Towards Reliable Link Prediction with Robust Graph Information Bottleneck
Authors: Zhanke Zhou
Jiangchao Yao
Jiaxu Liu
Xiawei Guo
LI He
Shuo Yuan
quanming yao
Liang Wang
Bo Han
Venue: Submitted to ICLR 2023 (2023)
Keywords: Robust link prediction
Inherent edge noise
Graph representation learning

Abstract: Link prediction on graphs has achieved great success with the rise of deep graph learning. However, the potential robustness under the edge noise is less investigated. We reveal that the inherent edge noise that naturally perturbs both input topology and target label leads to severe performance degradation and representation collapse. Here, we propose an information-theory guided principle, Robust Graph Information Bottleneck (RGIB), to extract reliable supervision signals and avoid representation collapse. Different from the general information bottleneck, RGIB decouples and balances the mutual dependence among graph topology, edge label, and representation, building a new learning objective for robust representation. We also provide two implementations, RGIB-SSL and RGIB-REP, that benefit from different methodologies, i.e., self-supervised learning and data reparametrization, for indirect and direct data denoising, respectively. Extensive experiments on six benchmarks with various scenarios verify the effectiveness of the proposed RGIB.

Under review as a conference paper at ICLR 2023
TOWARDS RELIABLE LINK PREDICTION WITH ROBUST GRAPH INFORMATION BOTTLENECK
Anonymous authors Paper under double-blind review
ABSTRACT
Link prediction graphs has achieved great success with the rise of deep graph learning. However, the potential robustness under the edge noise is less investigated. We reveal that the inherent edge noise that naturally perturbs both input topology and target label leads to severe performance degradation and representation collapse. In this work, we propose an information thhhor- principle, Robust Graph Information Bottleneck (RGIB), to extract reliable supervision signals and avoid representation collapse. Different from the general information bottleneck, RGIB decouples and balances the mutual dependence among graph topology, target labels, and representation, building new learning objectives toward robust representation. We also provide two instantiations, RGIB-SSL and RGIB-REE which benefit from different methodologies, i.e., self-supervised learning and data reparametrization, for implicit and explicit data denoising, respectively. Extensive experiments on 6 benchmarks of various scenarios verify the effectiveness of the proposed RGIB,
INTRODUCTION
As a fundamental problem in graph learning, link prediction (Liben-Nowell & Kleinberg, 2007)
has attracted growing interest in real-world applications like drug discovery (loannidis et al. 2020
knowledge graph completion (Bordes al.] (2013), and question answering (Huang et al. 2019).
Recent advances from heuristic designs (Katz, 1953 Page et al. 19999) to graph neural networks (GNNs) (Kipf & Welling 2016a Gilmer et al. 2017 Kipf & Welling 2016b Zhang & Chen 2018 Zhu et al. (2021) have achieved superior performances. Nevertheless, the poor robustness in
imperfect scenarios with the inherent edge noise is still a practical bottleneck to the current deep
graph models (Gallageer et al. 2008 Ferrara al. 2016 Wu et al. 2022a Dai et al. 2022).
Early explorations improve the robustness of GNNs for node classification under label noise DDai et al.) 2021 Li al.) (021)) through the smoothing effect of neighboring nodes. Other methods achieve a similar goal via randomly removing edges (Rong et al.] (022)) or actively selecting the
informative nodes or edges and pruning the task-irrelevant ones (Zheng al. 2020 Luo et al.) [021] However, when applying these noise-robust methods to the link prediction with noise, only marginal improvements are achieved (see Section 5). The attribution that the edge noise can naturally deteriorate both the input topology and the target labels (Figure ((a). Previous works that consider the noise either in input space or label space cannot effectively deal with such coupled scenario.
Therefore, it raises new challenge to understand and tackle the edge noise for robust link prediction.
In this paper, we dive into the inherent edge noise and empirically show the significantly degraded performances it leads to (Section 3.I. Then, we reveal the negative effect of the edge noise through carefully inspecting the distribution of learned representations, and discover that graph representation is severely collapsed, which is reflected by much lower alignment and poorer uniformity (3.2 To solve this challenging problem, we propose the Robust Graph Information Bottleneck (RGIB) principle based the basic GIB for adversarial robustness (Wu et al. 2020 (Section 4 Conceptually, the RGIB principle is with new learning objectives that decouple the mutual information (MI) among noisy inputs Å, noisy labels Y, and the representation H. As illustrated in Figure ((b) RGIB generalizes the basic GIB to learn a robust representation that is resistant to the edge noise.
