-supervised contrastive learning method (e.g., GRACE) perform poorly when learning with the bilateral edge noise. An intuitive explanation is that SupCon entirely trusts and utilizes the noisy Y while GRACE does not adopt V. From the data perspective, RGIB-SS) performs a hybrid mode of supervised and self-supervised learning. effectively extracts and utilizes the informative signals in Y without entirely absorbing its noisy signals. This is achieved with the synergy of two self-supervision terms follows.
(1) Supervision term I (HH;))+[(H;;))
(2) Self-supervision alignment term 7HH;; H2). Here, the main differences with common con- trastive learning methods are data augmentation and loss function. Specifically, RGIB-SSI. utilizes the hybrid augmentation algorithm and self ooff-adversarial loss. Compared with the fixed augmentation manner of other contrastive learning methods [20, 54]. i.e., performs random perturbation on node features and edges, we propose a hybrid augmentation method with four augmentation operations. The motivation here is to encourage more diverse views with lower MI I(A1; A2) and to avoid manual selection of augmentation operations. Besides, Prop. 4.2 provides a theoretical analysis of hy brid augmentation from information- thhoorticclt perspectives. Empirically, compared with fixed augmentation, the hybrid brings a 3.0% average AUC promotion on Cora (please refer to Tab. 8). This means that stronger augmentation scheme with more diverse views can help better deal with severer edge noise. Besides, the self-adversarial loss further enhances high-quality pairs and decreases low-quality counterparts. It refines the signal and brings up to 2.1 % promotion.
24

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G¹ = T (Gl0')
g = (G]
Latent space H
9
Observed graph
Augmented graph
Augmented graph
00
d
hj
h
hij O
hjj of
Figure 10: An illustration of the RGIB-SSL model.
(3) Self-supervision uniformity term (H)) (H2). As for the uniformity term, the under- standing part in Sec. 3.2 shows that a severer edge noise brings a worse uniformity. learn more robust representation, adopt the Gaussian potential kernel for implementation, which is usually not considered other contrastive learning methods. The ablation studies in Sec. 5.2 also illustrate that the uniformity term essential, especially in dealing with label noise. Besides, the uniformity of learned representation also enhanced (see Fig. 6), and the various query edges tend to be more uniformly distributed the unit circle, especially for the negative edges.
add this uniformity term to form the final loss of RGIB-SSS and
C.4 Baseline implementations
All baselines compared in this paper are based on their own original implementations. We list their source links here.
NeuralSparse, http://sst.s....t...../////o///a
PTDNet, httpg//.t.rr..ggtt....oo/////pop
GCN /bcectl...peeg/gsatoo.c.ooo//s/s
GIB, httggs///gghhbb.oooom///ssssssss
VIB, http:t/e./ggtta...ccooo////sssss PRI, hea._//ggeegggaaabooooo///ssssre
Peer loss functions,
SupCon, httppp///gtttii.cco///obooooooss
GRACE, http::///ghaab..ooaaa/A/AA//////
25

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D A further empirical study of noise effects
This section is the extension of Sec, 3 in the main content.
D.1 Full evaluation results
Evaluation settings. In this part, we provide thorough empirical study traversing all combinations of the following settings from different aspects.
3 GNN architectures: GCN, GAT, SAGE.
3 numbers of layers: 2, 2,4, 6.
4 noise types: clean, bilateral noise, input noise, label noise.
3 noise ratios: 20%, 40%, 60%.
6 datasets: Cora, CiteSeer, PubMed, Facebook, Chameleon, Squirrel.
Then, we summarize the entire evaluation results of three kinds of GNNs as follows. As can be seen, all three common GNNs, including GCN, GAT, and SAGE, are vulnerable bilateral edge noise.
Tab. 27: full evaluation results with GCN.
Tab. 28: full evaluation results with GAT.
Tab. 29; full evaluation results with SAGE.
Data statistics. The statistics of the 6 datasets in our experiments are shown in Tab. 13. As can be seen, homogeneous graphs (Cora, CiteSeer, PubMed, and Facebook) are with the much higher homophily values than the other heterogeneous graphs (Chameleon and Squirrel).
Table 13:
Dataset statistics.
dataset
Cora
CiteSeer
PubMed
Facebook
Chameleon
Squirrel
# Nodes
2,708
3,327
19,717
4.039
2,277
5,201
# Edges
5,278
4,676
88,648
88,234
31,421
198,493
Homophily
0.81
0.74
0.80
0.98
0.23
0.22
D.2 Further investigation on the unilateral noise
Loss distribution We visualize the loss distribution under different scenarios further investigate the memorization effect of GNNs. As shown in Fig. 11, two clusters are gradually separated apart with clean data, but such learning process can be slowed down when training with noisy-input data, which confuses the model and leads to overlapped distributions. As for the label noise, the model cannot distinguish the noisy samples with a clear decision boundary to separate the clean and noisy samples apart. Besides, it is found that the model can gradually memorize the noisy edges according to the decreasing trend of the corresponding losses, where the loss distribution of noisy edges minimized and progressively moving towards that of the clean ones.
26

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Eleam, epach-100 100
Ret D- epoch-100
Cleat, epoch 100
Ratio S.A, eparh-100 100
Denalty
Density
Densiue
ssllivv
negative
negatiive
no
upcch=500
Danaby
10
10
Dersty
10
AysuiO
10
positive
postivv
clean
dlenn
negafiive
negative
nony
Clean, #poch-
Ratiom0.4, epoche1000
Clean, epoche
Ratine0.4, spach-1 1000
Donsky
Denaty
Dnnpty
positive
poskive
clean
Clean
nogative
Training Loss
Training Loss
Training Loss
Training Loss
(a) Training loss of positivehnegative samples with clcan (b) Training loss of clcan/noisy samples with clean data
data (left) and 40% input-noisy data (right).
(left) and 40% label-noisy data (right).
Figure 11: Loss distribution of the standard GCN with 40% input noise (a) and 40% label noise (b).
Visualization edge representations shown in Fig. 12, the dimensionality reduction technique, T-SNE, is utilized here to measure the similarity of the learned edge representations h.j [h. h with edges Eg in the test set Erest Compared with the representation distribution of clean data (Fig. 12(a)), which present comparatively distinct cluster boundaries, both the input noise (Fig. 12(b)) and the label noise (Fig. 12(c)) are with overlapped clusters of the positive and negative samples. Namely, the GCN model is confused by the two patterns of noise, and fails to discriminate the true or false edges with their representations.
30
30
positive negative
20
negative
20
nogative
20
2
Companent 2
10
10
10.
0-
0-
-10
-10-
-10
-30
-20
-10
10
20
30
-30
-20
-10
10
20
30
-30
-20
-10
10
20
30
Component:
Component
Component
(a) Clean data
Input noise
(c) Label noise
Figure 12: Visualization of the representations of predictive edges through T-SNE, The edge repre- sentations are learned on the Cora dataset by 4-layer GCN with the standard supervised learning.
D.3 Decoupling input noise and label noise
For further and deeper study of the bilateral noise, we use the edge homophily metric to quantify the distribution of edges. Specifically, the homophily value Ithomo of the edge Eij is computed the cosine similarity of the node feature v; and æy, i.e,, hohomo cos(xs; xy). As shown Fig. 13, distributions of edge homophily are nearly the same for label noise and structure noise, where the envelope of the two distributions is almost overlapping. Here, we justify that the randomly split label noise and structure noise are indeed coupled together.
27

---

40
40
nolsy Inputs
noisy
noisy noisy inputs
Density
30
clean inputs
Density
clean labels
Density
10
10-
10
1.0
0.2
Edge homrphily
Edge homophily
Edge homophily
(a) Homophily distributions imput (b) Homophily distributions of out-(c) Homophily distributions noisy edges
put edges
edges
Figure 13: Distributions of edge homophily with random split manner.
Based on the measurement of edge homophily, we then decouple these two kinds of noise in simulation. Here, we consider two cases that separate these two kinds of noise apart, i.e. (case input noise with high homophily and label noise with low homophily shown in Fig. 14, and (case 2) input noise with low homophily and label noise with high homophily Fig. 15.
noisy inputs
15
noisy clean labels
noisy noisy inputs
Density
clean inputs
Density
2.0
Density 10
10
26-
10-
10
0.6
Edge homophily
Edge homophily
Edge homophily
(a) Homophily distributions input (b) Homophily distributions of out-(c) Homophily distributions noisy
edges
put edges
edges
Figure 14: Distributions of edge homophily for unilateral noise (case 1).
30
30-
10.0
noisy inputs cleon
noisy labels clean labels
noisy labels
Deraty
7.5
Density
Denstr:
noisy inputs
5.0
10-
10
2.5-
0.7
0.4
0.6
0.5
0.0
0.2
0.6 Edge homoshily
0.4
0.8
1.0
0.0
0.2
0.6 Edge homophily
0.4
0.8
1.0
Edge homophily
(a) Homophily distributions of input (b) Homophily distributions of out- (c) Homophily distributions of noisy put edges
edges
edges
Figure 15: Distributions of edge homophily for unilateral noise (case 2).
D.4 Large-scale datasets
We conduct an empirical study on two large-scale datasets, i.e., PPI (56,944 nodes and 1,612,348 edges) and Yelp (716,847 nodes and 13,954,819 edges). Here, first evaluate the performance without any defenses (i.e., standard training) with a 4-yayer GCN that learns on different noisy cases. As shown in the Tab. 14 below, the added edge noise also degenerates the performance of GNN on these two large-scale datasets.
Table 14: Empirical results on PPI and Yelp dataset.
dataset
clean
20%
40%
60%
PPI (standard training)
9076
8505 .8766
8486
8482
Yelp (standard training)
.9143
.8708
.8694
28

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Next, we evaluate the effectiveness and efficiency of the proposed methods on these two datasets with 40% inherent noise. As shown below, RGIB can also bring promotions compared with standard training. Besides, RGIB indeed brings extra computing costs, and specifically, RGIB-SSS can be comparably more expensive than RGIB-REP. However, the extra computing costs are not so high and within an acceptable range.
Table 15: Empirical results on PPI and Yelp dataset.
method
PPI (test AUC)
PPI (training time)
Yelp (test AUC)
Yelp (training time)
standard training
8486
0.24h
.8708
2.40h
RGIB-SSL
.8613
0,25h
.8875
2.71h
RGIB-REP
.8565
0.24h
8909
2.52h
D.5 Node classification task
We conduct experiments with 2-layer GCN on Cora and Citeseer datasets with random label noise on nodes. the empirical results are shown below, we justify that the RGIB framework can generalize to the node classification tasks with inherent label noise in nodes, where the two instantiations of RGIB also significantly outperform the standard training manner.
Table 16: Combating with label noise for node classification on the Cora dataset.
method
clean
Ey
20%
Ey = 40%
cy
60%
Standard training
.898
.868
.720
.322
RGIB-SSL
.900
.876
.786
.388
RGIB-REP
.894
.862
.760
.312
Table 17: Combating with label noise for node classification on the Citeseer dataset.
method
clean
Ey
20%
Ey = 40%
cy
60%
Standard training
.776
.746
.608
.278
RGIB-SSL
.784
.770
.646
.324
RGIB-REE .776
.754
.654
.364
D.6 Combating feature noise
Although our RGIB methods focus on edge noise, we conduct the following experiments to verify the point. Specifically, compare standard training, RGIB-REP. and RGIB-SSS using 4-layer GCN on six datasets with various ratios of feature noise. Here, the noisy feature is generated by adding 4%% random noise within the range of [0, to the normalized node feature.
The evaluation results of mean AUC are shown in below Tab. 18 Tab. 23. As can seen, the noisy node feature also significantly degenerates the GNN's performance, and the degradation becomes severer as the noise ratio increases. Interestingly, compared with the standard training manner, RGIB-SSS and RGIB-RED can also promote the robustness of GNN against feature noise. For example, when learning with E = 10% feature noise, RGIB-SSL can bring 16.6% and 6.5% improvements in AUC on Cora and Citeseer datasets, respectively.
We speculate that the graph representation H encoded from the node feature X, regularizing H in RGIB objectives can also balance its dependence on X and thus shows some potential robustness, even though we originally designed these objectives to handle edge noise. The above experiments show that our RGIB principle also has some merits in combating feature noise. We should note that this might be an initial verification, and more comprehensive study will be conducted to have rigorous conclusion in future explorations.
29

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Table 18: Combating with feature noise Cora dataset.
method
cf 2%
c/ = 4%
cf = 6%
cs 8%
c, 10%
Standard training .8528
.8629
.8077
.7220
.7353
.8681
.8770
.8679
.8184
.7812
RGIB-SSL
.9120
.9075
.8999
.8419
.8575
Table 19: Combating with feature noise on Citeseer dataset.
method
t = 2%
f 4%
tʃ = 6%
tʃ = 8%
tʃ = 10%
Standard training
.8289
.7630
.7292
.6786
.6491
RGIB-REP
.8543
.7804
.6832
.6591
.6455
RGIB-SSI.
.9079
.8606
.7977
.7201
.6916
Table 20: Combating with feature noise on Pubmed dataset.
method
' = 2%
f== 4%
f = 6%
' 8%
6, = 10%
Standard training .9193
.9064
.9054
.9008
.8904
RGIB-REP
.9344
.9340
.9252
.9060
.8929
RGIB-SSL
.9553
.9433
.9446
.9328
.9150
Table 21: Combating with feature noise on the Facebook dataset.
method
E, = 2%
E, = 4%
Ef = 6%
Ef = 8%
E, = 10%
Standard training .9823
.9804 .9858
.9807 .9816 .9809
.9738
.9691
RGIB-REP
.9864
.9756
.9700
RGIB-SSL
.9857
.9840
.9793
.9760
Table 22: Combating with feature noise on Chameleon dataset.
method
C 2%
c== 4%
C = 6%
cj 8%
C = 10%
Standard training .9724
.9642
.9552
.9493
.9328
RGIB-REP
.9766
.9712
.9626
.9547
.9374
RGIB-SSL
.9765
.9751
.9696
.9632
.9500
Table 23: Combating with feature noise Squirrel dataset.
method
e, 2%
e, 4%
eʃ 6%
es 8%
e, = 10%
Standard training
.9620
.9548
.9581
.9465
.9424
RGIB-REP
.9676
.9638
.9613
.9584
.9502
RGIB-SSI
.9653
.687
.668
.6355
.9541
30

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E Full empirical results
E.1 Further ablation studies on the trade-off parameters À
We conduct an ablation study with the grid search of several hyper-parameters in RGIB. For simplicity, we fix the weight of the supervision signal one, fe., A 1. Then, the objective of RGIB can be formed as Lets + R1 + X2R2, where the information regularization terms R1/R2 are alignment and uniformity for RGIB-SSL, while topology constraint and label constraint for RGIB-REP, respectively. As the heatmaps illustrated in Fig. 16, the A are better in certain ranges. Neither too large small value is not guaranteed find good solution.
0:50
0.0
684
0.01
0.5
0.4
0.85
5.7
034
0.7
10
100
0.01
0.1
1.0 10
100
04 0.
02 03 04 D5 063T080911
Au
(a) RGIB-SSL, grid search in range of (b) RGIB-SSL: grid search in range of
{0, 10 100 100 10², 10t}
{0, 0.1, 0.2, ... 0.9, 1.0}
0:80
0.0
0.80
0.01
0.4
0.72
5.70
0.7
030
0.8
0.88
100
0.56
0.01
0.1
1.0
10
100
00
0.1
03
04
0J
0880913
Ay
(c) RGIB-REP: grid search in range of RGIB-REP: grid search in range of {0, 0.1, 0.2, 0.9, 1.0]
(0, 10 10-1 ,100, 10², 10"}
Figure 16: Grid search of hyper- parametsssssters on Cora dataset (c 40%).
In addition fixing the hyper-paramette hyper-parn parameerrseserss within the training procedures, i.e., using the optimization schedulers. We attempt different schedulers in total, including constant, linear, sine, cosine, and exponential. As shown in Table 24, the selection of optimization schedulers greatly influences the final results, but there is no gold scheduler that consistently performs the best. Nonetheless, the constant and sine are generally better than the others.
constants, also attempt dynamic tuning of these
Table 24: Comparison on different schedulers. Here, SSL/REP are short for GGIB-SSL/RGIB REP. Experiments performed with 4-layer GAT and 40% mixed edge noise.
dataset
Cora
Citeseer
Pubmed
Facebook
Chameleon
Squirrel SSL
method
SSL
REP
SSL
REP
SSL
REP
SSL
REP
SSL
REP
REP
constant
8398
.7927
.8227
.7742
.8966
.8416
.9644
.9788
.9844
.9488
.9293
.9320
linear(-)
8427
.7653
8167
.7559
.6455
.8239
9669
.9529
.9444
.9699
.9316
.9265
sin(.)
.8436
.7924
.8122
.7680
8637
.2755
.6855
.9944
.9477
.9444
.9325
.9282
cos
8334
.7833
.0888
.7647
8879
.8372
.9477
.9629
.9388
.9444
.9251
.9303
exp(
.381
.7815
.085
.7569
.8617
.8177
.9471
.9613
.9422
.9418
.9316
.9299
31

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E.2 Combination of RGIB-REP and RGIB-SSL
We attempt to jointly combine and use RGIB-REP and RGIB-SSL simultaneously. Specifically, the -EEBB-RE-RR---SSES integrates both data reparameterization technique and self- learn- ing method and the final objective for optimization equals minimizing Eq. 4 and Eq. 3 simultaneously, Then, we compare based on a 4-layer GCN on all six datasets with e = 40% bilateral noise.
with standard training, RGIB-REP, and RGIB-SSI.
As shown in the Tab. 25, the combination mances RGIB-REP and RGIB-REP in most cases and outperforms these two instantiations on the Facebook dataset. Although we believe that a more careful finetuning of the hyper-parameters will bring further improvements to the WGIB-REP+RGIB-SS of the instantiations in practice to keep the learning objective simple and sufficient to combat the bilateral edge noise.
BGIB-REP+RGIB-SA
achieves considerable perfor-
combination, we suggest using one
Table 25: Method comparison among (1) standard training, (2) RGIB-REP, (3) RGIB-SSL, and (4) the combination of RGIB-REP and RGIB-SSL.
method
Cora
Citeseer
Pubmed
Facebook
Chameleon
Squirrel
Standard
.7419
.7380
.8748
.9500
.4966
.9406
RGIB-REP
.7966
.7519
.8834
.9770
.9621
.9455
RGIB-SSL
.8554
.8277
.8918
.9711
.9592
.9426
RGIB-REP-RGIB--SS
.8351
.8270
.8880
.9819
.9570
.9431
E.3 Further case studies
Alignment and uniformity of baseline methods. The alignment of other methods is summarized in Tab. 26, while the uniformity is visualized in Fig. and Fig. 18. Specifically, the mean alignments values on different datasets (i.e., the x-axis) and various noise settings (i.e., the y axis) from clean to extremely noisy. Besides, the and y axes in the uniformity figures are the projection dimensions as the common TSNE plots. Here, we keep the same ranges of and y axis from 0 to 128 in each plot to provide a fair and consistent overview of representation distributions under various noisy scenarios.
The intuition is that the alignment measures how the representations can be influenced when encoun- tering topologiaal perturbations, while uniformity describes how the representations distribute on the hyperplane. In other words, when taking the edge noise into consideration, the alignment actually measures the stability of GNN when the input graph is with extra edge noise in the testing phase, and the uniformity measures the denoising robustness of GNN when learning with inherent edge noise in the training phase.
As introduced in Sec. 3.2, desired representations should be invariant to the topological perturbations, i.e., with higher alignment. On the other hand, they should also be uniformly distributed on the hyperplane to preserve as much information as possible. Based on the concepts of alignment and uniformity, we then provide quantitive and qualitative analyses of the learned representations under edge noise in Sec. 3.2. As shown, with a higher ratio of edge noise, the alignment and uniformity can be damaged to a greater extent. Thus, these two metrics can help to comprehend the learned representations by GNN under edge noise. Thus, both metrics are indeed closely connected with the edge noise here, and we reveal that a severer edge noise brings a poorer alignment and a worse uniformity.
We have the following three observations. First, the sampling- based methods, e.g., DropEdge. PTDNet, and NeuralSparse, can also promote aligmment and uniformity due to their sampling mechanisms to defend the structural perturbations. Second, the contrastive methods, e.g., SupCon and GRACE, are with much better alignment but much worse uniformity. The reason is that the learned representations are severely collapsed, which can degenerated single points seen from the uniformity plots but stay nearly unchanged when encountering structural perturbations. Third, the remaining methods are not observed with significant improvements in alignment or uniformity. When connecting the above observations with their empirical performances, we can draw a conclusion. That is, both alignment and uniformity are important to evaluate the robust methods from the perspective of representation learning. Besides, such conclusion is in line with the previous study [35].
32

---

Table 26: Alignment comparison.
Cora
Citeseer
method
clean
E 20%
== 40%
= 60%
clean
== 20%
E 40%
= 60%
Standard
.616
.687
.695
.732
.445
.566
.689
.696
DropEdge
.606
.670
.712
.740
.463
.557
.603
.637
NeuralSparse
.661
.620
.691
.692
.455
.469
.583
.594
PTDNet
.335
.427
.490
.522
.330
.397
.459
.499
Co-teaching
.700
.691
.670
.693
.464
.581
.683
.706
Peer loss
.424
.690
.722
.629
.529
.576
.580
.598
Jaccard
.608
.627
.588
.703
.471
.596
.674
.703
GIB
.592
.635
.652
.692
.439
.524
.591
.623
SupCon
.060
.045
.033
.024
.086
.077
.060
.065
GRACE
.466
.548
.582
.556
.454
.551
.589
.609
RGIB-REP
.524
.642
.679
.704
.439
.533
.623
.647
RGIB-SSL
.455
.543
.578
.615
.418
.505
.533
.542
Learning curves of RGIB. We draw the learning curves of RGIB with constant schedulers in Fig. 19, Fig. 20, Fig. 21, and Fig. 22. We normalize the values of each plotted line to (0, 1) for better visualization and have the following observations.
For RGIB-SSL, the uniformity term, i.e., HH),, converges quickly and remains low after 200 epochs. Similarly, the alignment term, i.e., I(H1; H2) also converges in the early stages and keeps stable in the rest. At the same time, the supervised signal, i.e., I [H; Y) gradually and steadily decreases as the training time moves forward. The learning processes generally stable across different datasets.
As for AGIB-REP we observe that the topology AZZ A) and label constraints / (Zy Y) can indeed adapt to noisy scenarios with different noise ratios. As can be seen, these two regularizations converge more significantly when learning on more noisy case. That is, when noise ratio increases from to 60%, these two regularizations react adaptively to the noisy data A, Y. Such a phenomenon shows that RGIB-R -EP with these two information constraints works as an effective information bottleneck to filter out the noisy signals.
33

---

posiuve samples
positive samples
positive samples
positive samples
84
120
32 negative Samples
54
32
84
28
120
negative samples
negativee Samples
negative samples
64-
56
12
64
26
120
32
64
120
32
54
56
120
52
54
95
120
standard
(b) Dropedge
(c) Neural Sparse
PTDNet
positive samples
positive samples
positive samples
positive samples
32-
32
54-
128
128
128
64 negelive samples
32 negative samples
64
96
128
negative samples
negative samplees
128
128
52
64
96
128
(e) co-teaching
(I) peer loss
(g) Jaccard
(h) GIB
32
32
32
32
14
54
64
64
26
26
96
96
128
128
128
128
negative samples
nepative samples
nepative samp los
negative samples
64
96
64
64
126
(i) SupCon
(i) GRACE
(k) RGGI-REE
(I) RGIB-SSL
Figure 17: Uniformity on Cora with e = 40% bilateral noise.
34

---

posiuve samples
positive samples
positive samples
positive samples
120
54 negative Samples
32
84
120
negative samples
negative Samples
negative samples
64-
12
64
26
120
32
64
120
32
54
56
120
52
64
95
120
standard
(b) Dropedge
(c) Neural Sparse
PTDNet
positive samples
positive samples
positive samples
positive samples
32-
32
54-
128
128
128
64 negelive samples
32 negative samples
64
26
128
negative samples
negative samplees
32
128
128
52
64
96
128
(e) co-teaching
(f) peer loss
(g) Jaccard
(h) GIB
32
32-
32-
14
64
54
34
26
36
56
56
128
128
120
128
negative samples
negative samples
negative samples
negative samples
54
56
64
96
120
(i) SupCon
(i) GRACE
(k) RGIB-REP
(1) RGIB-SSL
Figure 18: Uniformity on Citeseer with = 40% bilateral noise.
35

---

1.c
10-
88-
120
grenent reg. or..eat
Lowa
2
gewrerteettoommmmmm raga arbation
www.geeenttn.m.mmm. repuatoation
Low
0.4
1egg adication
boreyr regeidecation
0.0
00
Epoch
Epoch
Epoch
clean
(b) 20%
(c) 10%
(d) 60%
Figure 19: Learning curves of RGIB-SSL on Cora dataset.
wwp.aay stran consrrint
trpelogy cannirii consbrinn
jo consiriint
100e
ucal
02-
420
220
1000
(a) clean
(b) == 20%
(c) == 40%
(d) 6 60%
Figure 20: Learning curves of RGIB-REP on Cora dataset.
10
0.
1000
pervieed lass
granent oo
granett roga arizaion
speriited loos
grrner Enoglarizzion
06
pervited 100k
gramont hoguareztte
0.4
fambly roga
Earmddy regelsetaation
02-
420
500
2000
200
432
930
000
220
000
200
1000
200
422
900
1000
Frozh
Fport
Fpoch
Faoohh
(a) clean
(b) 20%
(c) = 40%
(d) 6 60%
Figure 21: Learning curves of RGIB-SSL on Citeseer dataset.
per l9st
sesspervite loss
copoogg conatruitt
top.oogyyyyy cotubrant
0.6
conntraiee
0.2
0.-
5-
0.4
0.-
0.2
02
420
220
000
1007
422
202
1000
(a) clean
(b) 5 = 20%
(c) 40%
(d) 6 606%
Figure 22: Learning curves of RGIB-REP on Citeseer dataset.
36

---

E.4 Full baseline comparison with bilateral and unilateral noise
The entire evaluation with 10 baselines and two proposed methods are conducted keeping the same settings as in D.I.
Results each dataset are summarized as follows.
Tab, 30, Tab. 31, Tab. 32: full results of GCNIGAT/SAGE on Cora dataset.
Tab. 33, Tab, 34, Tab. 35; full results of GCNIGAT/SAGE on CiteSeer dataset.
Tab. 36, Tab. 37, Tab. 38 full results of GCN/GAT//AGE on PubMed dataset.
Tab. 39, Tab, 40, Tab, 41 full results of GCN/GAT/SAGE on Facebook dataset.
Tab, 42, Tab, 43, Tab. 44 full results of GCNGGAT/SAGE Chameleon dataset.
Tab. 45, Tab. 46, Tab. 47 full results of GCNGGAT/SAGE Squirrel dataset.
Table 27: Full results of GCN with edge noise.
layers
clean
20%
bilateral moise
40%
60%
20%
input noloe
40%
60%
20%
label noise 40%
60%
Cam
9183 0071
8870
0108
8430 0144
7959
9091 0085
8043 0085
8005 0092
9020
D841
8793=
1099
8723= D103
8686
8111
77119
0157
0526
.7490
8281
8053
0213
X060:
7526--00270
8068:2
1.004
(8355-10155
8254
$7671
861511.009
85851
8834 0155
5649
8674::
8317 0217
78642
0145
738022
7085-20146
5054::
.708
77534
7965=0124
7950
7689= 07440
.8161 11663
7355
1.087
.71101..022
.710611.168
7720:1
0161
7460:1.0101
77212:1.0100
7900__013 0131
77:1-01157
76618:10029
Pubmed
.9737:8.0011 19473---000
-9271 #00077
#.141..0011
9590-1.002 0022 9468 H 0022
.937
0337-1-0016
96466.0024
-6337=0002
9597-8.0022
19178:t 0084
8870=-00441
88788
0031
1841
8854- 0051
1859 0031
.851 + 0040
9030= 0082
9070c: 0862
.9081
88500+-00066
0003
3855-1-0025
8652-1-000
9112-
9033:2.0036
Facebook
9887-1 0008
9880-1.0007
986661.000 0007
994332.0010
9878-1 0008 9852-1
0006
9834-1-001
9892
0006
9888
0008
9888:1 0007
9820110000
9819:1
9668 01
962711
9886:
1981911.000
.998.1100013
96091100138
13689
9764:1.0034
9502 1006
.969:1.0100
9863
0013
9865_.0012
9876:1..010
9696---0000
9657
97784-4-00
9775-
09797 0002
961662
0013
9496220090
9274
9608 0038
9433st 0261
9368
0271
9686
0020
0580=100
0021
9362:2 0035
197522: 0033
0002
95111500077
9226
9666--00045
9450=5-0177
9276--
0229
9752
0027
9766-
0035
9745:2.0040 0040
Squirrel
.9661 0761:1.001) 00110011000000000000000000000001
9621 1. 1.0018
119:1.0000
951411.0011
9610:1
0028
9490
0031
9401
0000
9761-0001
0015
0731=0001
0010
9722-1.001
9925 0011
9430
0006
94061.0001
9386 0015
9416:1 0042
9395 0011
9411-1
9720 0016
9720=
0016
9710: 0021
940811
965622
37

---

Table 28: Full results of GAT with edge noise.
layers
eleam
20%
bilateral noine
60%
input noise 40%
20%
label nouse
9076 0070
8786 0155
8489 0131
8286 0130
00144 0070 5825- 0097 5850: 0077
8991 0058
8841
0097
8830= 0101
83214101811
8005
56161
8530=012
8357
8650 0157
80671 0302
75142
0389
7053
0452
8414
0127
7891
0537
7678: 0641
8376 0097
8154= 0069
8024=0069
Cliesoor
8911
8596-200125
-338--00177
-2274-0111
0096 85264
05124-0174
8762=00076
8650=00022
8648-0166
8386-.0133
8126---0157
-7775---020
25518400185
8191- 0092
0045--0010
.9912
0073
8174=.0172
-9998
-9534-0156
8299 0098
7807
7757340020
.7199
79702 0134
7860:2 0107
0126
1963=00129
0163
7801=
Pubmedd
9406:2 0033
9173
0028
899412000
0000
88841 0015
9355:1 0024 9176:: 0035
91001
0030
9306__0033
9221
0030
9232
0034
8960 0068
8610.1 0015
84311.0002
.8339.1.001
3668.1
0010 8547. 0050 8476.1
0037
8817=.0012
.8722 0000
8696
0062
83151.0059
81161.0073
.80011.000
8001.1
0306
8180=00066
Facebook
0874
9800 0005
985---000
9836
9878- 0006 5861: 98584
0008
9872=-000
9864
0000
9857=000X
9857
9820+-0001
-920+
0014
D011
9827
0046
9860- 0007
9908
+ 0025
-968--00311
.9577
0314
5804-
0018
1938 0014
99100 0036
9854
0015
9860
0007
9867-00112
Chameleon
9770 0044
9725
0027
965011.0188
9625 0018
9767:1 0026 9747:: 0020
973911
0018
9746:
0023
9743=
0017
9711=004
0041
9734 0047
9721 + 0015
96532.0013
9605 0011
9741 0028 9686:: 0010
9674.1
0027
9740-0053
9738
0027
9712-
0047
99422 0052
9659
0029
9573
99821..00 0054
96111 0033 9543.1 0075
9474.1 0074
9722 0013
-688
0055
9698
0065
Squirel
99400 0011
9690
96335--0011 --0017
9588--0025
9902-
0008
9090 0010
9659-1-0011
0014
9719=.0015
.901 0010
9686 0012
99201
0046
-936--
9335
0947
9455:t
94154-006)
96822-00134
9690-0012 9686- 00121
19578--0000 0067
9507
93094.0164
.924+-00099
0065
99199
0073
9520-00070
-5507=0162
Table 29: Full results of SAGE with edge noise.
layrr
clean
20%
bilateral moise
60%
input noise 40%
label moise
40%
Cora
9045 00666
8733 0101
8520 0137
8409 0056
5006ct
0074
1857: 0093
8917 0063
8868- IN70
8695
0106
8564- 0091
8664
8225
7833
7595
0259
$607
8437:1
8387410118
8046
0074
7920=0191
8426
7787
0433
0251
.71802 0738
8356
0232
7747:
0561
3000:: 0421
8158
7707=
0235
7660=0113
8648 0099
.8279+-0099
0120
8329 1: 0093
791411.0101
2686.1.0213
.753911.0119
8171110173
8226.1
0141
.8157.1 0203
8068=00115
7705-0172
8390=
7708.-0016
77244.0026
2086 0145
3997:4 0183
01445
7903=.0219
7690=00201
7554=-009
Pulveed
8095 0014
9136
0012
90942.0015
9033 0040
9295
0025 9378ct 0022
9410s 0023
8817=
8793 0044
8757=10041
8466 0055
8627
0056
8661-1.00
0061
86191 0073
87154 0080
8801 0082
9013-1
0057
8386 0085
8330=00044
8268=0092
831411000
8105
88331
85381
0199
0071
Facehook
9982- mon
9988
0008
9827 0008
.988
0012
5681-1 DO08 5862- 5M622 1008
9867=
9840
0006
9825-00014
9859-
.969
5649-4 2015
9815-1
0015
9828 0024
9751
0055
9600.1.066
961661 0091
9915:1.0077
9672.1 0057
951771 0226
9761
0136
9788
0045
9709-0151
Chameleon
9786
9683
0016
9605
95%6
07544t 0019 97344
97634
0018
9700=00019
0026
9603=000
9924-20000 0025
9644
0005
9550+2.005
9505
0010
9721 0017 9921 0021
9721
0016
9678__0122
(012
0641
(012
9605-100 1051
9729 0022
9606
0046
9541
2:05
9470 0035
9679. 0031
9658:1 0041
9640:1 0059
9673 0026
9641-
0038
9625- 0042
Squinel
9745 0015
9680 0015
9626-1.0015
95701 0016
9937.1 0010 9736:1 0010
9721.1.000
9692
0015
9643=0012
9606 0017
9554+-0100
9957
0016
.951
963744
0092
9630--
9007 0090
9663-
00020
9612-
0009
9612-00020
9682 00:45
9555
1:0054
0053
9600
0012
9574
0207
38

---

Table 30: Full results on Cora dataset with GCN.
OCN
byyers
mathod
bilatera noise
Inpur noise 40%
lahel noise
20%
400
00%
20%
60%
2016
40%
000
Standard
88800+-0108
8430+-0044
7959-..1900
.991 c. 0085
.5433-1
1-433--000 0085
.8905 +: .009
.9020 H- 0041
8893 .009
.8223- 0103
DropEige
857331-0126
130551.0175
80912.001 0100
9016:1 0085
.692.1
01.20
.863 1: 0099
.016
1: 0059
1. 8791
3660:1
0085
NearniSparse
53692-0000
4397+-0101
8215=0055
9118-
2500-4-007
01:1
.849
8086
89925
0009
8797 20073
5743- 0088
PTDNet
3852-1.0127
2180--0006
9095
0108
2926-11000 1.000
8822
6: .0096
9000 0080
8801
H. 0067
35702-1.00 0144
Co teaching
$857.1.0200
8:191.0196
0237 8026
9081
1 0132
8939.1.00
0100
8901 1: 0101
9022
1: 0203
8929
1. 0126
5699
1. 0115
L-2
Peer loss
.867 4917---0115
84724 8257
3910-
3993-
0133
9015
+ 0021
8874-8
0163
28134
0291
Juccard
5912-4.0199
.8661+.0168
7964-
0162
9016-1-000
899661
9094
+0000
8856
0093
5721-
0135
GIB
8857-1.0250 0296
1.326
8037 0309
9125
0065
3973.1 0141
8937 1: 0150
9083
1: 0116
8828
1. 0136
3307
01.16
SupCin
8827 4827-+00125
4451+
8125=
9006
00744
35777 0073
8846
01
8876 0082
3668
- 0088
2722
0092
GRACE
8588 1. 01.58
.8200
0204
7967-_0076 0076
8671
8102 1-
.048
6: 0019
8900
0092
8867
01009
3896
0074
RGIB. REP
8915-: 0091
.816+-0133
8358--0077
9137
0960
3958- 0048
.891 +
9066
0107
89582-013
2035- 0087
KGBB (GIB-SSL
92724 0091
9901 0:145
5892-.0000 0107
0049 9342
9108
+ 0063
.9014-2
0017
9473 + 0064
0051 9427
53800
0084
Standard
81111
.24199.0025
0970-10077
8027
- 0580
J786
1 0526
.. .498
K281 1: 0123
8054
0.13
3000-1.00 0172
Droplidge
8017.1.0180
7035
7303 0235
8338
- 0132
7526.1 1 0077
7454 0423
8363
1 0110
8273 0106
S1
0141
SearalSparse
8190+--010
.7318+ 01379
7293-
7794-
7657
.5244
+ 0094
8246
0111
8211
0106
PTDNet
.5559
0218--02100
82141
01144--0122
2270
- 0624
8460 1: 0128
8214-1.0156
8138 0151
Co-teaching
3137
7.7991.0372
7030 0475
S045 0609
7:71 0566
77530 0500
8446
1 0219
8209
2 048]
3157-1.00 0246
L-4
Peer loss
Juccard
.185 4-22
8143.1.000 0218
74468 .1388
749811.041
04118
7024 0103
7018=
8051
$200
1 06644
0772
7838.1
75066
+ 0025
1 0858
77577
7611
.3225
8289
1 0177
+ 0001
8036
8064
H .027
1: .029
81 0227
X009 + 0193
GIB
74855.0518
7188=
8002 0607
8059:1 0566
8357 0133
8137
8157.1 0270
SupCon
8240-1-0147
01:47
281944.0261
7490--0020 0200
$359
0124
X300
+ 0218
.8025
+ 0220
8491
+ 0120
8215550015
X256-+.0108 0108
GRACE
77722
0207
694011.022 0248
6929
00000 01:0
7877
0211
3107
1 0188
6975
1: 0024
K331
1: 0106
8237.
0252
8119:110000 0246
RGIB-REE
831311
0098
7966110010
7391
2410
0071 8634
83134
0136
8158
8554 0149
83182
0151
8297 0150
RGIB SSL
5930-4-0000 0072
63167
.339
0100
9024- 0097
35774
0152
0156 .8421
0006 9314
.9244
0000
9241
Standard
7940:1.0133
.4129
0226
7177
0207
8035
7973:1
.7546
R249
1 0010
7925
7901
Dropfidge
79472
7353-4.011 0130
6909
0208
7800
0560
7488-1 0187
7170 0187
K2S8 4 0148
8187 0076
7990:1.00 0513
NearalSpurae
2001000001
7334-2-
8434
7018 0930
8082 0418
7497 0386
7803 0741
8415 0167
8160
0181
5081 01D5
PTDNet
8007-1.0166
7600
0225
7167=
0155
7873311
75500 0511
8318 1 011
.73331.058
0538
7899
0166
Co-leaching
7010:1.0132
73871
7151= 0262
8079-00469
8058:1
7623
8296
+ 0365
80202.0177
7976: 0175
L-6
Peer loss
Juscad
.8055 0211
8002.1.01.99
7:9221.0246
0196-10086
02110005500
11991_0368
0331
5011 0204
800111.0166
.760 9: 0272
7660 0364
8409
K277 1: 0138
4 0016
8005
1.99990.889
0229
7918-1.000 0240
7901 0230
GIB
7961 21063
7584-200313
7201 0300
0451 8036
5067:1 0331
7639
0400
8347 0111
80162
0156
7947 0147
SupCon
8092-1 0242
.336
0415
8021
.7434 0257
8273
0202
8181
0234
8157- 0135
GRACE
7576.1.00488
71871.00243
6860
0213
7693
0161
7171 0223
6896
1: 0272
8309
1 0347
1.310
8102-1 0234
RGIB-REP
810011-0137
743991
0221
7040 0192
8282
0125
7837
0142
7623
0144
8365
- 0163
8247
0142
8340:1 0119
RGIB SSL
8823 0126
.8080
0240
7357
03342
8632
0187
3578
1
73300 0483
9184
1. 0070
912001.0108
9126.1 0081
Table 31: Full results on Cora dataset with GAT.
GAT
layers
method
20%
bilateel noise
40%
60%
20%
inpus noise 40%
60%
20%
labal noise
10%
60%
Standed
.878611-0155
8.839
0126:1.0120
9014:2.0070
$825:1 0097
8890 0077
8991
0088
8841:: 0097
$830-1 0101
DroeFidge
8741 0114
8279
8101 0141
8930 0056
8770:t
0063
8586
0110
89541
0093
8788:2
0083
8717: 0106
NearalSparse
8820 01334
0151
8248:20112
9051
88842
8983
8855
0129
PTDNet
0709-1-0155
84872.010
8314:10077
0075
90232.0105
5838.1
0116
8827
0090
9019 0086
8772 0083
8726:1 0060
Co-teaching
8881 0148
8871
837822010
90692.013
5806:1 0173
8939 0125
9289 0781
9100 0358
8069 0245
L-2
Peer loss
Juward
8867 0189
8809-1.0180
8492
88622
8334-1.12
83432 0140
90661.0143
90262.0135
0135
0142
882311
8008 0169
8937
2 0073
9010:1
9057
0212
89191
8948
0196
0194
892911
8918 0217
GTR
8876:1-019
8564
83752
0204
92602 0330
9092
9167
0313
9007 0128
8928
8915 0098
SupCun
87091100119
89621.0122 0121
8300.1.013
89571.0115
5827-1.0131
8905
0116
8331 1 0065
87301 01
8652-101.8 0145
GRACE
8286-1 0224
7564 0229
7328 0213
8238
0215
76151 0379
7309
2 0097
8833
0100
8805 0125
8807 0119
RGIR-REP
8759-1 02
8174
82692
0134
90062
88132
8991
8810 0125
RGIB SSL.SSI
9142 0092
.88781.0133
8777-10018
.2341.0053
8973:1
0067
8866 1 0148
9389 1 0053
9347 0088
9011 0057
Stundard
83234-01881
0015
2941
36162.0107
8454
8377
0114
8530
0221
8357
0082
#2814.0114
Doo
82374-0157
.7782 0072
75152 0050
85482 0144
5005
0990
8000 0151
8516
0100
8373 0142
8374 0157
NeuraSSparee
8309--01154
.7984
7769
0135
0174
8450
0110
8277
2 0148
8500 +
0123
X395
0122
$348 0121
PTDNNe
8364110147
8045 0996
8309
8370 0117
Costeaching
83941
8001 05500
78952
3696
8196
8344
0163
8423
0180
8534
0145
8174--0011
8544 0169
Let
Peer loss
x3444-0276
.000 (1254
0142
8520=
0201
.353+
0145
8576
0316
8533
0012
8527 10075
5444 0205
Jaceard
8319-1-0337
8001 = 0233
7937.2
0107
86342
0130
84491
0272
8406
1 014
8537
0111
8407
0154
8354 0152
GIR
83524.0167
811120024
79452
6279
8860
8579.1
8434
8140 01
SupCan
832442-012
8013-
37762
0145
8620
.541
0098
8381
5318= 0121
GRACE
7707 0347
6711 0695
6656.1
0578
7707
0267
71544 0066
7146-1 0037
8040
0447
7988 0292
8321 0230
RGIR-REP
8374 0153
77872200144
75522
0657
86322.018
8370 0118
81542
0147
8480 0181
83322 0129
$259 0163
RGID-SL SSL
8760:1.0112
84691.0101
0176
88652.0125
8553.1
8349
0030
9163
99751 0087
900611
0087
Standard
80674-0000
2032440045
0452
8414+
0127
38914
0537
.7678
+ 0854
8376
8154 0009
8024 0169
DropEAge
0111
.7355
7110
0388
0198=
7514
0615
7348
8499
8312-0016
8112-
0079
NewalSparse
8169--00100
.7726-2.0165
71.49-1.599 0596
24135.0183
7997+
01.99
.2273
0092
5400
0013
8257 0012
.149 0224
PTTNNe
83074-0166
77402207144
71452 0621
8253-
0514
8009 0231
7759 8330
84641 0006
8334 0168
$1.59 0148
Co-teaching
.019--0912
11596
70070-20455 0455
84962.0165
7909-
8204 0394
$215 0369
L=b
Jucesrd
loss
.132-4-0836
.155 n400
.7537 0446
.75724:0411
2070-1.0521 0521
712321 0454
0415
8405
0142
7788--0647
7947 0661
.7679
7760
8771
0737
8374
8405
+ 0229
0160
82451
8318
0067
8049 0224
3108 0180
GBB
.188-2 0386
.759920449
70142
0442
34522
6392
0709
7923
2- 0718
8366 0137
0100
5000 0222
Suption
7736-4-0629 6434
61152.0617
7535-
085
7102-1
8040
0384
.8609
GRACE
5748 5949
5611 060K
5675
0768
6125
0782
5537
+ 0607
5637
0583
3588
0692
61762 0961
RGIR-REP
81484-01558
.7553=2.179)
684221
0264
0029
5001
.7433
8663
8366
0107
4174
5192 0179
RGMB-SSL
86134
8194
0133
86572.0115
82131
0189
8045 0153
9049
0059
89601
014]
3985 0117
39