Technically, we provide two instantiations of RGIB based on different methodologies, i.c., RGIB- SSL and RGIB-REP (1) the former utilizes contrastive pairs with automatically augmented views

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(b) The basic GIB (left) and the proposed RGIB (right). solve the Under review as a conference paper at ICLR 2023
clean data
noisy data
noisy inputs À
noisy labels P
Y
Clean Noisy
Clean Noisy
HI) Ø I(A 3 ((:HHA)
optimal H
clean edges
..... noisy edges
4 = I(A ((A))))) (2 +) = I(Y:))
(a) Illustration of inherent edge noise. The GNN takes the graph topology A as inputs, and predicts the logits of unseen edges with labels Y. The noisy Á and Y added with random edges for simulating the inherent edge noise as Def. 3.1
+ here indicates the mutual information. intrinsic deficiency of basic GIB in tackling the edge noise, the RGIB learns the graph represenation H via a further balance of informative signals ①, 3. 4 regarding the H.
Figure I: Link prediction with inherent edge noise (a) and the proposed RGIB principle (b).
to form the informative regularization in self-supervised learning manner (Section 4.2); ) and (2)
the latter explicitly purifies the graph topology and supervision targets with the reparameterization mechanism (Section 4.3). Both instantiations are equipped with adaptive designs, aiming to effectively estimate and balance the corresponding informative terms in a tractable manner. For example, the hybrid augmentation algorithm and self-adversarial alignment loss for RGIB-SSI, and the relaxed information constraints on topology space as well as label space for RGIB-REA Empirically, we show that these two instantiations work effectively under extensive noisy scenarios and can be seamlessly integrated with various existing GNNs (Section 5). Our main contributions are summarized as follows.
To our best knowledge, are the first to study the robustness problem of link prediction under the inherent edge noise. We reveal that the inherent noise can bring a severe representation collapse and performance degradation, and such negative impacts are general to common datasets and GNNs.
We propose general learning framework, RGIB, with refined representation learning objectives to promote the robustness of GNNs. Two instantiations, RGIB- ESLL and RGIB -RE,, are proposed upon different methodologies that equipped with adaptive designs and theoretical guarantees.
Without modifying the GNN architectures, the RGIB achieves ofate-of-theaante results on 3 GNNs and datasets under various noisy scenarios, obtaining up to 12.9% AUC promotion. The distribu- tion of learned representations is notably recovered and more robust to the inherent noise.
2 PRELIMINARIES
MM Notation. We denote V= [o as the set nodes and E (Gi 1==1 as the set of edges. With adjacent matrix A and node features X, an undirected graph is denoted as G= X), where Ai =ty=1 means there is an edge Cty between i/ and uy. Xt.. E RD is the D-dimension node feature of 14- Link prediction is to indicate the existence of query edges with labels Y that are not observed in A.
GNNs for Link Prediction. We follow the common link prediction framework, i.c., graph auto- encoders (Kipf & Welling (016b)), where the GNN architecture can be GCN (Kipf & Welling 2016a)) GAT /Veliković et al. 2018), or SAGE (Hamilton et al. 2017). Given a I--layer GNN, the graph representations H E R for each node E V are obtained by a L-layer message propagation as the encoding process. For decoding, logits Φe., of each query edge eij are computed with readout function, e.g., the dot product Φe., = =h h,. Finally, the optimization objective is to minimize the binary classification loss, i.e., min Lets = E ggtrain log (a( )) -1)((1111))) (11-1;)) (og)olo
where (.) is the sigmoid function, and Vij for positive edges while yıg 0 for negative ones.
Topological denoising approaches. A natural way to tackle the input edge noise to directly clean the noisy graph. Sampling-based methods, such as DropEdge (Rong et al.) 2020), Neu- ralSparse ZZhnng et al. 2020). and PTDNet (Luo et al.] (021). are proposed to remove the task- irrelevant edges. Besides, as GNNs can be easily fooled by adversarial network with only a few perturbed edges (Chen et al. (2018) Zhuu et al. 2019; Entezari et al. 2020), defending methods like GCN-jaccard (Wu al. 2019) and GIB (Wu al. 2020) are designed for pruning adversarial edges.
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Label-noise-resittare techniques. For tackling the general problem of noisy label, Co-teaching (Han) et al.] (2018) lets two neural networks teach each other with small-loss samples based on the memo- rization effect (Arpit et al. 2017). Besides, peer loss function (Liu & Guo, 2020) pairs independent peer examples for supervision and works within the standard empirical risk minimization framework. As for the graph domain, label propagation techniques proposed by pioneer works (Dai al. 202T] Li et al., (2021) 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.
AN EMPIRICAL STUDY OF THE INHERENT 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 Section I.1tiion..nnnd. and then investigate the negative impact of noise through the lens of representation distribution in Section 3.2
3.1 How DO GNNS PERFORM UNDER THE INTEREEN EDGE NOISE?
Since existing benchmarks are usually well-annotated and clean, there is need to simulate the inherent edge noise properly to investigate the impact of noise. Note the data split manner adopt by most relevant works (Kipf & Welling 6016b Zhang & Chen 2018: Zhu et al. (202T) randomly divides partial edges as observations and the others as prediction targets. The inherent edge noise, if exists, should be false positive samples and uniformly distributed to both input observations and output labels. Thus, the training data can be with noisy adjacence A and noisy labels Y. i.e. the input noise and label noise. We elaborate the formal simulation of such an additive edge noise as follows.
Definition 3.1 (Additive edge noise). Given a clean training data, i.e., observed graph G XA and labels Y {0, 13 of query edges, the noisy adjacence À is generated by only adding edges to the original adjacent matrix A while keeping the node features X unchanged. The noisy labels Y are generated by only adding false-positin
edges to the labels Y. Specifically, given a noise ratio ca. the added noisy edges A' (À = A+A') are randomly generated with the zero elements in A as candidates. It satisfies that A' 3 == O and Ea = (onzoro(A)]lpoozerre (/n/////z) Similarly, noisy labels are generated and added original labels, where Ey = [nonzero()") ((Ioo(/rr)))o))()))) (oooorrool))))))
With the simulated noise, an empirical study is then performed with various GNNs and datasets. As shown in Figure 2 the edge noise causes a significant drop in performance, and larger noise ratio generally leads to greater degradation. It means that these common GNNs normally trained with the stochastic gradient de- scent are vulnerable to the inherent edge noise, yielding a severe robustness problem. However, none of the existing defending methods are designed for such a cou- pled noise, which is practical for real-world graph data that can be extremely noisy (Gallagher et al. 2008 Wu et al. 2022a).. Thus, it is urgent to understand the noise effect and devise the corresponding robust method.
0,90
ECN Cora
GCN ClN-Ceeeeer
CAT-Cora
GAT-Clbeseer
SAGt-Cora
SAGE-
Teat AUC
0.30
0.75
0.70
clean
C-2016
C-40%
C-60%
Figure 2: Performances (test AUC) with simulated edge noise. For fair evaluations, results are obtained by repeating 10 times with 4-layer GNNs on each dataset.
3.2 UNDERSTANDING THE IMPACT OF NOISE VIA REPRESENTATION DISTRIBUTION
Denote GNN as (:)(:) with learnable weights w. Node representations are extracted by forward pass as H = fw (A, X) while the backward propagation with stochastic gradient descent optimizes the GNN as by contrast When encountering noise within and Y. the representation H can be directly influenced, since the training neglects the adverse effect of data corruption. Besides, the GNN readouts the edge logit Φe,, based on top-layer node representations h; and h;, which are possibly degraded or even collapsed under the edge noise (Graf et al. 2021 Nguyen et al. 2022).