---

Table 32: Full results on Cora dataset with SAGE.
SAGE
layers
method
20%
bilateral nosse
60%
20%
input noise 40%
60%
20%
lahel noise
Standand
.7333400111
$520--0137
8469=0055
9006: 00074
3857:t
0093
8997
0063
8868
4: 0070
8695
0106
8564:
0091
DrogEidee
8844--010
8600+
5478=0099
9013=00080
5026
90050+ 8088
.8985 + 0009
8725-22000
3059-
0091
NeuralSparse
.821 4812
8550-+00000
5491=-0105
9020- 0090
25300-1
0067
.89494 0009
8885
+ 00111
8681
#0000
2638
0015
PTDNer
88002-000 0109
85%6 0106
8474=
9040= 0101
0085 8925-
8947 00700
8002 0096
8721
+ 0066
8586- 0075
Costeaching
67944-0097
8569
3557=
89555
4
8759
0291
8632-
L=2
Peer loss
Iaccard
58104-0102
81282-0170
33594-0132
88135400155
2480=00558
8474=-0081
9042-0009
9080 0209
2975 0210
3306-4-0177
0174
8960
8897 2 0045
8576
8922 0112
8695
8768 0251
0183
3539-4.018
X596:2 0133
GiB
8766
8679H.0125
8546-
9089
0252
5004:
0098
8998
+ 01547
8958 0082
#721==00101
X6144 0125
SupCon
0114
8537
8499 0076
5004:
89434
8916 0004
8742
0092
5601
0993
GRACE
81234 0113
29784
7244-0037
5010
0160
7992 0120
8250
8228=
0170
5361- 0102
RGIB- REP
5748.200054
8484 .0149
x380--0100 0102
9016=00072
0074
3876 0075
89144 000%
8803
000%
8628
0088
2449: 0017
RGIB-SSI.
91024 0074
89672.0144
8003=
0076
9196
0072
0090 5059:
0084 9053
0073 9278
9174
0089
91632
0076
Standard
0279
.78334 0093
7595=
0259
8607=
0110
-437
0099
8387
0183
:309
0000
8046
0074
7920 0191
Drogäide
01444-0166
.99444
01.30
7842- 0149
8734
8628
0144
8466
0111
8199
0115
1059
NearalSparse
82924-0157
22300+.0105
7573=
8703
3549-1-018
3396 0061
8418
0148
8025
0148
2069-1.000 0106
PTDNer
8310::0049
7847420174
7000
8626=
0141
3007= 0090
8650
0114
8435
0012
8125
0017
7961 0231
Co toachaching
8237:400139
.8855+
7658=00543
X660 0168
X533+
0149
8424
0196
8500
0545
#117=-0187
2088
0275
L=4
Vverloss
823011-0107
.7863-1-0772
7626-10094
0294
3609
X534
0179
.845
0234
0142 8363
8124-1.0191
79011 0231
Jaccard
0153
2850+=0115
0248
0255
3465+ 0227
84%
0233
0300
80- 00084
7976- 0288
GBB
8286-1-0059
79474-0230
7729-
8886
8 0280
2457-4
01358
85744
0207
85744
00135
7947- 0229
SupCon
8205+ 0143
7809 0176
7383 0218
8568= 0115
54504
0153
8445
0187
8436
0105
8190
0170
7741 0129
GRACE
4242 01245
64244
0290
61724
6696
0144
6434
0884
6376220255
0251
(438 0449
RGIB KEP
K214+-0112
.782440043
0202
8634--0121
8420+--014
0144
.852
0121
.367
0:149
80NT+.0187
7991+.0130
RGIB-SSI.
8837 0065
.87282.0116
0148 8613
8960 0109
S8174
0119
8825
0113
0008 9130
9041
= 0075
0023 0072
Standand
7787-1.0423
742201.025
0251
7180: 0248
8256
0222
.947.11
0561
8005
1 0421
R158 1 0168
77077
0235
7660:1.01
0134
Drogkidge
9035--0222
.2398 0550
7176
0389
8262-.015 0153
8193-
0679
8089 + (220
X340 0101
7793
H 0091
7799- 0144
NeuralSparse
7953-1.0177
72922 0238
8384 0120
82M4
798001..770
8214 1. 0107
.7988
5 1.036
76622 0160
PTDNet
0951-2.0151
-76944.0109
7352- 0202
8311=00143
3267- 0078
1100+::144
8222 + 0121
78825- 01178
7745- 0231
Co teaching
7887-4-0477
27645-4.0332
7212--0832
$306 0225
7791- 0595
8800 +: 04439
8524 + .0256
.7720 01263
77887-1 0266
I=6
Peer loss
7781110451
.7445 0236
7192: 0277
0234 8300
5020
0621
5043 0441
8309 1: 01
.7734
0388
7652
0262
3779-4-0447
.72833-- 0134
3073-
8148 + 0186
.7707
+ .1243
- 0204
GIB
7814+-0093
7349-00437
0437
8366--0194
0194
5106-1
-800 0617
8172 + 0255
.7806
0266
7689-1.000 0180
SupCon
7879:110056
701911.028
6673:10017
8219
0:69
7668
1 0666
715991.0717
8242 1 0159
7880
.7666
7696:1.010
GRACE
4806-t-0100
6455
3967=
0248
6949
-1 0181
6536-
0365
6614 0399
7239
+ 0231
2033
0160
7014 0111
RGIB REP
01146
.7157-5.0725
7099
0473
8391
22115
S149
.7927
8358 + 0100
2974-8.0100
2046
0135
RGIB-SSL.
8062- 0130
8450
8306 0108
8746 0091
3634
0099
860 00032-00100
0089
8931 0108
5940:2
0076
Table 33: Full results on Citeseer dataset with GCN.
CCN
layers
methud
201
bilateral nouse
20%
input noise 40%
60%
20%
label nouse
40%
Standard
.8051+-00844
.855-- 0154
8254-t 0016
8767
00977
3615
.8585 0085
88344
0135
8629
0104
667442.0123
Droplidge
0108
8112-00188
0011
3557-4-077
.8483--00 --164
8862
8695-
NearalSpurse
8605 1. 0019
8.02.1.0138
8239
0069
8801
0085
2634.1 0109
88614:1.00 0014
8827
1: 0094
9753
.720:1.0150
PTDNet
8646- 01.5
4.04--0011
8223
0142
8905
- 0095
3647-
0112
8571 +: 0116
8813
+ 0074
8734 0125
25708 0697
Co Co-teaching
8089
00000
.8416. 0215
8278
.8659
8833
+ 0221
8844-8
0081
2847-1.099 1094
Peer
8724- 01.59
.8374+01152
8308-0000
0062
8783
3065 0152
8638
0131
8909
0144
8653
0276
3308: 0212
Juccard
8082-4.0166 0164
.1250
5292- 0137
5725
0090
35743- 0186
8696 015%
8849
0231
8718:50015
X680-1 0154
GIB
8825 9:019
.4776 0333
8306
0155
8819
- 0381
3000-1 0318
8688 + 0136
8914 0253
0122 3681
.3707-.0144 0144
SupCon
8344- 0101
81733--0155
8140
8325
0136
8436 0128
8337
003)
GRACE
8450+ 0123
7898-0008
0081
8529-
0133
.7967
0171
8007
8626
0122
3699-1
0156
RGID REP
8585 1 0088
8167=.0113
$751
0074
8637-4.015
2637
0122
0019 8800
8795
9: 0101
86338
56384.00775
3641-1 0095
RGIB-SSL
9199 0091
.957
8759= 0987
9271
= 0855
5018:: 0090
8942 0014
9495
- 0009
0515 0044
0480:2
0062
Stundard
J8644-0145
73300 0201
7085 0146
8054 0240
3708
+ 0325
77883 + 0258
7900
+ 0014
7800
0174
36699- 0240
Droplidgs
3635 76006
.333-1.0170
01990
0106
3730 1
.7473
1- 0060
7937
8 00155
.7853
3632-4
0142
NearalSparse
776511.0123
.7397 0168
7148
8093
7809
0160
.7668
1: 0289
7968
1 101188
.7921
7752-1 0209
PIDNet
.7964-0131
-7425++-8200
77853=00130
9119=
0101
3811+ 0191
.7658
+ 00177
2200
+ 00141
.7765
H- .0124
76222- 0215
Co teaching
7533 7533-1-0181
.7238 0245
.131 0157
8059
- 0263
3753
0408
.7668
1: 0232
2094
1: 00311
.7877
7913-1 0043
L-4
Peer loss
7423-1.0255
7335-1.0213
0242
8106
0250
77677
0400
.7653
0315
7091
1 0241
.7990
0289
(751 0245
Juccard
7473-400160
.3524+.02044
0204
-107-
0172
8176-
0283
-776-4-044
0477
.7725
+ 0432
1001
-- .099
7689-4 0310
GIB
.759911 0036
7388.1.0240
0210
8070
0398
3717 0612
.7798
1. 8611
7996
1. 00000
.7852-5.0188 00186
.7649
0275
SupCion
7554+ 0196
7458-
0176
7299
0218
M010= 0099
37677
0111
7655
+ 0164
5024 01008
.7983
0012
77077 0166
GRACE
76324 0224
72422
0219
6844-
.8300
0232
7509
+ 0011
.2630
.0195
0007
RGIB
7575 1.031
73112-
0227
$299
0134
19030
.7771
1: 0178
8083
1.0152
.7866
1.024
7945 0203
RGIB-SSL
8094-2.0108
2427
8137 0170
8747
1 0144
2461
0109
8245
2: 00142
9204 0085
.228 0099
59250-1 0071
Standard
728554.0089
2110+-00222
0016
17200 0161
3460-
0401
.7212 .4700
.5000 00311
.7741
+0157
.668- 0229
Droplidge
7683-1.0133
.7191 71915
2006
0504
7645
0:41
7442 0160
.7009
1: 0182
8007 1: 0152
7823-
7656 1. 0196
SeuralSpurse
76411-0180
.731 1.091
7036
0387
7913
0152
7476.1
0279
.739
1: 0235
7793 1 0391
.77392
0188
7668
0256
PIDNet
36614.0153
3012- 0385
7982--013 0134
-728-
0164
.7464
0158
.7933 0012
.77424.015
7382-4.00 0151
Co teaching
7395
..199
7188-10255 0248
7813-
0188
. 0175
7330 0496
8004 1: 0017
.7885
1. 0075
.7011 0215
L-6
Peer loss
Juccard
7396.1. 0087
7415-4-0444
7105 0258
.7162 0307
3135 0111
7
7166: 0250
17933 0263
77883=
74499
7512.1 0488
7501
.7369
1: 0509
3. .443
79225 4 .016
7915 0169
.7825
77748:1
6- 00455
0299
.77 10 0298
7695 0239
GIB
05311
.72611 0268
0224
7748-00653
.722-1 0697
.1276 1504
79455 1.018
7.031.00221
768111 0251
SupCon
.714-4-0022
7415+-0116
72000
02331
1907
1007=-0000 0161
75444 0109
10133
.7445
+ 0031
.7659
1014
77573 0180
GRACE
5995 4.021
6901 0210
6916-10152 0152
7041
1 0260
2027
0247
7752
+ 0220
.7739
9 .0151
3773 4 0051
RGIB- REP
7725 0077
7:29-.0002
7232-10010
7781
0218
7559
774151.023
0221
7900 0252
.7797.2.0181
.78661 0194
RGLB SSL
8417-400199
299554.0148
7633
8379
3026-
0173
.7793 0007
8964
5964
906244 0006
0098
40

---

Table 34: Full results on Citeseer dataset with GAT.
layers
methed
20%
bilateral noise 40%
00%
20%
input 40%
60%
20%
label moise
40%
00%
Standad
.886-+-0125
820722012 0121
8689
0099
8526-t
30
851224
876244
8650+-0102
5648: 0166
Dropfilge
836641-01 0013
8100:1000 0098
8750= 0079
8496:1
0101
8512 0121
8820:1 0086
8679 0112
5673:1
0114
SpuralSpurae
8571 0101
843
83224 0092
8743- 0117
85774
0067
8580
0135
8833-1 0080
8734
0076
9657 0089
PTDNet
8802-1.0107
83311.0113
81574.0075
887441100
86931:
36691.0112
Co-texching
8638-1 0220
20124
3199
87202.0122
0128
8521::
0139
85100 0224
8924
0122
88882 0365
2919 0305
Peer loss
86374-0125
88378 0070
0120
8721=-0122
8520 0173
8559 0216
8878
0185
86552 0288
5611 0258
L-2
Juccard
86151-0197
.83991.0222
01244
8841
8556:1-011
8498 1 0309
6543
0143
8676.10099 0195
8661 0256
GTB
8610:1
84622:0114
83242 0316
8909
8781
8779
0144
8741
SupCan
8495-2-0100
8178
315522 0099
8611
8086
8454
0111
8303
+ 0137
8499
K179 0185
GRACE
8092-1-002 0221
.7564
7479.10226
0030
7628 0240
7433
1 0245
8758
1 0146
8768 0073
$654:1 0172
RGIB-REP
8545 0108
8310=
81372 0091
8736
0107
8566: 0997
8503
4 0159
87784 0093
8696 0081
86142 0084
RGIB--SS.
9106:t
0058
86772
91722
94194
9410
0090
Standard
8026--0157
11755-0044
0515
8191
5043-
0105
7912
8174+
7998% .014
7934 0156
DropEdge
80614 0079
7624 0211
74342
0132
7977 0178
0163 7814
82674 0148
8103 0178
8057
0148
NesralSpurae
.798---0122
.7761
= 0172
7550 0129
82832.013
0130
8088
0088
7911 0011
8259
0119
8138
0092
7066 0109
PTDNet
8000+-0013
7597 0185
8254 0105
8132+ 0089
7950 0143
S137 0243
M082
0094
8036 0139
Co-teaching
8016:10184
7521
7547 0150
Led
Peer loss
Jucard
80644-018
8068-1-022
7771 02273
75442
0186
0191
83462.0145
1258:-01244
5108 0122
8063
0138
7901
7945
1 0073
8160+
8206
1 0168
0529
8036 0176
7225- 0207
7009 0215
GIB
8170 02.30
78844 0541
7645 6047
84222 0365
81124 0212
7973
2 0505
810744 034
8080 0155
5010= 0177
SupCon
0125
7478
79722
7852 0201
GRACE
7319:1.0433 66112.0395
64491.0670
7316:2
0261
5947
0660
6000
0507
7773
1040
7739 0475
7882 0328
RGIR-REP
7-017-0107
77343 0164
0121
0155
7006
0157
7377. 0008
8108
7946 0180
7035 0131
RGIB-SSI.
0145
30292
0098
8592 0120
8251
0132
8145
+ 0110
0091 5084
9101 0076
9102
0117
Standard
+7077
.3333+00270
71339
0251
2920-.0134
7800+ 0107
.741 + 0126
7963
+ 0129
7883 0162
7801
+ 0161
Dooe
0088
0197=
7116:2
0119
7254
7640
7826 0186
SeuraSSpuree
7704+-0099
7242:4.0133
0138
8047
7647
7248
0096
+
7875-20176
7880= 0148
PIDNEt
.788
7505++.2233
7286-4.025
0257
7927+.0287
7822+ 0132
.7579
+ 0555
S000 + 0085
7977
78900 0145
Costeaching
.8899 0141
1799
7236-2
6292
79642
7899 0181
7740 0185
7933 0406
7918 0348
7079 0245
L=6
Peer loss
7846- 0214
.1499=
.71872 0259
7979-.0172
7955-
.77%
.7665
7912-
Jaccard
JM024-0017
7157=.0248
3036
-008 0226
.7733
8- 0245
7964+
0226
.7936+
0256
7847-20211 0218
GIB
7818440020
.735
7137220416
0416
8161
7955 0183
7766.
8007 0155
7958 0166
7794 0044
SupCon
73700+-0524 7160=
6670-2.0442 0442
7607
7729
0356
699
0597
7810 0219
.7522
01
7591
0962
GRACE
5068110118 0104
5108
0319
5058
0096
5288
5068 0178
RGIR.REP
78174:0129
7254:2
0188
7983
0160
7769
0168
7620
0176
7981
+
27111
78174 0164
RGIB-SSLL
X275 0148
7989 0136
76812.011 0140
8261
0096
8024 0087
7800
8855
+ 0103
8918
8040 0119
Table 35: Full results on Citeseer dataset with SAGE.
SAGE
layers
method
20%
bilateral nosse 40%
60%
20%
inpur noise 40%
60%
20%
label nouse
40%
60%
Standard
8138.1.0143
8:041.0011
8279:1 02999
8644:2
.100
5608
1 0064
8847 0066
R560
0120
8533:2.0112
8112:1:0122
DropEdge
865411.0111 0018--01188111
8593
0129
8503-1013
8834
0150
8791 0108
8806 0101
8809 07 0085
8734-2 0064
8710:2 0170
NearalSparse
9658
0079
8:77..00099
8437-0000 0069
8746--0010
0100
367441
.8722
00133
01700---110
8645
PTDNet
8620:1.012 0125
8088.1.0133
3477=10078
8740
0094
8675 01.0
8800
0034
8637 1: 0138
15761..0147
2530:1.007 0073
Co-teaching
8506 0143
843612.0179
8323=
0144
8652-
0133
560611 0067
8650 0247
8767 0069
866012 0101
A392.1 0363
L-2
Peer
8825
851411 0208
0157
.84001.0120 .120
8291 0163
8319-.0188 0186
8661
8691
0199
3693
1:
8875
.8335 6- 0151
0152
.863 1 0278
86161.0122
8535
$113110161
3563 0241
GaB
85744--0092 0092
95774-0134
5323-50125 0125
8693-.0188
0186
2631- 0310
88339 8400
8000+ 0197
85762.01822
2403-4 0157
SupCun
83441.0106 0010
82413.0123
8168
0111
8485
0133
3493 01447
8854
0161
8457
4: 0109
8391
1. 0096
8323-1.0113 0113
GRACE
8283 0295
831911.019
8253 0208
8361
0190
8273.1
0222
8351
1.314
8431
0193
8422
0119
3400.1 0233
RGIB
8514+
8213-00112
8614-
0118
3527-4-011
0105
.8664
0153
8592
8533-
01334400134
5455-
RGIB SSL
9000 1. 0000
1.019
8916:1.077 0078
0076 9045
2915
1: 0097
.892
0041
0082 9143
9075 0080
9087-1.00
1000
Standanl
7914:1.0101
76861
2339
8171
0173
8226.1
0141
815T 0203
80SS 1 0016
7781
1:: 0136
.70611 0172
Drogfidge
788911 0182
7850
7678 0206
8300
0130
S0TT
0953
8271 1: 0115
8058 0118
7899
12: 0173
7826-1
0082
NearalSparse
79311-0115
7746.1.0195
7639 0175
8331
0119
8212.1
0176
.8324
10144
8111
1 06
7904
1. 0229
.747.1 0242
PTDNet
7911-1.0105
7804
7563-
8299- 0177
$25911
0119
.334
R121
1
7892
7845 0166
Costeaching
7078-1 0189
767012 0730
7557=
8226-2011
0172
$2071 0202
8256 0261
8233 0203
8083-2 0218
7836: 0347
Led
Peer loss
lucard
7913.1.0116
700111-018
.71101.0300 0307
7691
02069
7556:1022
7626-1018
8197 0264
8359-
80441 0189
03155
8200 0064
8239
8135 1: 0131
8167 0119
7972
0090
0157
7744 0227
7 0195
GIB
7031
7750
7601
0291
8190
81061
8211 0371
8151
79472
0766
SupCon
7870:1.0130
7672.1.0145
0615
8191
10811
8006 1. 021
7982
1. 1.137
.788
0122
.7783:1.113 0172
GRACE
6196:10053 6404-1.0255
6308:
6286
- 0287
63651 0223
6196 0157
6495 1 0202
6431 0210
6111 0191
RGIB REP
78514 7703-4.0123
7562-.0200
$195
8151
.813 0221
8008
6 0118
.7863-
.7798 0212
RGIB
85451
8/8221.0147
8352. 0127
8525.1
.8561 0141
8967
1 0088
8866
10611
3903:1.00
Standand
7708.100168
722331
0614
7204:
8086
7997:1
7872
0145
7907 0219
7690
0201
7564 0096
Doog
7756 0143
7485
8104
7290= 0761
8070
5060:1 0165
8016 0195
7961
+ 0174
7769
- 0149
7534
0212
NearaSppasee
0177
77564-2.000
0076
8039 0144
5008 0157
8016
0223
2002
+ 0199
.7662
.117
7588 0160
FTDNee
7844-1-0116
75255
0270
7435=
8070 0130
81181 0108
8085 0135
7048 2 0165
7830 0182
76342 0210
Co-leaching
7770.4 1746
7244-1.0643
7268-
8087=001%
3016
7864
8011 0904
767122000
7676-2 0232
L=6
Peer loss
Incard
.796611.0188
7800:1 0057
7213-1.0628
72677
0606
7294-.0233
7248:
8135=00225
8166-
0225
2012.1 0206
5088 0177
.71441.000 0208
7980 0250
8005
7000-1 0077
1. 2009
.7749 0269
77161.0316
7672-1 0098
7617 0147
GBB
78538-0158
73522
0808
72472 0332
8180= 0988
5073 0241
8049
0378
796T 0223
77100 036
7576-2 0097
SupCun
761911.0176
.11331.4426
7040-00211
7864
7800
.75291.00
1419
7775
.77151.0206
771
751771 0235
GRACE
5608-1 0371
1767220441
0494
6591
0344
6600
0166
6670
0792
6966
1 0367
667k 0683
6079 0476
RGIR-REP
0076
7427
7981=
0104
7999
0149
.7844 0738
7959 0137
7865
0163
780122 0169
RGIB-SSL
8372-1.0169
.822-1.0159
8184
8441
0075
0078
8252-1
0129
82881.01144
8795
8802
0100
8749-1.011 0178
41

---

Table 36: Full results on Pubmed dataset with GCN.
QCN
layers
method
20%
bilateral moise 40%
0076
20%
input 40%
60%
20%
label
40%
60%
Stondard
194734 0019
92714
9141=-004]
95900=-0000
(M12
9468-+
.957 + 00166
9646
0004
9637=2002
0597::
0022
Drogfidge
9394-1.0005 0025
9155-2.002 0027
8994:2 0036
9467= 0022
9300 0021
9146:1: 0002
9504 0026
9558:2 0027
0519:1 0019
SparaiSparse
94792 0021
9251
9120= 0029
9558
0019
93154: 0033
9269
0014
9654 0015
05254 0023
05884 0017
PTDNet
9467.1 0018
9964
91111:0022
9320:1
9272.1.00
9651
961611.002
9584:1
0021
Coleaching
9500:1 0085
99352.0100
0096
9510= 0098
0331:1 0054
9255 0013
9676
0018
95602.011
0127
9608:1 0098
L-2
Peer loss
Juccard
9500:2 0014
9496.1. 0039
.9325 0057
9339220004
9140= 0138
9235:
9558 0068
9556
0019
.927741 0067
9307 0019
9283 0000
.250:1.0115 0050
9615
9550 0012
0175
0521 0178
9532
0086
95911 0054
0545:2 0144
GIB
9500-1 0205
9509
9300:1
9228
9559
9575
0597
SupCon
9345:2 0020)
9257+
0131
0015
9345 0030
9214 0023
9625
9520
0018
0506: 0010
GRACE
9341
23011-0002
931911.002
0027
91542000:9
9409 0046
9331
0032
9225 1 0078
9489
0004
9516 0027
9511:1 0022
RGIB-RET REP--REP
95374 0013
9368
0018
9270 085
0570
0022
9467: 0023
9365 0016
9606
0013
9690
0023
9671 0015
RGIB--5LL
9585
9471 0033
9301 1200
9425
9305:
9121 0125
9719 0000
9734
0015
9711
Staadard
8870+
8748
-641=0041
8854=00051
8759:
.661
0040
9000
0082
9029
0022
9070:
0062
DropFige
8711
8487
8045
0962
0682-0015
3456:1 0076
8376 0046
9311
0071
9201 - 0091
0340:1 0077
NearalSparae
8008+0000
0733
x610= 069
8911
0090
S722
0041
8649
0044
9272 0108
91
0117
9089
0084
PTDNet
8872+
8733
8623=
5903=
0087
8776
0078
860P 0055
9219 0112
9099 0104
5033 0101
Co-traching
0000
8760
8638
1011 0045
57004
8606
9115 0075
0291
0327
0119 0124
L-4
Peer
luccrd
0130
0036
8815+-0039
8803 0060
8566=-0057
8512 0136
-017
-
8987 0221
3811
5764:1 0099
0127
8659 0073
9136+ 0116
9008 1 0110
9135
9101 - 0044
0116
9210: 0095
9096.10132
GBB
8809 0239
8729
0205
85442 0651
80322 0356
83084 0063
8618 0317
9017
+ 0089
91142
0065
9064 0059
SupCon
0061
8718
8525=00108
8739
8558
9131 000
9108
0095
91622
GRACE
8912-1 0034
874911.000
0098
8588=-0942
8810
0034
8795
0099
8503 0040
9234 0088
9252
0053
0135:1 0043
RGIR-REP
90174 0044
8834 0082
1652=-0038
9008
0633
8822 0054
8657 0015
9357 0078
0341
0062
0112 0062
RGIB-SSL.
9225 0125
.8188 0065
8097 0655
9126
0044
8589
0052
8693
- (049
9534
+ 0036
0028 9604
9613 0023
Standard
8870=
.831
0032
8640-
0636
8855
- 0025
8742+
0029
8652 + 00441
90000
0083
9112
0159
5063 0036
Dro
0633
8421
8144
8342=0008 0058
8623
5407
8328
9140 0014
9102
0082
0972
NearelSparse
8814+ 0049
35%
8020
-603=-0000
8792=
0172
3:25
0031
.6112 8046
9140 0065
20 0047
9080:
0077
PTDNNr
8907
8610+
50026
3318=
8791 - 0040
35708- 0032
86619 00300
.123 + 0004
9086
0017
9105 0012
Co-teaching
0126
8698
8568= 0150
8773= 0825
87674
0113
X691 + 00955
9101 0372
0311
+ 0290
0204 0162
L-6
Peer loss
8820
8633- (879
8534=-004
8734
8775
8668
9155
9172
0088
9251220107
Jaccard
8882+
3655
-525=-0025
#711=-0031
37944--012
0126
.8670+-023 0200
9045
0013
9118
0108
9065- 0059
GIB
0195
8718= 0070
5767 10
8623 0114
9081
+ 0131
91
0147
9155 0106
SupCon
07166
3627
8705
0108
5733
0026
8643 0044
9231 0081
0294
- 0075
9218 0092
GRACE
0778
8664
8584
8675
8591
9206 0064
9190
0110:2 0049
RGIB-REP
8846 0071
88155
0035
8640=
0043
8818
87164
0021
8646
8047
9161
+ 0081
0156 0075
9179: 0095
RGIB--SLL
0002
.857
00099
3633 0036
8891 0057
8731 0034
8659 005
9450
0042
9488
0065
9484 0027
Table 37: Full results on Pubmed dataset with GAT.
GAT
layers
methed
20%6
bilateral nosse 40%
20%
injut nolse 40%
60%
201
lihel nosse
40%
0006
Standard
.9173-1 1123--0000 0028
.8984 1: 0003
J888
0033
9255 H.
9176:1 0033
.9102
0000
.306 1: 0008
.921 11: 1.0030
-93221.000 0034
DropEidge
.102-4
0.03
8970
$837.1.00
0034
9208:1.000
9177:1
9374
.9099
0000
9267:1.0019
NeuraiSparse
.9006+-000
0021
8992+
00031
38.00-.0000
0829
9179-.0012
52210--
0082
9141 6: 5038
9310 0009
9297-40000
5252- 0021
PTDNet
9119:1.000 1-015
.8933 - 0030
3844
0037
9068:1
0007
9210:1 0026
9143 6: 0025
9311 4: 0001
9280
0034
9267:1 0004
Co-traching
9163--
.8924
00550
3881220100
0100
9058- 0089
5180-
0089
99336
+
9964
0207
9276- 0249
L=2
Juccard
loss
.1288 1 0057
9156-4-0115
.890
8965
0094
3083
883111.000
5823-4.00
0085
9147-
9042
0092
8055
9171-
018111.01755
0089
.928
9172
1 0214
0070
9400
9301
11 0009
0225
9296
9355 .0127
+ 0084
9219- 0222
9282-1.00 0109
GBB
9136- 0061
89965
= 10
2881
0115
9052
0313
52088
0097
9383
+ 0208
5000
+ 0009
19291
+ 0005
9277 0089
SupCon
SK02
.8881
00037
886334.000
0027
0025 9026
5080-1
0033
9202
.9123-
5659
0027
GRACE
.7822
0893
7914:1.770
0767
8053-1.000
0498
7993
0695
87911
1 0042
1926
12557
8916:1
0211
RGIB-REP
9190-4
0025
91344
- 2017
3939-..000
0056
9250=.0055
9114 0024
9099
9002
9511 0036
9506
1002
S276 0022
RGIB SSL
9223 4 0020
.954
H 00032
.9600-.000
0027
0021 9183
9001
0023
.897 0.0029
9408
4 4.0027
.407
2 0022
99788
0018
Standard
96101100045
.8134
1: 0012
8339.1.0048
0048
1668110004
8547:1
0050
8476
1: 0037
8817
0012
8772 0040
3696:1 0082
DropEddge
8600-1 0034
8462
= 0028
83251.
0044
86052.000 0041
0057 8557.
8454
1: 0009
B852
1 0082
8851
12. 0055
SS71
0034
NeuralSpurse
8691-100034
.805
.0025
8402-4.000
0055
9012
0042
8572-4
8048
8159
9008
8906
0009
88006
6-076
.526-
- 0025
PTDNet
8614110008
.858 1: 0006
8372-1.0333
0338
86101.003
0035
$511:: 0000
1860
1: 0027
5900
1 0052
8855 0000
5385
1: 0004
Co-teaching
8688:1 0125
8450 0065
8413:1
1001
8666
0066
8S36:1
0103
8468
1 0118
8791
1 0107
8946
0335
5968
0132
L-4
Peer loss
Juccard
.8133 0613-1-00440
.8993
8800
+ .0105
0082
8265-1.01
$342-4.010
9000
01:36
0195 8677
8652
0061
14951
85011
0065
.8711
81851
1: 9009
10
8861
8351
1 0078
1 00110
5839
0. 0008
00000
3350
5555
0510
0119
0009
GiB
8616-4 0096
8534
+ 00443
0010
8914
0009
.8423
4 0020
8823
+ 0111
0000
:300 0076
SupCon
8534410061
.292
5 0022
3018
0.000000000 0064
8625
L 0065
.447- 0069
88300
0141
5768
1 0057
3696
1. 00032
9629
01 1005
GRACE
8355 0177
8202
-. 0387
5016:1.00
0909
8139
0344
8078
06/9
7822
1. 0856
7878
1 0839
$1851.070
8008 0794
RGIB
8056--00043
8443
5055
8339-40003
0033
3674
0055
8528--000
8322
9006
8909
0084
8925
8833- 0002
RGIB SSL
8891 1: 0037
.8651
1: 0000
8-80-1.0000
10066
0039 8878
3650
0014
8889
1: .0055
9276
1 2012
9277
926111 0024
Standand
8315 0059
8116 0073
8040-1.000
0092
33744.0003
$201
0021
805T
0000
8450
0086
8414-100116
8113:1 007]
Dige
8468-1 0407
8236 0050
5004:1
0071
8655
0037
8303-1
0044
7015
1 0334
8501
1 0058
85681: 0968
8523-1: 0381
NeuralSpurse
9426-1.0035
.823 1 0002
3098
0067
8323
0342
$258-1 0090
7977
1. 0299
8622
0062
8563
0091
8577.1 0075
PTDNet
838411.0019
8117=.0384
0360
8508
$251 0284
8204
0014
1
8408
0251
$452-1 0085
Co-teaching
8360:1 01
8111 20123
3081 0113
8301 0101
8246: 0398
8071
1 2312
S729 0000
86471
0107
$474 0079
La6
Peer loss
Juscard
8354:1-0128
8333-1-01-40
8811 = 0073
111611.0111
5044:1 0082
5011110104
8408:2 0221
0112
$277
0424 8340:1
8216
8133
1: 0336
0497
5501
8508
1 0132
1 0100
850T 0210
8563 1563
8304 0108
8310 0089
GIB
8316:1.0125
8266
0055
80642
8350
$0061
8274
8551
1 0140
8-031
0154
8761 0072
SupCun
2218-1-0236
.7763 1: 0006
7113-1.0241
8137
3025
0286
.2555
0297
8543
1 01.9
0.300
35:00:1.00 007]
GRACE
.77331 0737
.729
- 0344
68211.1217
7015 1340
6053
0605
6643
1009
S212
1 0062
5455
0459
5743-1 0549
RGIB REP
8324-4-0039
8162
0088
793444.006
#248-+.007 0070
S145-L
0088
.7892
4- 0107
8518 +
8449 0083
5492-1.015 0113
RGID
8738-1.0051
8443.1
0033
8124.1.084
8709
S432-1
0016
8187
.9500110000
9235-1.0031
42