For quantitive and qualitative analysis of H under edge noise, we introduce two concepts (Wang) & Isola 2020) here, i.e. alignment and uniformity. Specifically, alignment is computed as the
For generality, we provide the full empirical study in Appendix D
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distance of representations between two randomly augmented graphs. It quantifies the stability of GNN when encountering edge noise in the testing phase. A higher alignment means the GNN is more resistant and invariant to input perturbations. Uniformity, measures the denoising effects of GNN when learning with edge noise in the training phase. A greater uniformity implies that the learned representations of various samples more uniformly distributed on the unit hypersphere, preserving as much information about the original data as possible.
the other hand, qualitatively
As shown in Table 1 and Figure 3 poorer alignment and a worse uniformity are brought by severer edge noise. The learned GNN (-) is more sensitive to input perturbations as the alignment values are sharply increased. and the learned edge representations tend be less uniformly distributed and gradually collapse to individual points as the noise ratio increases. Thereby, it is discovered that the graph representation severely collapsed under the inherent edge noise, resulting in the non- discriminative property. This is usually undesirable and potentially risky for downstream applications.
Table I: Mean values of aligmment on Cora dataset, that are calculated as the L2 distance of rep- resentations of two ran- domly perturbed graphs ie., N Σ H H2ll2 where Hi Jmn AA X) and H3- Jw A22. X).
positive samples
positive samples
positive samples
positive samples
96-
128
120
16 negative samples
negalive samples
negalive samples
negative samples
dataset Cora
Citeseer
clean
.616
445
(a) Clean data
(b) E 20%
(c) E 40%
(d)
60%
8 20%
.687
.866
g 40%
.695
.689
Figure 3: Uniformity distribution on Cora dataset. Here, we map the
representations of query edges in test set to unit circle of IR² followed by
E 60%
.732
.696
the Gaussian kernel density estimation, as in (Wang & Isola 2020).
4 ON TACKLING THE REPRESENTATION COLLAPSE VIA ROBUST GIB
As aforementioned, the inherent edge noise brings unique challenges to link-level graph learning. Without prior knowledge like the noise ratio, without the assistance of auxiliary datasets, and even without modifying GNN architectures, how can learned representations be resistant to the inherent noise? Here, formally build method to address this problem from the perspective of graph information bottleneck (Section 4.1 and design its two practical instantiations (Section and
4.1 THE PRINCIPLEE OF ROBUST GIB
Recall in Section 3.2 the graph representation H severely degraded due to incorporating noisy signals in A and Y. To robustify H. one can naturally utilize the information constraint based on the basic graph information bottleneck (GIB) principle (Wu et al. 2020 Yu et al. 2020), i.e., solving
min GIB - - -(H;Y), Ý), s.1. T(H; <.
(1)
where the hyper-parameter constrains the MI / [(H; A) to avoid H from capturing excess task- irrelevant features from Á. The basic GIB (Eqn. [ can effectively withstand the input perturba- tion (Wu al. 2020). However, is intrinsically susceptive label noise since it entirely preserves the label supervision with maximizing I (H; Y). Empirical results in Section show that becomes ineffective when learning with inherent edge noise simulated as Definition 3.1
The basic GIB decreases 1 Ā(²) by directly constraining / (H;; A) to handle the input noise, as illustrated in Figure ((b) Symmetrically, the label noise can be hidden in [(H; Y(Ā), but trivially constraining [(H; Y) to regularize I (H; ||A) is not ideal, since it will conffict with Eqn. I and also cannot tackle the noise within I(A; Y). Thus, it is crucial to further decouple the dependences among A, Y, and H. while the noise can exist in areas of T (H; YJĀ), T(H; AlY), and [(A; Y H)
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uniformity H (((H))
H(H)
A1
H
topology
Å
H
Y
Å
Y
Å
Z
H
Zy I(P:Z,)
A
2
1(A:Z,)
aligrmen (CMy Hq)
(a) RGIB (Eqn.
(b) RGIB-SSL (Eqn. 3
(c) RGIB-REP (Eqn.
Figure 4: Digrams of the proposed RGIB and its two instantiations (best viewed in color).
Analytically, (():(( H Y is constant and redundancy ((HHAY Y) can be easily minimized. Thus, the I [(A; ( can be approximated by the other three terms, ie., (H), (H;YY) [|A] and I I(H; AY) Since the two later terms are also with noise, a balance of these three informative terms can be a solution to the problem.