---

Table 38: Full results on Pubmed dataset with SAGE.
SAGE
layers
method
20%
bilateral nosse
60%
20%
input noise 40%
60%
20%
lahel nouse
Standand
1913-420093
9094- 1M15
9035-20044 (0440
9295 0005
9578:t:
0022
9410::
0013
88174: 0014
87952
0044
8757 0941
DrogEide
9142-4 0026
9188-
9014--0038
9179-_0022 08020
9178-+
0013
.106 + 00013
9012 + 0001
9001-20001
0018
5011-
0020
NeuralSparse
-918- 0030
9094-.00334
9271-0001 0019
9271
99299-1
00026
.9254 +- .0117
.9033 + 0002
8925
H 01350
2364-
0052
PTDNer
91144 0026
91384
0030
9104 0637
9267= 0020
92114
0027
9229
8033
9030 0008
8941
+ 0110
5368 0633
Costeaching
9164 00311
-9104 8M27
9092=
9362
9958:
99238-
8827
9054
0236
3356-2.01
L=2
Peer loss
Iaccard
9108 0123
9100+--010
991744.0044
9167420105
9044-00087
9109 0631
9236 0098
9208-
0079
9928----000
9259: 0175
0001
.907
9308 0144
80766
9005
8861 4 0052
8785
8833-
0086
0092
3909-4 0058
33424 0048
GBB
9173 0049
9125 0211
9096 0020
9338
0152
92944:
0125
9299
0194
5880 0125
0129 8865
3853+ 0058
SupCon
0026
9106
9259
9105::
9250
9031
0015
8929
GRACE
83174
0063
8291
8336=-0000
0068
8288-0007
0071
33442
0074
0052 8300
8330+ 0046
8344
0077
8338- 0075
RGIB- REP
.9922
0022
9136
2007
9119=-0057
9228
0051
9522
0051
.951
9005
+ 0005
9015
0051
5000 0043
RGIB-SSI.
92764t 0052
92655
0030
9313= 0651
9273
005
9305:
0040
9354 00025
9304
0017
0294
- 0049
0374
0066
Standard
0027+-0056
8663-
0061
8619=
0073
8715= 0080
2001
0082
9033
00357
8386 0085
8330
0104
8368+
0092
DrogEidee
4767
N740
0066
8816-
5016::
90.36
0044
1059
SearalSparse
.8225
9946
9821
3982
9146-1-000 0002
00H1 -1100+:000
.847 + 0015
8535-
0091
8472-4 0079
PTDNer
8842 0045
8853
(0199
8846 0952
8885
0055
5057 0055
9127 8055
8464 0105
8458
00666
2408 0065
Co toachcc
8078-4-0104
47752+0000
0122200122
8727= 0079
29711
0159
.901
+ 0058
8001
+ 0240
3498
0338
X5 x514+ 0988
L=4
Poer
96834-0116
.8717
0100
3065=-0117
$736-
0075
3997
0079
.9095
6 0019
85 0151
8514
0015
2423
Jaccard
8646-2
-1255+00141
8625=00140
0725
9017:
0205
9155
0164
8378
- 0137
8365-
0118
X275- 0685
GBB
1206
5641-00255
8824
- 0101
3918-
0095
.921
9: 0045
8479 0105
8410-10151
X330--01100
0120
SupCon
8768 0086
8687
0100
8518 0198
8907
0054
8957
0104
8879
0113
5865
0046
8811
00/2
5810 0044
GRACE
7142
6641
8484
7227=
0440
0516
6998
0806
6647
0535
4640- 0424
RGIB KEP
5746--
0049
.8705
-758--0011
1800
0045
2904-
0048
.844
0051
8504
8466
2533+_0011
0118
RGIB-SSI.
93155 0036
9257 0035
9214 0033
0046 9276
0039 9310:
9286
0035
9463
- 0000
0057 0448
0021 9430
Standand
8314.1.008 0081
.1051.0279
8333:10008
8226
0335
8518.1
0105
8566
1. 0199
R242 0104
8161
1 0071
5090:1.00
01.99
Drogkidge
8781+-0041
3665--
0065----0070
8356- 0081
3800
0050
37211
0122
8622
+ 0055
8500
+ 0000
8698
H 0074
8426
1096
NeuralSparse
8611110001
.8589 1.001
8482:1
0083 S694
8858.1
1 0068
.8899
1.05
8368 1 0004
8344
L 0090
8353
008)
PTDNer
0073
1525--006-
3430
3627-
0115
8739--0100
8232 + 0006
8195-2.010
(017
5198- 0060
Co teaching
8385
4-854
42202--8800
8349--0187
8265
- 0385
-007-1.0164
.6612+-019 0009
8259
+: 0052
8243
0074
2143- 0409
I-6
Peer loss
Jaccard
833911.0166
8355 0010
.810++
8111
02285
0272
8347
0124
008
8317
0:12
3364-
3610.1.000 0135
8605 1: 0023
866T
+: 0199
K277
82500 0128
1: 0224
M217
8260 0197
+ .1.50
.201.1 0016
X1400: 0192
GIB
8309-4-0000 0062
.8195+.0329
8236-
6206
8226-
0335
3527-4-00
.867
0344
8330 0142
M237
0128
$141+ 0165
SupCon
8288 1000
.827611.031
8174:1.000
8617-.0122
8443.1 0528
8365 1: 0350
8745
1 0071
86921.0152
2661 0076
GRACE
J290
.7151+0036
6737 0522
7102
0296
7392+
0257
6671
0444
7120 0166
20042 0588
7736- 0286
RGIB REP
8492
0057
0083
8373- 0079
8439
0063
3527
4-
.8536-+.012
8275 + 0218
8148
0009
8211 0145
RGIB-SSL.
8915 0060
.9971
8956- 0646
8948
0079
9006-
0042
8995 0040
9313 0042
9311 0036
9294- 0053
Table 39: Full results on Facebook dataset with GCN.
CCN
layers
method
201
bilateral nouse
60%
20%
input noise 40%
60%
20%
label nouse
40%
Standard
19880-4 0007
9966- 18007
9843-2 0010
DOUS
9952--
99834-
0011
99922 +: 0006
9898
0100
5888-1 D007
Droplidge
9966
0007
.985-+.0000
9821= 0009
9768-
0010
.9705
9001
9783
9979-3
5881-1 0007
NeuralSparse
.977-1.0000
0007
98611
10.0
9637-10000
9772
0007
9632
0007
.9752 1: .010
9788
1: 0006
9885 9.
9638-1 0005
PTDNet
9876:t 0008
9862
0003
9838
0008
9776-
0007
9548-
0008
9722 6: 0011
9859 + 0006
5786 0009
5886-1 0006
Co Co-teaching
9871
10052
.9722
8104
9876-1
0009
0018
+-
.9750
.916 0226
9973-4.012 0021
59825
0009
Peer
9755 0095
9757
9895=0000
0696
9797
9855
0051
9720 8019
9763 +
0752
0110
9774- 0034
Juccard
97339- 0044
0616
0000
9739
0098
9907 0040
99733-2 0150
9729- 0000
9775
0014
5729-1:
0042
GIB
9712 0 0009
.9735
0.0021
0154 9720
9796
0245
9620 0173
.907 + 8024
9724 0017
9774-8.1400
1404
99177 0086
SupCon
0008
9748-
2014
9665-
9724
5409
9783
0000
05692.011
0924
GRACE
0120
3811+
8412=
0224
8950=
0081
2792-4
0119
.8530
8864
8826
0005
.5784:
0104
RGID REP
9881
0007
986661.000
.008
9846--0000
0009
9880-
0007
9934 0005
.9841 +: 0011
9895 1: 0007
9891
80000
5994-1.0000
0008
RGIB-SSL
9845 0005
9810 0005
9765= 0014
9840= 0011
9408
0012
9757
9802 0005
9859:2000
0010
58666: 0007
Stundard
992994-0020
05224--0424
9438
0402
9819
0015
9668
+ 0147
.9622
+: .0154
9882
+ 0007
9880
0007
5986
1006
Droplidgs
0028
.9622
0099
9473-1.0120
9803
0016
9685
.9531
#:012
.9733
0008
9771 8. 0008
99776
0009
NeuralSpurse
952511.000
.96381. 0009
9156-
0007
9712
0022
9691
1 0015
.953311.00
0001
9781 1
97611
978411 0008
PIDNet
97225
.96744-.0151
9480--0995
0096
9725- 0014
5008
+ 0089
.9243-
.0267
9979- 0003
9880--.0000
57783- D008
Co teaching
9620-1-0113
.9526
1.229
0439
9480=
03500
9712
0064
9007
1 0141
99144
00146
9762
1: 0000
9797-5.0167
96338 :: 0154
L-4
Peer loss
9607 90161
.95361
0503
9450
0437
9758
0011
9003.1
0204
9622
10016
9769
1: 0085
.950 0128
9734 0137
Juccard
97944-
9379++9447
9784--0122
0127
9002-
0295
.9638
+ 0140
97V2
+ 0016
9725-
40064
5158- 00225
GIB
.973 0182
9608.1..54
1400
9796
- 0260
9667
1 0200
.9650
1. 0033
9742
1 .0033
.9003
1. .0023
9971 0000
SupCion
9588- 0067
0100
92972
0121
0010
0121
5647
- 0058
95174 0115
9401
+ 0115
9647
- 0164
1957
+ 0144
9553- 0133
GRACE
8899
0100
8865
0226
3315--010
.3355
0015
0036
897244 0004
0125
RGIB
9632 0026
.970011000
0002
9519-0002
9833
1
.98221
1: 0057
9684 0006
9883
1.007
5659
0007
RGIB-SSL
9829- 0023
9911
11:11:00000 00025
9643-20002
9821
= 0019
5707
0021
9668
5: 00300
9957
+ 0010
9981
1t: 0008
5657
0007
Standard
3798-1-0011
9609--0018
9268-+01199
9764- 0034
9502 0096
.9599
+ .0100
9963
+ 0003
.9665 4: 0012
9976-1 0010
Droplidge
97741 0030
063551.0061
9500
.100
95
0034
9579
1 0083
9411
1: 005-1
9556
1 0014
964221.000
0006
9559 1 0012
SeuralSpurse
97741 0020
9596
9435 011
9449
2 0029
9496:1
0073
9400
0087
9561
2012
956212
0011
9465
0016
PTDNet
9767
.6605+00133
0013
9419--0173
9555-
0034
9568
+ 0064
9416---014
-0445
9500
+ 1001
9460--.000
00014
9569 0009
Co teaching
9638.1.0000 0051
956511.023
9126:-0261
0261
9520= 0051
.968
1 0140
.988911
1. .022
9474
1. .068
9658
1044
9561 0169
Peer loss
9699.1 0044
9557
01141
0221
9554
0653
949911
0135
9645
0234
9696
1 0063
9581
11 0200
9650:1 0021
L-6
Jaccard
0002
.95334
00137
9507
0064
9616
0115
0000
.028
1 .000
9925-1.000
00755
94133
0057
GIB
9619:1.0188 00145
960221
0196
0226
9511
0533
955211
0261
.9555
2. 0316
9556
1: 0099
9578:1.00
0052
SupCon
9462-4 10000
-292--0113
9000--0775
9555
-011
20100
.411
+ 0088
9350--0000
.0077
.95
10-1 0122
GRACE
8939-1.0113
3363-1
0220
$259=1.000 0019
9071
0077
3552
0228
.8274
6029
.992
4 0093
.9126
5. 0015
5977-L 0100
RGIB- REP
97741
0031
9680-1.0097
0099
0133
9780
0024
9619.1
0069
.9233..000 0083
9968
1 0008
9863
0010
96772- 0011
RGLB SSL
9731 0023
96411+.007
9295-200122
9763
00334
9923-4
.9451
9845
+ 0008
9943-4
0012
5847- 0011
43

---

Table 40: Full results on Facebook dataset with GAT.
layers
methed
20%
bilateral noise 40%
00%
20%
ínput 40%
60%
20%
label moise
40%
00%6
Standad
.9699 0009
5836-2
0007
9878=#
9861-:
0006
-958-+-000
9872:4
98644 0000
5857: 0008
Dropfilge
9645 0007
9553 0010
9517-1000
0009
9652.2 0007
9636:1
0067
9953 4: 0000
9755 0007
9743:1 0010
9739 0012
SpuraiSpurae
9661 + 0007
9681 = 0000
96382 0006
9765- 0008
96524 0067
9744
0005
9667:4 0010
96574 0006
0656: 0007
PTDNet
9762-1.0007
2.000
9735.
9766
9645:1
9766-1
9759.1.00
9757-1.0000
Co-texching
9723 0006
9685-2
0054
9762:1
0639
963222
0057
9720:1 0001
9741
2: 0046
9701
9789 0230
9792-2 0288
L-2
Peer loss
Jaccard
9762 0032
9971 00:38
9644 0020
.9661 1. 20005
9736.1.00
97252 0076
0078
9556.
9699 0020
0064
9758.1
0720 0104
0180
9780
9708
+ 0103
0012
9692 0194
9720 0100
97.9 0064
9807 0207
9701 0087
9713 0078
GTB
9679:1.
96162
0007
9709
0564
9746:
9738
9799 0094
SupCan
97184:: 0078
9646 0088
96892 0087
9743
8008
9737
0009
9629
9770+
0038
9456 0033
9419 0029
GRACE
8377:1 0009
67161200233
8519110224
83441
8302
0271
8487
1: 0407
8768
1 6457
8370 0416
8794 0319
RGIB-REP
9865 0006
9854 00099
9837 0006
98724
0007
9855: 0007
9855
4: 0008
9671
1 0007
9861 0009
9659 0006
RGIB--SS.
9788
9764
9784
9764
9784
0009 9791
Standard
99574-0015
0850
9820
0014
9855
5627
0019
9773
9873
9874
9874 0005
DropEdge
98391 0009
9701 =
9767: 0024
8000 0836
9772
0022
9706
+ 0000
98574
9845
0000
9644 0007
NesralSpuree
98544 0009
.8112 0019
9768 0027
9899_
9700
0028
9736 0000
9872 0009
9869 0007
5868 0007
PTDNet
9848+4-0039
9830==0008
9745 0076
0012
57877 0039
97524
+ 0007
9672 0005
9868 0010
9872 0007
Co-teachin
9789:1-0019
9977
97172.011
07032_0050
97184
9783.1
0750
9646 0014
Led
Peer loss
Jucard
9344 0029
9757 0079
9630=20105
.963 - 0067
9687 0046
96831.0015 0015
9610=-0066
0162
9616:1
2610
0064
0184
9783
9655 1 0151
0073
9677
9689
0053
0005
9638
9625 0000
9691
9604 0036
0205
GIB
9609 31
9747
0017
9709
6204
9721
0207
9741 0190
9740 0111
97474 0018
0768 0093
0799 0016
SupCon
0156
9464
0254
91342
0309
9824
9600
0500
9482
GRACE
7803 0649
794722
0600
76182
1058
8259
0660
7588
0963
7260
84401
0370
8448
8122 0403
RGIR-REP
9655
0014
9899
0016
9783
0034
0838200019
0019
9806
0013
9765
- 0037
9677 0008
9871
1 0008
0867 0007
RGIB-SSI.
9751 0033
9007
96022
0043
9767 0030
9683
0053
9603
2 0002
9755
+ 0028
9772
0032
9762 0047
Standard
.905 4:025
9668
03321
9677
0314
9804
0018
9738+
0044
9710 0036
98544
0015
9860
0007
5867 0012
DropEdge
9643 0641
0638
9500
0274
0942
9606
9735
9736
9736 0009
SeuraSSpuree
97464 0077
9667=
9560-2-00
0122
9740=
9668
0056
9652 0043
9735
0017
9742+
9733- 0030
PIDNEt
9909-1-0037
96799
9582-1.220 0296
9754-.0012
9638
0045
9645 +: 0009
97.38 + 0041
9741
0002
9256 0012
Co-teaching
9745- 00:34
9686
9625 0408
6084 97472
9833 0108
9738
+ 0015
9622 0724
9676
0077
9731 = 0215
L=6
Peer loss
9000+-0014
.133
95W92.089
9716
5776-
9766-
9671
+
96055
9768
Jaccard
9789-4-0084
9651
0337
9673-.0007
0010
9731
9602 0064
9607
0004
0251=.0030
GIB
98392 0037
.977
0420
0584210174 0374
9868 0200
9760
0135
9791
- 0347
9648 0054
9640
0946
9774 0029
SupCon
9428++-0194 0194
87794.044 6347
0171
9459
0378
9312 0544
3465
+ 0173
9484
0194
5023 0833
GRACE
72344 0667
60752
1089
74152
1242
1345
62161
70221
RGIR.REP
9779 0031
9711
9431:
8392
9753
9666
0087
96142
9847
98482.0013
9856- 0009
RGIB-SSLL
366334 0107
0005
9254 0206
96762.0055
0057
9502
0087
9584 4 0067
9751 4 0065
9675 0052
97133 0105
Table 41: Full results on Facebook dataset with SAGE.
SAGE
layers
method
bilateral nouse 40%
inpur noise 40%
label nouse
20%
60%
20%
60%
20%
40%
60%
Standard
[9888.1
1008
.92271.0008
9788:1 0012
9881 12 2000
9962:1
0007
9862:1
0008
996771
0006
.98001.000 0006
962551.000 0014
DropEddee
9745-1 0008
.90012.00)
9765 0011
9766
0907
9749:1
0011
9750
0008
9761 1: 0008
9739.2 0010
97191 0011
NearalSparse
97533
0008
.61994.001
9683
9675
.9659
9011
.967
0007
9640
9627-4.000
PTDNet
9755
0007
.92331..000
9779-10011 0012
97 0007
9761 0008
9754
8007
9766 1: 0007
9743-1.00
0007
9726:1.000
Co-teaching
9759-1 0101
9722 0004
9728 0100
9702
0050
9747:11 0023
9739 8081
9764 0125
9768:1 0205
9700:1 0282
L-2
Peer loss
Jucard
97968 0082
9747-1 0018
.9720 0072
0009
9721
9790
0103
0005 9787
9729
99811
9760011
9952
.9787 1 0015
1 0102
9733
.921 1 0019
970611.005
994228.015
.97931.000 0031
0098
GaB
.9798 .089
97800+-0000
5765-
0020
9724-.013
01399
571674 0287
.977 + 8006
97500 2011
99115+.0007
5772- 0050
SupCun
9482 1. 0166
9206
9141: 0139
9793
0043
9782 0072
9796
1: 0011
9278
1 0018
9252 00774
91.661 0011
GRACE
911311-011
890711.029
9077 0669
9012
0113
8885.1
0177
9031
1.0104
9053
1 0126
90501.00173
9067 0137
RGIB
9800-+ 0009
2002
9878--0000
0009
9903-4
0007
.9833
0011
9875
0000
9860--0000
5855-- 0011
RGIB SSL
.81001 0007
18291.00000
9811 11 0000
9838
- 0007
.921
1: 0008
.9823 .004
0008 9641
.983
1.007
9641.1 0010
Stanhanl
9824-1.000
0015
.783-1.0025
9698
00000
9869
0015
98151
0024
9815
1005
9844
0015
9817
0007
9609
0009
Drogfidge
961811 0026
9764-1 0014
9722
0022
9625
0015
9601
0043
9600
0003
9649 0008
9627
0017
961511
0012
NearalSparse
9811-1 0031
.97661
0021
9710:1
9636
0023
9600.1
0026
9615
0016
9518
1 0008
9628
1. 0014
961331 0025
PTDNet
9823110002
99777
9738 0500
9623-
0041
9409:1
0051
9607
2002
9652
1 0007
9632
0009
961111 0024
Costeaching
960011 0009
9653-2 8075
9659 0110
9600 0098
96141
0090
9626
0017
9654 0253
9706 0156
9678:1 0763
Led
Peer loss
lucard
9668 1.0051.0105
962111-001 0019
.91551
9676.1.0061
9716: 0630
9772 0068
9681 0079
9662 0145
0094 967511
961111
0014
9762
9600
0012
0056
9671
9725
1 0201
1 0107
9609
67261
1. 0088
0063
9660:1 0032
9705
GIB
0074110000
9758-1.020
9733
9508=
9616:1
96792
0171
9765
010
9714
0018
0107
SupCun
916911.0144
109911.019
9001
9326 0208
9188 0100
9185
9295
1. 0014
8328
1, 0158
9025-1.01.8 0148
GRACE
8683 0292
8855 0696
854$=
8585 0424
8731
11 0261
8413
0364
R108
1 0213
8527
0375
8376.1
0330
RGIB REP
9922 0016
9984
0000
9750-
9843 0014
9911
.900
.000
4857
1 0004
9933
0005
58330- 0019
RGIB
981711.0016
.99711.0014
.997
2013
9920
0011
9626
0010
Standand
9751
9603 0365
9616:
9715
9677:1
9517
0238
9761 + 0136
9788
0045
9749
0151
Doog
9657:t 0009
96762 0058
9578-0044 0149
9734-
0050
6616:1 21
9665 - 0013
9631
+ 0013
9606
0046
0651
0145
NeariSparse
9640- 0075
9577+-0222
9672- 0061
9647
0048
5w994 0156
96733
.006
9632
0017
9618
0057
96152
0051
FTDNee
9668-1 0029
9633-1
0142
96172.0111
9774
0828
9693:1
0053
9643 0055
96291
1 0015
4736
0010
9708 0037
Co-teaching
9644-1-0116
96991-0463
9730
977811
9583
9742
9744
0106
0770:2 028
L=6
Peer loss
Incard
9644.1.014 01411
9750:1 0119
.602211.244
9666-1.
0453
9681--0085
9686
0988
9666 0261
0167
961 0096
9718.1 0053
.524 0258
9683 0583
9655
9614
0170
0220
9784
8769
0110
0028
9650:1 0256
9781 0178
GBB
9739- 0197
9656
0564
97492
9743= 0351
971 0034
9615
0391
960T 0153
0779:2 00)
07412 0222
SupCun
9007-100112
88591.0115
8730
0140
8997=
0142
8910:1
0104
90061.
0215
91461.0114
91661.0143
9185
GRACE
8207-1 0165
7725
0640
7901 1065
8018-
8234-1
0547
7960
0915
8061
1 0367
8767:2
0306
81031 0351
RGIR-REP
9770.1 0040
9557
9674_ 0149
9749-
0046
9683 0072
9554
0093
9K38 0003
9829
0015
3614 0029
RGIB-SSL
9773 0032
.97201.0031
9678:.0030
47812.0025
9739:1 0023
969211.0000 0050
9774 0016
9786.2.000
0012
9786-1 0017
44

---

Table 42: Full results on Chameleon dataset with GCN.
QCN
layer
method
20%
bilateral moise 40%
00%6
20%
input 40%
60%
20%
label
40%
60%
Stondard
197334 0025
9696
9657=_0029
77744-00117
976244
9754+ 40022
9775
0018
9769=20018
9755:
0636
Drogfidge
0030
9640
0025
9579 0029
9646
0026
9600 0024
9661 1: 0002
9656-1
0018
9664 0029
96511.001 0031
SparaiSparse
9941 0032
96974 0130
9633= 0630
9682= 0026
9646: 0023
9627
0022
9679
001
9669: 0022
0651 0679
PTDNet
9748.1
0033
0031
9679
9619:1
.627
0005
9690
9674
96-811.000
Coleaching
967911 0059
96412.
0086
9676:2 0088
9767 0096
971 0084
97094:
0106
9700
0218
9729 0015
9745:: 0272
Peer loss
9647 0057
96622 0006
9698220099 0059
9781 0043
97844: 0014
9759 8005
9762 0159
9695-2 0000
9716:1 0036
L-2
Juccard
968111 0100
.9657
9660:1
9611
0190
.999:1 0005
.974 0028
9711
1 0010
9754
2 009]
9750:1 0086
GIB
9649-1 0052
9634-
9209
9:1
9766
9674
9660
9778
SupCon
9625: 0024
9677+
00744
0632
9757=-0011
0018
971 0023
9697
+ 0014
9749
0013
9755 0077
0750: 0630
GRACE
9145.1.0105
8978:1.0100
8915 0033
9881
9081 0121
9005.1 0062
1933
1: 0012
9171
1 0012
9181 0054
9196.1 0058
RGIB-REF REP--REP
975441 0026
9698
0026
9652
0797 0025
9757 0016
9744 00138
9604 0023
0798
0016
97834 0028
RGIB-55LL
9719
9658
0025
9595
9709
06644
9786
9797 0022
Staadard
1636-4-0033
94%-
9274
0018
94334
0261
9308 0271
9686
0000
9580
0001
9962-4
0035
DropFidg
9568- 0044
95482
0030
0679
9433
0055
9453
0058
9580 0000
9579
0033
0578:1
0633
NearalSpurae
9599220000 0026
9297
9402 0657
9609
0027
9540
0178
9334
0060
9583
0017
0583
0000
9571
0091
PTDNet
-0007
95144+00856
9424 0997
5010
1M224
94574
0194
9300
0050
9585
+ 0022
9576
00150
5068: 0635
Co-traching
0915
95161_0190
9483 0874
0524
9446:1
9447
0756
9642
9650
0058
0f11:: 0950
Le4
Peer
luccrd
9543 0090
950011 0061
9538.1
0223
9267=-032
0144:
9507 0056
-558--0633
94361 0286
94874
0230
9412
9364 0355
9631
9601
0166
1: 0001
9501
9659 - 0020
0055
0569: 0227
9557.1 0070
GBB
95341 0006
9561
+: 0292
9131 0367
9605
0060
9531 0231
9416 0513
9651
+ 0061
05872 0100
0489 0629
SupCin
0:46
8143
9536:
9564
0580
0477 0038
GRACE
8978 01445
89871
8050
8949
0930
8994
0141
9007
0045
8964
0043
9053 + 0110
9074
0136
9076: 0113
RGIR-RFP
9723 0035
9621
0016
9519 0652
9708
0027
960 0052
9480 0055
9785 0001
0797 0017
0785 0012
RGIB-SSL.
9665 0022
95924
9500=
0955
9658=
0022
9570
0047
94%
6009
9730
0017
0752
0050
9744
0632
Standard
9962-+-0042
9511
9286-
0067
9656 0045
9450
0177
9276
H: 0229
9752 + 0027
9766
+ 0033
5145
0040
Dro
0002
9484
8053
9331=
0481
9331
0002
0599
0027
0547
0924
NearalSparse
9644- 0035
9550+
0013
9411=
9538=-000
0035
9468 0038
9387
0053
9525
+ 0033
0541
0051
9535- 0637
PTDNer
5005
9585-+.0005
00033
5284--0000
0088
9637
9637 0024
9553- 0001
.9391
H: 00344
9933
+ 0007
9559
H 0012
9543-1 0632
Co-teaching
9664 0058
956222
0130
9319 0110
9505=
0121
9534:
0222
9271
- 0308
9679 0198
USST=
0112
05574 0782
L-6
Peer loss
0128-4-0120
9551
9308=
9552=
9441
9958
0528
9575 0164
9667
0250
95912
0236
Jaccard
9734+-0003
9549++905
9292=-0006
9637=-0035
0035
9495--022
0213
.457
0350
2500+
9550
0128
94B3- 0066
GIB
9800:2 0201
9621
9329
9544 0212
9547. 0395
9456 0231
9619 0072
0112 0905
0536:2
0124
SupCon
00574400054
9452+.0073
9352= 0846
9534
8968
94544
0068
9575
0 0023
9618 0086
9627=
0055
9557= 0067
GRACE
9004110008
0030
$0831
8052
9154
9166
0151 0882
RGIB-REP
9666- 0028
9580
9467=
0067
9662-
9406
0041
9453 8049
9740
0033
9776
0021
0753: 0611
RGGBB-SSL
99633 0032
9416 0036
9177 0082
9627 - 0037
94814
0057
9276 0003
9712 0017
9142 0022
9/12-22 0052
Table 43: Full results on Chameleon dataset with GAT.
GAT
layers
methed
20%6
bilatera nosse 40%
20%
injut nolse 40%
60%
201
lihel nosse
40%
0006
Standard
.9725 1: 0027
.9650
00188
962551.000
0018
9967
.0025
9944-L
00000
.9759
6: 0018
.946 6 0023
99433-..001
2011
99111 0041
DropEidge
9716.1
9630
9563:1.000
0023
9747 0026
9717:1
.9708
.97661
969811.0035
NeuraiSparse
.53334-000 0029
9667
90025
9615-4.0223
0028
27622400028
59488---00
0028
.944
5015
9702
+ 0002
9742-4.008
00.33
91044 0042
PTDNet
9926 0028
.967 -. 0024
9620-1
0027
9768
0023
9735
0021
.739 6: 0022
9670
0021
9631 Ht: 0023
9611
0058
Co-traching
96174
9622
2047
9622-2.0000
0051
9624-
0084
5645-t
9664
9610
5638
L=2
Juccard
loss
.9675
.962 1: 0063
.9655 1: 0062
.9666
0100
5683-4.000
9697110073
9602-
96521..013
01.0
8046
9621---011
964991.000
0112
0051
9611
.6166
1: 0123
0078
950
96.3
3 0006
0045
96.26--000
961221.0000 2000
-0074
9620- 0089
9603
1. 0134
GBB
59228 + 0018
9639
0115
9625
0208
9681
- 0091
5643-t
0257
9601
+: 0226
9644
+ 0119
9688 0099
9631 0115
SupCon
30009 1-
.9611
+ 0195
9607
0032
97524
0011
5021-4
0026
.701
0000
9689
0057
.908
78829
14994
7396.1.000 0693
7992
0371
1960.1
0071
7913
1: 0118
7905
0261
.7965
0322
789111.00 00.00
RGIB-REP
553364 0091
9668
- 9002
9629-1
0026
5769--0011
5752
0016
2015
97006
0001
9755-t.002
S952 0041
RGIB SSL
.835 4-037
.993
- 0033
9636-9.0041
0002 9640
9587
0048
.9545
6: 0.00
9009
0005
.9728
2: 0022
99199
1046
Standard
.921 9721-100035
9862
L: 0023
000511.0033
9741
1 0022
9686
0030
9671
1: 0027
9740
0007
9738.1
0027
9712
0047
DropEddge
9666-1 0031
.9572
0034
946911 0043
9659
0028
9598
0024
9896
1: 0040
9749
1. 0024
9747 0016
9718
0031
NeuralSpurse
9687-4.0027
.960
9536-1.000
0033
9699
9600
0036
9500
+- 0017
97748
+ 0002
.756-
0002
9127 0035
PTDNet
9907-1.0041 00411
16401..0031
9510-1.003
0034
9708 0027
9602
0025
9548
1: 0055
9754 0025
9761
1. 0021
9708
1. 0011
Co-teaching
9726-1 0096
9648: 0000
9639
0088
9551
1 0057
9903
0027
9556-
1: 0003
9691
1 0009
9664 7565
9622 0242
L-4
Peer loss
Juccard
.945 1-
9776.1.0089
.892
.9114
0098
0008
9663110012 20
9668
0103
9729 0024
9637
0041
9677
0211
0088
1: 0008
2012
.9499
95871 0078
1- .0033
951911.000
9531
1. .0996
9973
9587 0068
95731.0137
GiB
3000-4-0144
0144
9646 00441
3641+-0089)
96.99.01197
.1197
5644
0189
9686+-000
96666 6044
9609
+ 0064
SupCon
96141.0033
.537
0000
20125
9675
0002
9631
0062
9555
1: 0008
9594
11 0131
011233.0136
5286
11 0102
GRACE
5429 0465
6085
11332
6273-1.1475 1475
6178 0940
5839
1022
6513
1. 1344
5531
1 0139
6195 1058
6333 0818
RGIB
9704-4-0035
9645
5022
9591-.0030 0033
9713-
5654-4
9572
0047
9741
0040
.745
17745-00022
9718 0039
RGIB SSL
.900 1 0034
.942 10
3389-1.0059
9526
0037
9451
1.511.000 0045
.3500
1: .083
9619
1 0007
.9615
1: 0042
0038 9613
Standanl
965991 0029
.9533
00036
9482-1.000 0054
9644
0033
9543-L
0075
9474
0074
9722
0013
9688
0055
9698 0065
Dige
9461 0082
.9295
- 0063
9067:1
0218
9469
0059
9703-1
0087
1183.1000 0063
9698
1 0006
9689
0046
9604 0037
NeuraSSpurse
.963
9003-
.9391 1 00007
9185
1. 0157
9473
0149
9338
0013
.9500
1 0007
9688
0007
9705
1. 0026
9696
1 0063
PTDNet
9508-1.0075
.959
0222
9288
0153
9535 0002
94241
1 0043
9721
1 0013
9717
0036
96971
Co-teaching
960911 0033
.9677 0129
9501 0141
9709 00
9569
05
9487
0155
9673 0785
9
0260
9611 0176
La6
Peer loss
Juocard
9973611.0116
9680-1 0057
.9607 - 0055
.9570
0006
0101
98151.0185
9659 0000
9555
9563
0155
0145
9463
9469
1 0066
1. 0100
9001
9750
1 0126
0124
9701
9732 0091
0196
9772 0120
9682 0078
GIB
9657:1.0198
9666
9612220165
97762_0272
9610
96741
9587
9514
9585 0138
SupCun
918111.0185
.31011.017
901511.019
91551
9450-1.0157
91941.0229
9111
1 6006
937011.0174
92/1 1. 0144
GRACE
5681 1:: 11-85
5107 - 0257
5147110061
5599
5046:1
0124
5167
- 8502
5850 0627
5884 0795
5518 0404
RGIB REP
9612-1-0000
9410+
9258-4.0166
9531#
0054
9417-1-000
0058
.933
9716
+ 0049
9747-
9707 0056
RGID
9367-1.0058
9233
9199.1.003
0015
9226.1 2056
.2005
0,000
1
.552
9557-1.0083
45

---

Table 44: Full results on Chameleon dataset with SAGE.
SAGE
layers
method
20%
bilateral nosse
60%
70%
input noise
40%
60%
20%
lahel noise
Standand
96382-0006
9605-
8813
25362-0028
9754- 0019
.9744t:
9762
0018
9700
0019
9670
0016
0607:2
0049
DrogEidee
9004+ 0025
96144-0003
9556=-0022
9677=-002
9669-t
99673 0033
9630 0001
9678-20005
0033
9625-
0047
NeuralSparse
.9099
0031
99844
00029
9650- 0024
9751- 0025
9751
57335-
00022
.9745 + 80021
970 + 0014
9661 +0005
9906--
0051
PTDNer
96744 0035
9994-
0025
9538=
9762 0022
9637 0033
9651 + 0029
9600
+ 0016
96582
0015
05984
0042
Costeaching
9581 1040
95599
9543-
9453
9429
9585
9595
00199
9525- 0809
L=2
Peer loss
9638-4 0096
-95814
0098
9612+00021
9541 0091
9561
0075
9557
0104
9625
0146
9506
0155
9587-4
Jaccard
9677. 0074
0012
9689 0056
9503 0007
9540 0190
9517 0015
9902 0010
0522
0027
0545: 0118
GiB
9K60 0006
98993-
0214
9530= 0021
9671
0007
00000000000
9639
0275
9537
+ 0055
0525
0044
9586 0068
SupCon
9636-1
9506
0585:
95014
0004
0530
0037
GRACE
8215-: 0332
8327+-0178
8372=-024
8369-
0146
8251
+ 0441
8295
8610 0055
8585-
0114
5641 0137
RGIB- REP
9099-
0028
00055
5577==0055
9765- 0022
5741 0018
.970
0014
9739
0017
9715
0025
96700 0637
RGIB-SSI.
97401 0030
97001
0027
9686
0025
9752 0020
97374
0031
9741
2 0035
978P
0016
9791
- 0028
0781
0029
Standard
96455
9550+
0015
9505=
0010
9721
0017
972
0021
9721
+ 0016
9678
+ 0006
9641
+ 0028
0005
0051
DrogEide
96444400041
9985
0M12
9516-
0024
96524
0021
9663
00150
9614-
1038
NearalSparse
9649-..0022
997664.0000
0035
9495-40041
9619
0032
9002
0022
.9620 0001
.675
0017
96424.000
0031
9603
00:16
PTDNer
0036
99544
00154
56062-005
9624
0027
9622
0015
.650 00035
9687
+ 0004
96-6
0025
0596:
0055
Co toachcch
972224-0050
9665
0040
9582=
5083
0083
9K23-
0101
9611
0007
9617
+ 0255
9638
00070
5637- 0066
L=4
Poer
9542-1 0099
966331
0044
9196- 9600
-010-
0110
9667-1
.9601
0000
9657
4 0005
9679
0006
9661
0180
Jaccard
9439- 00776
9435-+
8147
2653--011
0115
9683+
0218
.697
0157
9606 0043
0517 00448
9688- 0688
GBB
9557+-0077
9038-0011
9646
6 0214
961 0200
99559
8005
9576 80026
9464
0080
5000- 0065
SupCon
9494 01:30
9351
0023
9383=
0960
0592=
0122
95384
0122
9512 0125
9478
+ 0009
0501 = 0126
04492 0099
GRACE
8040 0364
85424
8415
8266-
0345
8202=
8642
0533
0306
7792-
0441
RGIB KEP
9666
95824
-520=-092
0018
97194-000
0028
.722
9000
0001
9696
0023
9671
RGIB-SSI.
9725 0027
968722 0022
9655 0049
0021 9740
9707
0040
.721
+ 0044
0023 9748
9767
0051
97554:
0040
Standanl
960661.0000 0046
9541
9470
0035
9679
- 0031
9658
0001
.640
0009
9673 1: 0026
9641
1:. 0038
9625 0012
Drogkidge
0832-400028
9540++-0022
00124
9487=.0043
9592
0028
9582+
0035
9550
0299
9500
+ 0014
9561
H: 01150
99537- 0024
NeuralSparse
9525 1 0043
.933311.00 1.365
0033
0036
9597
- 0027
9569 1 0037
.550
0057
9589
1 0004
.952
L. 0025
99
15 0034
PTDNet
95364t
0042
9557++.001
5706= 00ES
9581+
.459
+ 00774
9580 0004
9559
00021
9429-- 0045
Co teaching
.503
0000
9906-+--0133
1929
9492-
0031
9457-4-000 0009
.900
+: 0015
9445
+ 0009
9465 H. 0055
9497-1.011 0015
I-6
Peer loss
9553 1. 00641
10009
9230- 9116
5697
- 0064
9656 1 0110
9652
1: 0066
9521
1: 0210
9595
0195
9589 11 0017
9416-4-0113
9555-+-0116
9535--0111
9610-
9463
0057
9421 + 00052
5443+
GIB
9609-1.0133
99579+.9000
9568--0070
9691-
9098-4-000
.592
+: 0279
9433
0065
9366-..007
0077
3347---0115
SupCon
3310:1.0107
.134.1.0087
9178:10101
9291
0114
92M11 0271
9181
1: 023
9342 0112
9375
9013:1 0087
GRACE
(479:
.7756+
7718
0880
7275
- 0909
79144
0755
810P
+ 8K55
7754
0258
2629 0508
7686- D437
RGIB REP
96336 0022
9555-4.0002
9455-
0038
9703
0022
9641
+
9607
50000
9683
+ 0013
9681
099118.0000
9664- 0042
RGIB-SSL.
9065 0030
9601 0045
9563- 0636
9662
0052
9664:
0040
9610
0002
9009
+ 0002
9716
0033
9008 0031
Table 45: Full results on Squirrll dataset with GCN.
CCN
layers
methud
201
bilateral noose
60%
20%
input noise 40%
60%
20%
label nouse
40%
Standard
.9021 0011
9519- BM22
9444 00124
9610= 0012
5490
0091
.9001 +: 6036
997444
0013
9731 4:
5712-4-.00 1011
Droplidge
9480-+-000
9270==0022
0022
0011
9426-1
.9399-.00
9622
0015
961244.000
0008
9604---000 0014
NearalSparse
961111.000
0019
9528.1.000
0027
9409
0023
9621
0017
9495
1: 0016
.9644 1: 0002
9741 1: 5013
9724-1.0015
9722-1.0010 1010
PTDNet
0602-400018
9813--
0016
9424-
0024
9619
0016
9488 0035
9303 +: 0002
9744 0011
9725-2000
0015
5718: 0009
Co Co-teaching
.0076
.9535
0044
9465
0059
9599
9525-4-001
.954
9629
+ 0047
9666
0 0304
-8631-11000
Peer
9610=t 0015
96004
0057
9451=0099
0097
9636-
0107
9491 0125
9456
8091
9694
9628
0154
9676-1 01:49
Juccard
961334
0022
0558---000666
9609
0014
9622- 0159
9594 0111
9337 0119
9529 4: 0017
6 0088 9805
9401 0010
GIB
9522 0- 00955
94778
0094
9489
0159
9494
0260
9461-4-00 0091
9432
0058
9457
+ 0099
9462
0017
.492-1 0060
SupCon
9494-1 0013
9439-
8009
9458
9421=
9549::
9489
9444 0011
9423
0015
9318 0009
GRACE
9292-4 0001
93164
0010
5290=-000
0012
9317-4
.927
8016
9054 0018
9058
0015
5052-1 0018
RGID REP
9614-1 0019
99544
00026
9471--0035
0035
9611-
0001
.912-4 0032
.9392
0043
.761 9: 0015
19753-.000
9945-- 0015
RGIB-SSL
19571
9476 0029
5402 (1200
9534 0016
9486
0027
.953
(0031
9752 0023
0749:2
0019
9733 0023
Stundard
99332-
99005- 0013
9286
0025
9416
0042
9398
+ 0011
9411
00400
9720 0016
9920 0016
5P7
0021
Droplidgs
9439-1.006 0063
930244
9265
0015
9426
0042
9336
1
.9358 6: .0033
.908
1 0008
0013
9998
0016
NeuralSparse
949111
.909 0025
0033
9169
9400
0028
.917
0062
9633
962611.0016
9625 0016
PIDNet
9485 00056
13326---0044
8800- 8048
9469-
0048
5400-- 00000
.9579+
50026
.6333 + 0018
9625-+00012
59226 0012
Co teaching
9461 100L-1.0000
.982-1.008
937.5 0051
9462
0101
9415
1: 0092
.306
1: 0033
9675 9: 0292
9641
11015
96555 : 0264
L-4
Peer loss
9157-1-0117
.9345-1.0117
9268 000
9562
0033
.9866.1
0058
.3356..
0013
9636
1 2012
9694
1: 0033
9696
0006
Jaccard
9443-4 0047
.932++.9077
9288-
0177
.9200
0055
9529 + 0022
95124.000
0008
9501- 0120
GIB
9472-1.0127 .0127
.9329
9002
0202
0200
9390
0045
.9106 1 0196
.9397
1. 0049
9641
1 .005
9628
1. .002
9601
1. 0012
SupCion
9473 0010
9348+
5501
5372
0010
9343-
0057
95815
+ 0003
95 16 0042
0595
0011
9511 0040
GRACE
9394
0008
9330+
0007
9563-
0816
9363
2017
0019
9174
0014
99166
0013
RGIB
9509 0016
.955
1071
9434:0005
0059
9895
9422
1 0059
.9905
1: 0150
9735
0013
9733
0015
9737-1.00 0009
RGIB-SSL
9499-2 0040
9426 2019
9425= 0021
9479
- 0051
9429:4
0017
.4229
.0024
.727
+ 0015
.929
22 0019
57126-t
0011
Standard
9484-1-0049
38299--0008
9408--0029
9489
0057
9408
+ 0021
.9366 + .0022
.683 + 0008
9675-
0027
9656-1
0034
Droplidge
0160:1.0063
.9755
0002
0219100017
9475
2 0057
9409
1 0030
9365
1: .002
9669
1: 0002
9652
H: 0051
9638
0038
SeuralSpurse
9308-1-0049
9419-1.0004
9391 0029
9532
0060
2:1
0044
9388
1: 0016
9681
1 0000
9647
0018
9671 0019
PIDNet
9535 0047
.9451 0050
0233
9514--0000
0041
9422-4
0068
.9380
-0020
.690
00023
9667 0003
9649-4
0025
Co teaching
9180-1.0144 01.44
9954-11.000
9463-10076
9538-
0051
9411.1
0018
1. 2012
9671 1: .0179
9694
1. 0083
9651-1.01 0185
L-6
Peer loss
Jaccard
9539-1 0051
9519-4-0133 01.38
99459
9420
0017
0100
-400--00115
9405= (960
9445=
9885
- 0000
0133
.429
9446-1
1-
0012
9462
.9628
1- .013
0081
9697
9684 0182
4 .0014
.9677
963911
0002
0188
96734
9624 0189
0077
GIB
9147-1.010
99159
8026
0110110106
9131= 0269
9413.1
0001
.93661
1: 0238
9633
1: 2011
961911.014
9612-1 0064
SupCon
9590-1 0022
9507++00019
9463-10088
0928
9:80
9014
9519-
0018
.9644- 5002
.9038
+: 0005
95544
0044
5419-4 0012
GRACE
9399-1..00 0007
.9722
5006
9248 0017
9397
0012
9330
0009
9349
6 2017
9212
+ 0017
9218
0013
9211
0014
RGIB- REP
9556
0028
.999211.000
0060
9463 0063
9570
0018
9496
1 0060
9420 0056
9577
1 0029
.6761
0051
9662:1 0018
RGLB SSL
9462-4 0029
0418
0032 9509
94744-
0017
.449++ 0000
9642
0012
9659
0012
9647- 0010
46