(),Y) where I (AA; Y
Based on the above analysis, we propose the Robust Graph Information Bottleneck (RGIB), new learning objective to balance informative signals regarding H, as illustrated in Figure 4(a) i.e.,
min RGIB - -1 ((HYY), s.t. TH < HH)) TH. ((H < mr, T(:: 74-
(2)
where constraints on H(H) encourage a diverse H to prevent representation collapse (>) and also limit its capacity (<nH)) to avoid over-fitting Another two symmetric terms, I (H; YIA) and T(H; AlL), mutually regularize posteriors to mitigate the negative impact of inherent noise on H.
Note that MI terms like ;H;; Al are usually intractable. Therefore, we introduce two practical implementations of RGIB, i.e., RGIB-SSI. and RGIB-REP based on different methodologies. The former explicitly optimizes the representation H with the self-supervise regularization, while the latter implicitly optimizes H by purifying the noisy A and Ý with the reparamcterization mechanism.
4.2 INSTANTIATING ROBUST GIB VIA SELF-SUPERVID LEARNING
Recall that the graph representation is deteriorated with the supervised learning paradigm. Naturally, modify it into self- supervied counterpart by explicitly regularizing the representation H (shown in Figure p to avoid collapse and to implicitly capture reliable relations among noisy edges, i.e.,
min RGIB-SSL. A
()) V)) >uu (H) (H2))
AaH H2)
(3)
supervision
uniformity
alignment
where margins As Aus An balance one supervised and two self-supervised regularization terms. When A₂ = 1 and A 0, the RGIB-SSL can be degenerated to the basic GIB.
Note that contrastive learning is a prevalent technique in the sulf-super area to learn robust repre- sentation (Chen et al. 2020), where contrasting pair of samples with data augmentation plays an essen- tial role. In practice we follow such manner and calculate the supervision term by B (H; Y [|) A.-MA where A,, A2 are sampled from P()). Similarly, we have E (HH(H)) ~ (H, + HHH))); and a higher uniformity leads to high entropy H (H) proved in Proposition 4.1 As for [(H; A
([T(H; = - (T (H,; 1/2(()))) VY) =))==))))))(((((()( (H1; Y Las (H2; Y)).
we approximate it by the alignment term I(H;; H2) with the derived upper bound in Proposition 4.2
Proposition 4.1. higher entropy H (H) leads a higher uniformity. Proof. See Appendix C.1
Proposition 4.2. The alignment [HH; H2) is the upper bound of I Ā(:; Since I(H; A(Y) S
T(H; A) < HH, H2) + [A1; A2)) = (/2(T(HH:)))) Hz) +c), a constrained alignment estimated by MI I(H;; H2) is capable bound lower I I(H: Aly" and I : Proof. See Appendix C.2
However, directly applying existing contrastive methods like (Chen et al. 2020 Khosla al. 2020 can be suboptimal, since they are not originally designed for graph data and neglect the internal correlation of topology and target Y. Two following designs are proposed to avoid trivial solutions.
Hybrid graph augmentation. To encourage more diverse views with lower [(A1; A2) and avoid manual selection of augmentation operations, we propose a hybrid augmentation method with four augmentation operations predefined candidates and ranges of their corresponding byper-parameters.
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In each training iteration, two augmentation operators, (.) and (-), and their 81 and Bg are automatically sampled from the search space. Then, two augmented graphs are obtained by applying the two operators on the original graph G. Namely, G' = (G)o,) and G = (G
Self-adversarial alignment & uniformity loss. With representations H1 and H2 from two aug- mented views, we build the alignment objective by minimizing the representation similarity of the positive pairs (hl kgy and maximizing that of the vandomly-samples negative pairs (h by: mmn)
Gij # Cmn- The proposed self-alversarial alignment loss is Ratign -i=1 R, + +R, R where (hig "RA"=
h2))) -1 h2g ll and R. = =p"ss(h) (hl) hmm)) 17- |m h (1)) Impor-
tantly, softmax functions p-) and (-) aim to mitigate the inefficiency problem (Sun al. 2019) that aligned pairs are not informative. Besides, with Gaussian potential kernel, the uniformity
loss is as Rumis = -Et, e -l1,,, A 1 +e || - li, where Cij,Cmn are randomly sampled.
Xe JeLassthRRottinn Beoliggtin Rounis- Remark 4.1. The inherent noise leads to class collapse, i.e., samples from the same class have the same representation. comes from trivially minimizing the noisy supervision (H; Y) and results in degraded representations shown in Section 3.2 Fortunately, the alignment and uniformity terms regularizing representations can alleviate such noise effects and avoid collapse (see Section 5)
Optimization. As Eqn. the overall loss function of RGIB-SSL
4.3 INSTANTIATING ROBUST GIB VIA 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. We propose RGIB-REP that explicitly models the reliability of Å and Y via latent variables to learn a noise-resistant H (as Figure 4(c).. namely,
min RGIB-REP & A, Zy) (ZAT A)+ Ay (ZZ;
supervision
topology construint
label constraint
where latent variables Zy and Z are clean signals extracted from noisy Y and A. Their complemen- tary parts Zy and 4 are considered as noise, satisfying Y Zy Zy and Ā= Z + Z When Zy = Y and Z A, the RGIB-RED can be degenerated to the basic GIB. Here, the [(H; Zy) measures the supervised signals with selected samples Zy, where the classifier takes ZA (a subgraph of Á) as input instead of the original Á, i.e., H= fw (A, X). Constraints T AZ A) and I(y;; Y) aim to select the cleanest and most task-relevant information from À and Y
Instantiation. For deriving tractable objective regarding Z and Zy. a parameterized sampler (+) sharing the same architecture and weights as (.)(.) is adopted here. (o(.) generates the probabilistic distribution of edges that include both À and Y by P= P=o(pooH HH E (0, 1)MP where hidden representations He = f (A, X). Then, the Bernoulli sampling is used obtain edges of high confidences, i.e., A = SAMP ()))) Zy = SAMP (Pl) where I S | and |Zy| s |
Optimization. A relaxation is then conducted on the three MI terms in Eqn. 4 With derived bounds in Proposition 4.3 (i.c., RA and Ry) and Proposition 4.4 (i.e., Lote), the final objective is formed as L = XeCois RA + Ry, and the corresponding guarantee is discussed in Theorem 4.5
Proposition 4.3. Given the edge number of Å, the marginal distribution of Z is (ZA) =
Σn Pos (PI)) n) P(n)) [[ P.j. Then, we derive the upper bound T(ZA; A) < E| (ZAIA)I(((ZA))) = Een Pij log Pst +(1 -Pj))))) L-Psi
I-7 RA, where T con-
stant. Similarly, the label constraint is bounded as T T(Zv; 8) S Ry, Proof. See Appendix C.3
Proposition 4.4. Since I(H; Zy) 2 Ezv Za blog P. /Zy Z = (fw AA), Zy), the super- vision term in Eqn. Aean be empirically reduced to the classification loss. Proof. See Appendix C.4
Theorem 4.5. Assume the noisy training data Derain (A, X, Y) contains a potentially clean
subset D
Z X, Zÿ) and Zi-- z based on which trained GNN predictor s (.) satisfies
Fall Z 4: X) Zz t.. The error is independent from Danh and e - O. Then, for any X Xa, dy E
10, 1), ZA Zí and Zy= Z& minimizes the RGIB-RR REP (Eqn. 4g. Proof. See Appendix C.S
exp(] [n], -h3, and pa n0s (1tg) 12
exp(a- 1bj 22 11
"Specifically, (h, h2?)) EN, exp([pi n 15
NN exp(a-
hb 22
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Table 2: Method comparison with GCN under inherent noise, i.e., both the input and label noise exist. The boldface numbers mean the best results, while the underlines indicate the second- bbstt results.