---

Table 46: Full results on Squirrel dataset with GAT.
layers
method
20%
bilateral moise 40%
0076
20%
inpuc 40%
60%
20%
label moise
40%
00%
Stondard
19000: 0007
9635+.80017
9598=10025
9702-=00000
0008
9000=
96599-2000 0014
9719:
0018
9701
9686 0012
Drogfidge
9619:1.0014
95672 0000
9505 0022
9627= 0008
9503:1 0014
9951 1: 0012
9684 0015
966222
0026
9661 0018
SearaiSparse
9656-2 0009
9607
2012
9533= 0019
9662=
0011
9639:
0010
9584
0017
9709
0012
9697
0014
0685:2 0009
PTDNet
9655
.9022 0012
9516--0024
9668:
9630:1
9600
0001
96101.000
96981.00
2012
9689110016
Coleaching
9566:1 0034
9560 8066
9826:20026
0045 9505
0309:1 0092
95091: 8046
9591 4 0293
9880
0174
9360:: 0089
L-2
Peer loss
Juccard
.77221 0097
971551 0098
9726220009
965111.0053
9648:1009
96822 0106
9807 0202
6.640
9768.1
971 0064
0041
9704 0017
.6455.0077
9681 4 0121
9637 0081
96352 0069
9668:
2 0101
9627 0002
0681 0116
GIB
9737-1-0019
0069
0700
9965:100000
9638
9660
9611
9679 0061
SupCon
9471 0019
9386 0003
9353= 0026
0539
9463 0026
9411 + 0014
9629 0002
9631 0019
0636 0021
GRACE
7300:1 1193
7583-1.
1135
70152.1380
1383
7:94
1101
6903
1807
7052 1659
8109 1: 0382
7760 1064
7608 0620
RGIB-REF RIB-REP
9675. 0014
9631
2012
9587=
9692
0007
9660:
0013
9653 0016
9735 0012
9719
0013
9715 0010
RGIB-55L.
94212
0017
9418=
9304
9362
9776
9705
0057
9696 0024
Staadard
19581 0046
9436
9335
9592
0047
9455:t
9415
0061
9682
0080
9690
0028
9686
0021
DropFigg
2501-:0041
9340 0046
9334- 0059
9476
0854
9354::
0067
9297 0019
9539
0047
9515
0054
9510: 0037
NearalSparae
9524:20004
9383--0004
9320 0004
9401
9401
0071
9337
0045
9584
6044
9559
0001
9555
0052
0556- 0022
PTDNet
9416++
9510
9302 235
9447
1651
9393-
0053
95015
- 0131
95544
+ 0029
US5T=
00125
9552 0042
Co-traching
0104
9024
9009
0821=
0074
9345:
9371
0001
95764
0134
0592
0064
9581 0087
Le4
Peer
luccrd
9530+ 0071
956111.0133 01.33
9506+.0145
9196: 2110
0000
9578
-500=-0063
0185
9411 0086
9583.1 0256
9445
9558 0000
9548
9573 0074
0128
0500
0572.1
0088
00K3
95934 0134
9554 0089
GBB
93514 0080
9271
0001
9098
0779
015
9504: 0740
9550
0748
9544 0003
0549
0073
9534 0060
SupCio
0044
9922
1305
9340
95144
95744
9557
0065
9556 0049
GRACE
5710:1 1110
5658
1143
6203 1324
5694
1202
5771 0945
6021 +: 1203
5391
+ 0694
5097.2001
0139
5029
0072
RGIR-REP
9541
1:411:000 0059
94324
0059
9116
0963
9526
0637
9401 0072
9389 0015
9681
+ 0078
9672
0079
0660 0048
RGIB-SSL.
93474
0038
00151
9260=
0652
0321
M222
93274
0020
9270 0008
9592
0031
9651
= 0014
9641 0021
Standard
9907-+-0050
98099.0164
9254
0089
9487
0065
9419:
0041
.225
+ 0073
9585
+ 0097
9520 0070
9507
0162
Drog
0047
9279
9181= 0078
2368=
9263:
9175
0127
9412 0002
9431
0076
9444 0090
NearelSparse
9399- 0088
9279+-0166
8167
9224=-0551
9423--000
0065
93424 0023
9213 0115
9468
+ 0118
9474
0102
9504 0051
PTDNer
9433-2.0105
9348
9213=
9460 0048
9:40-1 0002
.9273 +: 0094
.452 +: 0088
9541
#.0M2
9453 0077
Co-teaching
0059
0329+ 0231
9229
0392= 0093
9352: 0998
9261 + 0100
9406 0383
0445
0272
0490 0784
L-6
Peer loss
9330++-0166
9916
9349=00477
9271==011
9170
.057
9230 0221
0113 9341
9187 0925
Juccard
9577+-0012
9374
9297=00128
9869=-0111
9406--0011
9339+:0108
9390=.0151
9214#.0225
GIB
92041
9475
0127
9484 0211
01
9458 0307
9421 1 0133
9484
0101
0486-2 0206
SupCon
0209+-0000
9174
9107= 0052
9278 0957
92234 0054
9210 0008
9140 0072
0175# 0088
9182 0081
GRACE
50001.0009
54271_0961
60042.1210 1210
60131
5342
5064 0145
5017
5074 0190
RGIB-REP
9404 0067
9248
0218=0051
9437=
0084
93994:
9238 8063
9473
9486
0073
9397 0045
RGIB--SSL
9286 0033
924112.005
9165= 0055
9276 0023
9245
0015
9257 2: 8005
9903 2 0043
0586 0027
3499
:: 0067
Table 47: Full results on Squirrel dataset with SAGE.
SAGE
layers
method
bilateral nosse 40%
inpur noise 40%
label nouse
20%
60%
20%
60%
20%
40%
60%
Standard
9600:1.0015
162661.0015
957001..01 0016
9737-1.00
00100
9736:1
0010
.921 1: 0001
99992 1: 0015
9643-2.0012
960661.0017 0017
DropEddee
9657-1.001 0011
990021 0015
9535 0020
9622: 0010
9625:1
0069
96091:
0000
9593 1: 0009
953012 0009
9593:1 0007
NearalSparse
9572-4.00 0008
.91774.0012
9562-40011
9534-0000
9524-4
.9521
9016
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2011
9534
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9559-1003
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9630
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9501.1 0007
952411.000 0010
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9529
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Co-teaching
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9403 0650
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Peer loss
lucard
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2019
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Incard
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47


Reviews:
[{"review_id": "dqg1rVUOVJ", "reviewer": "Reviewer_Fboz", "summary": "This paper tackles the challenge of link prediction on graphs in the presence of edge noise, a topic that has seen little exploration despite the advancements in graph neural networks (GNNs). Through an empirical study, the authors reveal that edge noise can adversely affect both input topology and target labels, leading to performance degradation and representation collapse. In response, the paper introduces an information-theory-guided principle named Robust Graph Information Bottleneck (RGIB), which aims to extract reliable supervision signals and prevent representation collapse. RGIB achieves this by decoupling and balancing the mutual dependencies among graph topology, target labels, and representation, which creates new learning objectives for building robust representations in the face of bilateral noise. The authors present two specific implementations of RGIB, namely RGIB-SSL (which employs self-supervised learning) and RGIB-REP (which uses data reparametrization), for implicit and explicit data denoising respectively. The effectiveness of the proposed RGIB methods is validated through extensive experiments on six datasets and three GNNs under various noisy conditions.", "questions": "Most of my concerns are addressed in the appendix. I have no further questions or suggestions for this work.", "limitations": " Board impact is discussed in the appendix.  ", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The paper is well-written with clear motivation and structure. Overall interesting problem; good mathematical exposition; solid theoretical results and analysis. This paper has a very comprehensive experimental analysis including the experiments in the appendix. ", "weaknesses": "This paper is overall good. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer Fboz for the valuable feedback and the positive support of our work. \n\nAny further comments and discussions are welcomed!"}, {"review_id": "XzgmtcKC3M", "reviewer": "Reviewer_4Q8x", "summary": "The authors extend the Graph Information Bottleneck (GIB) to \"bilateral\" structural noise and label noise. That is, both the adjacency matrix and the labels are being randomly perturbed. The authors observe that the bilateral noise leads to \"poorer alignment and a worse uniformity\". To handle this noise, the authors decompose the term used in GIB and, due to its intractability, propose two efficient, practical instantiations.", "questions": "1. Would it make sense / be possible also to evaluate the method using an attack that optimizes for $\\tilde{A}$ or $\\tilde{Y}$, e.g., using first-order optimization?\n1. As you do not provide code, I wonder if the clean labels/adjacency are used in any way during training?", "limitations": "The authors should elaborate more and more prominently on what the computational requirements of their method are and how it compares to the other baselines.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The authors propose two methods that significantly and consistently outperform the baselines\n1. The approach is principled due to its roots in the powerful concept of information bottleneck\n1. The experiments are extensive and are convincing that the method has some merit for certain applications\n\nThe paper is generally well-written and logically structured.", "weaknesses": "1. The method seems to rely heavily on the assumption that the node features are clean. It would be good to study how sensitive the model is to feature noise.\n1. The authors could elaborate more on the assumptions etc., that make their instantiations tractable and when these assumptions are met.\n\nMinor: margins between lines 215 & 216 seem violated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer 4Q8x for the valuable feedback. We addressed all the comments. Please find the point-to-point responses below. Any further comments and discussions are welcomed!\n\n**Q1**. *The method seems to rely heavily on the assumption that the node features are clean. It would be good to study how sensitive the model is to feature noise.*\n\n**Reply:** We appreciate the reviewer's question about the node feature. **Although our RGIB methods focus on edge noise, we conduct the following experiments to verify the point.** Specifically, we compare standard training, RGIB-REP, and RGIB-SSL using a 4-layer GCN on six datasets with various ratios of feature noise. Here, the noisy feature is generated by adding random noise within the range of $[0,1]$ to the normalized node feature. \n\nThe evaluation results of mean AUC are reported in the additional one-page PDF file of our general response.\n\nAs can be seen, the noisy node feature also significantly degenerates the GNN's performance, and the degradation becomes severer as the noise ratio increases. **Interestingly, compared with the standard training manner, RGIB-SSL and RGIB-REP can also promote the robustness of GNN against feature noise.** For example, when learning with $\\epsilon_f=10\\\\%$ feature noise, RGIB-SSL can bring $16.6\\\\%$ and $6.5\\\\%$ improvements in AUC on Cora and Citeseer datasets, respectively. \n\nWe speculate that as the graph representation $H$ is encoded from the node feature $X$, regularizing $H$ in RGIB objectives can also balance its dependence on $X$ and thus shows some potential robustness, even though we originally designed these objectives to handle edge noise. \n\n**The above experiments show that our RGIB principle also has some merits in combating feature noise.** We should note that this might be an initial verification, and a more comprehensive study will be conducted to have a rigorous conclusion in future explorations. We will add the above discussions and evaluation results to the submission.\n\n**Q2**. *The authors could elaborate more on the assumptions etc., that make their instantiations tractable and when these assumptions are met.*\n\n**Reply:** Thanks for the constructive advice. The primary assumption of our work is that the edges of collected graph data can be potentially noisy. As in the submission, we present two ways of realizing the RGIB principle, which actually corresponds to some assumptions in building the objectives with the proper approximation. \n\n**RGIB-SSL assumes that the learned representation can be improved with higher uniformity and alignment.** As RGIB-SSL directly acts on the graph representation, it is more suitable for recovering the distribution of representation, especially when encountering representation collapse due to the severe edge noise.\n\nRGIB-REP explicitly purifies the input graph’s topology and target labels with the jointly reparameterized $Z_A$ and $Z_Y$. Here, the latent variables $Z_A$,$Z_Y$ are expected to be more clean and informative than the noisy $\\tilde{A}$,$\\tilde{Y}$. **RGIB-REP assumes that the GNN model can identify these latent variables and further benefit its learning procedure against noise.**\n\nEmpirically, RGIB-SSL is more adaptive to sparser graphs, e.g., Cora and Citeseer, where the edge noise results in severer representation collapse. RGIB-REP can be more suitable for denser graphs, e.g., Facebook and Chameleon, where the latent variables of edge data are extracted by RGIB-REP. More importantly, the two RGIB instantiations be complementary to each other with flexible options in practical applications.\n\n**Q3**. *Margins between lines 215 & 216 seem violated.*\n\n**Reply:** Thanks for this comment. The small margin is due to the automatic typesetting of the LaTex compiler, and we did not manually modify this margin in our draft. We will rearrange this part to be clearer.\n\n**Q4**. *Would it make sense / be possible also to evaluate the method using an attack that optimizes for $\\tilde{A}$ or $\\tilde{Y}$, e.g., using first-order optimization?*\n\n**Reply:** Yes, it is reasonable. In Appendix D.4, we conduct the adversarial attacks on $\\tilde{A}$, which is the same as you suggested. As shown in Tables 11 and 12, the adversarial attack that adds noisy edges to the input graph also significantly degenerates the GNN's performance. **Importantly, it is observed that the two instantiations of RGIB can also promote the robustness of GNN against adversarial attacks on graph structure.** RGIB-SSL and RGIB-REP achieve $4.1\\\\%$ and $5.5\\\\%$ improvements in Test AUC on Cora dataset with $\\epsilon_{adv}=20\\\\%$ adversarial perturbations.\n\n**Q5**. *As you do not provide code, I wonder if the clean labels/adjacency are used in any way during training?*\n\n**Reply:** We would like to kindly point out that we have provided an anonymous link to our source code. Please refer to line 732 in Appendix C.2. \n\nIn addition, any clean labels and adjacency are not used in training. On the contrary, the model directly learns from the noisy adjacency $\\tilde{A}$ and label $\\tilde{Y}$, which is practical as the collected data is potentially noisy in real-world applications. \n\n**Q6**. *The authors should elaborate more and more prominently on what the computational requirements of their method are and how it compares to the other baselines.*\n\n**Reply:** Thanks for the advice. We provide a detailed explanation in the following two folds.\n\n**(1) Noise information.** RGIB and all the baselines are run without any noise priors, e.g., noise type or noise ratio. The only required information for training is adjacency $\\tilde{A}$, node feature $X$, and edge labels $\\tilde{Y}$, not including any additional heuristics or assumptions. \n\n**(2) Training time.** Further, we evaluate the effectiveness and efficiency of the proposed methods on two large-scale datasets with bilateral noise. As shown in Table 14 of Appendix D.5, the extra computing costs of RGIB are within an acceptable range."}, {"review_id": "iRLv3qqDyS", "reviewer": "Reviewer_cjox", "summary": "This paper focuses on the robustness of GNNs under the edge noise. The authors disclose the influence of bilateral edge noise and the corresponding robustness issue via a series of empirical studies on edge noise. Based on the observations of bilateral noise, the authors propose an information-theory-guided principle, Robust Graph Information Bottleneck (RGIB) and its two instantiations, RGIB-SSL and RGIB-REP. Experimental results verify the effectiveness of RGIB instantiations.", "questions": "- I wonder how RGIB improves other graph representation learning tasks, like node classification?\n- Why the model performance on original graphs are not reported? Since the graphs without additional noises are not considered as completely clean, RGIB should be able to provide performance gain as well.", "limitations": "Please refer to the weaknesses.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is well written and structured.\n- The emprical studies and relative experiments are clear and convincing.\n- Experiments are comprehensive and complete.\n- The proposed RGIB principle is simple yet effective.", "weaknesses": "- There is a lack of more comprehensive studies on other graph representation learning tasks, such as node classification.\n- RGIB is proposed under the assumption of edge noise, however the motivation statement is not very convincing. The relationship between GIB and bilateral noise should be further explained.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer cjox for the valuable feedback. We addressed all the comments. Please find the point-to-point responses below. Any further comments and discussions are welcomed!\n\n**Q1**. *There is a lack of more comprehensive studies on other graph representation learning tasks, such as node classification. I wonder how RGIB improves other graph representation learning tasks, like node classification?*\n\n**Reply:** The answer is yes. Actually, we have conducted experiments on the node classification task with random label noise on nodes, which is the same as you suggested. Please refer to Appendix D.6 for details. **As shown in below Tables 15 and 16, we justify that the RGIB framework can generalize to the node classification tasks with label noise on nodes, where the two instantiations of RGIB also significantly outperform the standard training manner.**\n\nBesides, please refer to the broader impact and the general robustness of GNNs discussed in the Appendix. B.4 and B.5, respectively. These contents are also relevant to your question.\n\n| Cora (Table 15)   | clean    | $\\epsilon=20\\\\%$ | $\\epsilon=40\\\\%$ | $\\epsilon=60\\\\%$ |\n| ----------------- | -------- | --------------- | --------------- | --------------- |\n| standard training | **.898** | .868            | .720            | .322            |\n| RGIB-SSL          | .900     | **.876**        | **.786**        | **.388**        |\n| RGIB-REP          | .894     | .862            | .760            | .312            |\n\n| Citeseer (Table 16) | clean    | $\\epsilon=20\\\\%$ | $\\epsilon=40\\\\%$ | $\\epsilon=60\\\\%$ |\n| ------------------- | -------- | --------------- | --------------- | --------------- |\n| standard training   | .776     | .746            | .608            | .278            |\n| RGIB-SSL            | **.784** | **.770**        | .646            | .324            |\n| RGIB-REP            | .776     | .754            | **.654**        | **.364**        |\n\n**Q2**. *RGIB is proposed under the assumption of edge noise, however the motivation statement is not very convincing. The relationship between GIB and bilateral noise should be further explained.*\n\n**Reply:** We would like to further explain the motivation of RGIB and the relationship between GIB and bilateral noise in the following three folds.\n\n**(1)** **Conceptually, we would clarify that GIB (Equation 1) is intrinsically susceptive to label noise since it entirely preserves the label supervision with maximizing $I(H;\\tilde{Y})$.** As illustrated in Figure 2, the GIB decreases $I(H;\\tilde{A}|\\tilde{Y})$ by directly constraining $I(H;\\tilde{A})$ to handle the input noise. Symmetrically, the label noise can be hidden in the area of $I(H; \\tilde{Y} | \\tilde{A})$, but trivially constraining $I(H; \\tilde{Y})$ to regularize $I(H;\\tilde{Y} | \\tilde{A})$ is not ideal, since it will conflict with Equation 1. Besides, it cannot tackle the noise within $I(\\tilde{A}; \\tilde{Y})$, where the two kinds of noise can share similar patterns as the random split manner does not change their distributions in expectation. Thus, GIB cannot provide an ideal solution to the bilateral edge noise investigated in this work. \n\n**(2)** **Thus, it is crucial to further decouple the mutual dependence among $\\tilde{A}$, $\\tilde{Y}$, and $H$.** Based on the detailed analysis elaborated in Section 4.1 and Appendix B.2, we derive the RGIB principle that balances the three important information terms $H({H})$, $I( H; \\tilde{Y} | \\tilde{A})$ and $I( H; \\tilde{A} | \\tilde{Y})$. It works as an information bottleneck to filter out the noisy signals in both $\\tilde{A}$ and $\\tilde{Y}$, utilizing the supervision signals $I( H; \\tilde{Y})$ at the same time. Analytically, GIB only indirectly regularizes the MI term $I( H;\\tilde{A}|\\tilde{Y})$, as we introduce section 4.1, which can only solve partial noisy information. By contrast, RGIB takes all the related MI terms, i.e., $H({H})$, $I( H; \\tilde{Y} | \\tilde{A})$, and $I( H; \\tilde{A} | \\tilde{Y})$, based on which to provide a solution that balances these MI terms as a more strict information bottleneck. \n\n**(3)** **We provide two instantiations for implementing the RGIB principle, i.e., RGIB-SSL and RGIB-REP.** These two instantiations benefit from different methodologies, i.e., self-supervised learning and data reparametrization, for implicit and explicit data denoising, respectively. Note that these two methodologies are not considered in the original GIB. Besides, the GIB is highly coupled with the GAT. By contrast, RGIB does not require any modifications to GNN architecture. It can be seamlessly integrated with various GNNs and promote their robustness against bilateral noise.\n\nIn a nutshell, RGIB generalizes the GIB with improvements in both theories and methodologies to learn a robust representation more resistant to the bilateral edge noise. We will follow the reviewer's advice by refining the corresponding description to make it clearer.\n\n**Q3**. *Why the model performance on original graphs are not reported? Since the graphs without additional noises are not considered as completely clean, RGIB should be able to provide performance gain as well.*\n\n**Reply:** We have conducted this supplement experiment that evaluates all the baseline methods on clean datasets in Appendix E.1. As the below Table selected from Table 17, **the proposed two instantiations of RGIB can also boost the predicting performance when learning on clean graphs and significantly outperform other baselines in most cases.**\n\n| Table 17          | Cora      | Citeseer  | Pubmed    | Facebook  | Chameleon | Squirrel  |\n| ----------------- | --------- | --------- | --------- | --------- | --------- | --------- |\n| standard training | .8686     | .8317     | .9178     | .9870     | .9788     | .**9725** |\n| RGIB-SSL          | .8758     | .8415     | .9408     | **.9875** | **.9792** | .9680     |\n| RGIB-REP          | **.9260** | **.9148** | **.9593** | .9845     | .9740     | .9646     |"}, {"review_id": "L1E6M1azhw", "reviewer": "Reviewer_Z7rx", "summary": "This paper proposes to tackle the bilateral edge noise via mutual information. The authors start from empirical observations that existing GNNs are vulnerable to bilateral edge noises. To tackle this issue, the authors propose a robust graph information bottleneck which is information-theory guided. In practice, a self-supervised regularization and a purification mechanism are proposed. The extensive evaluation on various datasets as well as models demonstrates the effectiveness of proposed algorithm.", "questions": "1. Please clarify the motivation of hybrid graph augmentation and the highlight the contribution compared to those in [18, 47].\n2. Please include the evaluation under attacks [4, 8, 46] if possible, or discuss the generalization of the proposed algorithm against adversarial attacks.\n", "limitations": "The authors have not discussed the limitations.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written and well-organized. \n2. The introduced bilateral edge noise is an interesting and challenging problem in GNNs.\n3. The proposed robust graph information bottleneck is well-motivated by both sufficient empirical observations as well as theoretical analysis.\n4. The experiments are extensive, including various datasets and GNNs. Various ablation studies demonstrate the effectiveness of RGIB-SSL and RGIB-REP.\n", "weaknesses": "I have several concerns:\n1. In RGIB-SSL, the authors introduce hybrid graph augmentation which shows superiority over other contrastive learning methods. However, how the proposed augmentation outperforms other graph contrastive methods needs more clarification to provide some insights of this design.\n2. It is recommended to demonstrate the robustness of the proposed algorithm under attacks, such as [4, 8, 46].\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer Z7rx for the valuable feedback. We addressed all the comments. Please find the point-to-point responses below. Any further comments and discussions are welcomed!\n\n**Q1**. *In RGIB-SSL, the authors introduce hybrid graph augmentation which shows superiority over other contrastive learning methods. However, how the proposed augmentation outperforms other graph contrastive methods needs more clarification to provide some insights of this design.* \n\n**Reply:** We appreciate the reviewer’s insightful advice. Regarding the three information terms of RGIB-SSL in Equation 3, we elaborate a detailed clarification by comparing it with other contrastive learning methods in the following three folds.\n\n**(1)** supervision term $I(H_1;\\tilde{Y}) + I(H_2;\\tilde{Y})$. \n\n**Compared with the common manner of self-supervised contrastive learning, RGIB-SSL also considers utilizing noisy labels $\\tilde{Y}$.** We show that, although being potentially noisy, the labels $\\tilde{Y}$ can also benefit learning. Besides, our experiments show that both the supervised contrastive learning method (e.g., SupCon) and the self-supervised contrastive learning method (e.g., GRACE) perform poorly when learning with the bilateral edge noise. An intuitive explanation is that SupCon entirely trusts and utilizes the noisy $\\tilde{Y}$ while GRACE does not adopt $\\tilde{Y}$. \n\n**From the data perspective, RGIB-SSL performs a hybrid mode of supervised and self-supervised learning.** It effectively extracts and utilizes the informative signals in $\\tilde{Y}$ without entirely absorbing its noisy signals. This is achieved with the synergy of two self-supervision terms as follows.\n\n**(2)** self-supervision alignment term $I(H_1;H_2)$. \n\n**Here, the main differences with common contrastive learning methods are data augmentation and loss function.** Specifically, RGIB-SSL utilizes the hybrid augmentation algorithm and self-adversarial loss. Compared with the fixed augmentation manner of other contrastive learning methods [18, 47], i.e., performs random perturbation on node feature and edges, we propose a hybrid augmentation method with four augmentation operations (please refer to Appendix C for details). \n\nThe motivation here is to encourage more diverse views with lower MI $I(A_1; A_2)$ and to avoid manual selection of augmentation operations. Besides, Proposition 4.2 provides a theoretical analysis of hybrid augmentation from information-theoretical perspectives.\n\nEmpirically, compared with fixed augmentation, the hybrid brings a 3.0% average AUC promotion on Cora (please refer to Table 5). This means that a stronger augmentation scheme with more diverse views can help better deal with severer edge noise. Besides, the self-adversarial loss further enhances high-quality pairs and decreases low-quality counterparts. It refines the signal and brings up to 2.1% promotion.\n\n**(3)** self-supervision uniformity term $H(H_1) + H(H_2)$.\n\nAs for the uniformity term, the understanding part in Section 3.2 shows that a severer edge noise brings a worse uniformity. **To learn a more robust representation, we add this uniformity term to form the final loss of RGIB-SSL and adopt the Gaussian potential kernel for implementation, which is usually not considered in other contrastive learning methods.**\n\nThe ablation studies in Section 5.2 also illustrate that the uniformity term is essential, especially in dealing with label noise. Besides, the uniformity of learned representation is also enhanced (see Figure 6), and the various query edges tend to be more uniformly distributed on the unit circle, especially for the negative edges. \n\nIn the revision, we will refine and highlight these explanations for the hybrid graph augmentation.\n\n**Q2**. *It is recommended to demonstrate the robustness of the proposed algorithm under attacks, such as [4, 8, 46].* \n\n**Reply:** Thanks for the nice suggestion. As an adversarial attack, the Nettack [49] is considered in all the three works [4, 8, 46] you mentioned. In Appendix D.4 of our work, we conduct experiments based on Nettack that perturbs the graph structure, which is the same as you suggested.\n\nAs the Test AUC shown in the tables below, the adversarial attack that adds noisy edges to the input graph also significantly degenerates the GNN's performance. **Crucially, it is observed that RGIB-SSL and RGIB-REP can also promote the robustness of GNN against adversarial attacks on graph structure.** RGIB-SSL and RGIB-REP respectively achieve $4.1\\\\%$ and $5.5\\\\%$ improvements of Test AUC on the Cora dataset with $\\epsilon_{adv}=20\\\\%$ adversarial perturbations, demonstrating the robustness of RGIB against adversarial attacks.\n\nBesides, the broader impact of this work and the general robustness of GNNs are also discussed in Appendix. B.4 and B.5.\n\n| Cora dataset (Table 11) | clean     | $\\epsilon_{adv}=20\\\\%$ | $\\epsilon_{adv}=40\\\\%$ | $\\epsilon_{adv}=60\\\\%$ |\n| ----------------------- | --------- | --------------------- | --------------------- | --------------------- |\n| standard training       | .8686     | .7971                 | .7671                 | .7014                 |\n| RGIB-SSL                | **.9260** | .8296                 | **.8095**             | **.8052**             |\n| RGIB-REP                | .8758     | **.8408**             | .7918                 | .7611                 |\n\n| Citeseer dataset (Table 12) | clean     | $\\epsilon_{adv}=20\\\\%$ | $\\epsilon_{adv}=40\\\\%$ | $\\epsilon_{adv}=60\\\\%$ |\n| --------------------------- | --------- | --------------------- | --------------------- | --------------------- |\n| standard training           | .8317     | .8139                 | .7736                 | .7481                 |\n| RGIB-SSL                    | **.9148** | **.8656**             | **.8347**             | **.8022**             |\n| RGIB-REP                    | .8415     | .8382                 | .8107                 | .7893                 |"}, {"review_id": "wXXxuyOPCv", "reviewer": "Reviewer_4XLr", "summary": "This paper focuses on the robustness of graph neural networks (GNNs) in the presence of edge noise during link prediction on graphs. The authors empirically investigate the impact of edge noise on both the input topology and target labels, revealing significant performance degradation and representation collapse. To address this challenge, they propose a novel principle called Robust Graph Information Bottleneck (RGIB), which leverages information theory to extract reliable supervision signals and prevent representation collapse. The paper explores two instantiations of RGIB, RGIB-SSL and RGIB-REP, which utilize self-supervised learning and data reparametrization techniques, respectively, for implicit and explicit data denoising.\n", "questions": "If the SSL and REP versions are parallel methods, could the authors discuss the guidelines on method selection? Are they be simultaneously used while still improve? It would be better to add a section of such discussions with supportive case evidence.\n", "limitations": "See above.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Overall, this paper makes a valuable contribution to understanding and enhancing the robustness of GNNs in link prediction tasks affected by edge noise.\n\nThe main novelty is based on systematic study of Bilateral Edge Noise including its empirical influence, visualization, and two instantiations of Robust Graph Information Bottleneck with data reparameterization and data augmentation. \n\nThis paper also provides theoretical analysis for noisy dataset convergence relaxations. \n\n", "weaknesses": "This paper provides a novel method and analysis for the edge noise issue for the GNN, through the lens of information theory. The method is clearly explained, and the performance is supported with grounded evaluations with visualizations. I did not carefully check the theory proofs, but from the main paper, there is no obvious limitation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer 4XLr for the valuable feedback. We addressed all the comments. Please find the point-to-point responses below. Any further comments and discussions are welcomed!\n\n**Q1**. *If the SSL and REP versions are parallel methods, could the authors discuss the guidelines on method selection? Are they be simultaneously used while still improve? It would be better to add a section of such discussions with supportive case evidence.*\n\n**Reply:** We appreciate Reviewer 4XLr’s insightful comment. We would like to clarify the comparison of the two instantiations of RGIB in the following five folds.\n\n**(1)** **From the theoretical perspective**, RGIB-SSL **explicitly** optimizes the representation $H$ with self-supervised regularizations, i.e., alignment $I(H_1; H_2)$ and uniformity $H(H_1)$, $H(H_2)$. By contrast, RGIB-REP **implicitly** optimizes $H$ by purifying the noisy $\\tilde{A}$ and $\\tilde{Y}$ with the reparameterization mechanism to extract clean signals in the forms of latent variables $Z_{Y}$ and $Z_{A}$. The information constraints $I(Z_{A}; \\tilde{A})$, $I(Z_{Y}; \\tilde{Y})$ are directly acting on $Z_{Y}$ and $Z_{A}$ and indirectly regularizing the representation $H$.\n\n**(2)** **Besides, from the methodology perspective, both instantiations are equipped with adaptive designs for obtaining an effective information bottleneck.** RGIB-SSL utilizes the automatically augmented views $\\tilde{A_1}$, $\\tilde{A_2}$ in a contrastive manner to be resistant to input noise. RGIB-SSL is intrinsically robust to label noise due to its self-supervised nature. Besides, RGIB-REP explicitly purifies the input graph’s topology and target labels with the jointly reparameterized $Z_{Y}$ and $Z_{A}$. It enables to model the mutual patterns of edge noise from both input and label spaces. \n\nIn addition, we conduct a further comparison and analysis of the two instantiations that are summarized as follows. \n\n| Instantiation | methodology              | advantages                                                   | disadvantages                                                |\n| ------------- | ------------------------ | ------------------------------------------------------------ | ------------------------------------------------------------ |\n| RGIB-SSL      | self-supervised learning | automated graph augmentation; good effectiveness; can be applied in entirely self-supervised settings without labels. | with expensive calculation for the contrastive objectives, especially the uniformity; requires extra graph augmentation operations. |\n| RGIB-REP      | data reparametrization   | no needs to do data augmentation; good efficiency; the input/output constraints do not require extra annotations for supervision and can be easily controlled. | sensitive to the hyper-parameters $\\lambda$; less effective in extremely noisy cases; only applicable in fully supervised settings. |\n\n**(3)** **In addition, RGIB-SSL assumes that the learned representation can be improved with higher uniformity and alignment**. As RGIB-SSL directly acts on the graph representation, it is more suitable for recovering the distribution of representation, especially when encountering representation collapse due to the severe edge noise. By contrast, RGIB-REP explicitly purifies the input graph’s topology and target labels with the jointly reparameterized $Z_A$ and $Z_Y$. Here, the latent variables $Z_A$,$Z_Y$ are expected to be more clean and more informative than the noisy $\\tilde{A}$,$\\tilde{Y}$. **RGIB-REP assumes that the GNN model can identify these latent variables and further benefit its learning procedure against noise**.\n\n**(4)** **Empirically, we observe that the two instantiations of RGIB can be generalized to different scenarios with their own priority according to the intrinsic graph properties.** RGIB-SSL is more adaptive to sparser graphs, e.g., Cora and Citeseer, where the edge noise results in severer representation collapse. RGIB-REP can be more suitable for denser graphs, e.g., Facebook and Chameleon, where the latent variables of edge data are extracted by RGIB-REP. More importantly, the two RGIB instantiations can be complementary to each other with flexible options in practical applications, and we have summarized such a point in Remark 5.1 of Section 5.1.\n\n**(5)** **Finally, we attempt to jointly combine and use RGIB-REP and RGIB-SSL simultaneously, as you suggest.** Specifically, the \"RGIB-REP+RGIB-SSL\" integrates both data reparametrization technique and self-supervised learning method, and the final objective for optimization equals minimizing Equation 3 and Equation 4 simultaneously. Then, we compare \"RGIB-REP+RGIB-SSL\" with standard training, RGIB-REP, and RGIB-SSL based on a 4-layer GCN on all six datasets with $\\epsilon=40\\\\%$ bilateral noise.\n\n**As shown in the below table, the \"RGIB-REP+RGIB-SSL\" achieves considerable performances as RGIB-REP and RGIB-REP in most cases and outperforms these two instantiations on the Facebook dataset.** Although we believe that a more careful finetuning of the hyper-parameters will bring further improvements to the \"RGIB-REP+RGIB-SSL\" combination, we suggest using one of the instantiations in practice to keep the learning objective simple and sufficient to combat the bilateral edge noise. The above clarifications and evaluation results will be added to our draft.\n\n| dataset           | Cora  | Citeseer | Pubmed | Facebook | Chameleon | Squirrel |\n| ----------------- | ----- | -------- | ------ | -------- | --------- | -------- |\n| Standard training | .7419 | .7380    | .8748  | .9520    | .9496     | .9406    |\n| RGIB-REP          | .7966 | .7519    | .8834  | .9770    | .9621     | .9455    |\n| RGIB-SSL          | .8554 | .8427    | .8918  | .9711    | .9592     | .9426    |\n| RGIB-REP+RGIB-SSL | .8351 | .8270    | .8880  | .9819    | .9570     | .9431    |\n\n"}]

Author rebuttal:
### A General Response by Authors

**We would like to thank all the reviewers for their valuable comments on our work.** 

**We have received five reviews with positive ratings 7,6,7,7,7. We appreciate that all the reviewers have good impressions on our work**, including **(1)** interesting problem and powerful solution (Fboz, 4Q8x, cjox, Z7rx, 4XLr); **(2)** comprehensive and convincing experiments (Fboz, 4Q8x, cjox, Z7rx, 4XLr); **(3)** sufficient theoretical supports (Fboz, Z7rx, 4XLr); and **(4)** well-written and good presentation (Fboz, 4Q8x, cjox, Z7rx).

**In the rebuttal period, we have provided detailed responses to all the comments and questions point-by-point.** Specifically, we further clarify the assumption (Q2 for 4Q8x), motivation (Q2 for cjox), method (Q1 for Z7rx; Q1 for 4XLr), extension scenarios (Q4 for 4Q8x; Q1,Q3 for cjox; Q2 for Z7rx) and training details (Q5,Q6 for 4Q8x) of our work. Besides, we add new empirical evaluations with the extension on feature noise (Q1 for 4Q8x) and the integration of the two instantiations of RGIB (Q1 for 4XLr). The attached one-page PDF file contains the evaluations on feature noise (Q1 for 4Q8x).

Lastly, we would appreciate all reviewers’ time again. Would you mind checking our response and confirming whether you have any further questions? **We are anticipating your post-rebuttal feedback!** 

Meta-review: This paper proposes a novel principle called Robust Graph Information Bottleneck (RGIB) to extract reliable supervision signals and avoid representation collapse in graph neural networks under edge noise. The authors also explore two instantiations of RGIB, namely RGIB-SSL, and RGIB-REP, which leverage different methodologies for data denoising. The work provides empirical evidence to support the effectiveness of RGIB in enhancing the uniformity of learned representation and improving the alignment and uniformity of edge representation. Additionally, the authors propose several informative terms and selection mechanisms to further enhance the performance of RGIB.

Initially, reviewers raised several concerns regarding concept clarity, assumption validity, robustness evaluation, and other task evaluation (such as node classification). Those were well addressed by authors during the rebuttal, leading the unanimously positive reviews post-rebuttal: (7,7,7,7,6). AC sides with the reviewer consensus and considers this paper a clear accept case.
Decision: Accept (poster)