Cora
Citeseer
Puhmed
Facebook
Chameleon
Squirrel 40%
method
20%
40%
60%
20%
10%
60%
20%
10%
60%
20%
40%%
60%
20%
-00%
60%
20/8
60%
Standard
8111
.7199
.9770
.7864
77380
7085
8870
.8748
.864
.9529
.9500
99438
9616
3499
-9274
19432
9406
99386
DropEdge
8017
.7423
.7303
.7635
.7393
7094
.8711
.8182
8354
9811
.9822
99473
.9588
.9588
.4077
.9399
9377
9365
SpuralSpurse
8190
.7318
.7293
.7765
.7397
71488
.9088
.733
.6330
9825
9633
94456
.9599
.9997
.9402
.9194
.3099
.2977
PTDNei
8047
.7559
73888
.7795
7423
7263
8872
8733
8623
9725
9674
94855
9607
9514
9424
9485
9326
9364
Co-teaching
.8177
.7479
.7533
.7238
.9433
.760
9820
9450
.9955
.9516
.9611
9374
Peer loss
8185
.7468
.7018
7423
.7345
.7104
8961
8815
8566
9807
9536
9430
9543
9533
9267
99557
9345
9286
Jaccard
8143
.7498
.7024
.1473
.7324
8872
8803
8112
9794
9559
9428
9503
9533
9544
9443
9244
GIB
8198
.7485
77488
7509
73388
7121
8899
8729
8544
9773
9608
9911
9554
9561
9521
.9472
9329
9302
SupCon
.8200
.7819
.7490
.7554
.7458
.7299
8353
.718
8525
.5888
.5088
.9297
.9561
.931
9167
.9173
.9:48
.3011
GRACE
.7872
.6940
.929
.7632
.7242
6844
.922
8749
8558
8899
8866
.8315
.978
.8887
.949
.9394
9380
9763
RGIB-REP
8313
.7966
7591
7875
7519
7312
9017
8831
8652
9832
9770
9519
9723
9621
.519
.509
.455
.434
RGIB-SSL
89930
.8554
.664
J8427
9225
9829
9711
9655
9500
9425
5 EXPERIMENTS
Setup. In this section, we empirically verify the effectiveness of the proposed RGIB framework. popular datasets and 3 GNNs are taken in the experiments. The inherent edge noise is generated based on Definition. 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 the evaluation metric
as (Zhang & Chen, 2018 Zhu et al., 202T). The software framework is the Pytorch (Paszke eet al. 2017), while the hardware platform is single NVIDIA RTX 3090 GPU. We repeat all the experiments 10 times to obtain evaluation results with mean values and standard deviations.
Baselines. As existing robust methods separately deal with input noise label noise, both kinds of methods should be considered as baselines. For input noise, three sampling- bbased approaches used for comparison, i.e., DropEdge (Rong et al. 2020). NeuralSparse (Zheng et al. 20220) and PTDNet (Luo et al. 2021 Jaccard (Wu et al. 2019). GIB (Wu et al. (0200). VIB (Sun et al. 2022), and PRI (Yu et al. 2022), which are designed for pruning adversarial edges, are also included. Two generic methods are selected for label noise, i.e., Co-teaching (Han et al. 2018) and Peer loss (Liu & Guo, 2020). In addition, two contrastive learning methods are taken into comparison with adaptation to the link prediction task, including SupCon (Khosla et al., 2020) utilizing the full labels for supervision and GRACE (Zhu et al. 2020) that are optimized in a self-supervised manner without labels. All the above baseline methods are evaluated w.r.t. their original implementations
5.1
MAIN RESULTS
Performance comparison. As shown in Table 2 the RGIB achieves the best results in all 6 datasets under the inherent edge noise with various noise ratios, especially the more challenging datasets, i.e., Cora and Citeseer, where a 12.9% AUC promotion can be gained compared with the second-best methods. When it comes to decoupled noise settings shown in Table 3, RGIB also surpasses all the baselines ad hoc for input noise or label noise by a large margin.
Remark 5.1. The two instantiations of RGIB can be generalized to different scenarios with their own priority w.r.t. intrinsic graph properties that can be complementary to each other with flexible options in practice. Basically, the RGIB-SSL is more adaptive to sparser graphs, e.g., Cora and Citeseer, where the inherent edge noise presents a considerable challenge and results in greater performance degradation. The RGIB-REP can be more suitable for denser graphs, e.g., Facebook and Chameleon. Meanwhile, two instantiations also work effectively on heterogeneous graphs with low homophily.
The learned representation distribution. Next, we justify that the proposed method can effectively alleviate the representation collapse. Compared with the standard training, both RGIB-REP and RGIB-SSL bring significant improvements for the alignment with much lower values, as in Table 4 At the same time, the uniformity of learned representation is also enhanced: it can be seen from Figure [3) that the various query edges tend to be more uniformly distributed on the unit circle. As RGIB-SS SSL explicitly constrains the representation, its recovery power on representation distribution is naturally stronger than RGIB- REP, resulting in comparably better alignment and uniformity measures.
Full evaluation results can he found in Appendix

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Under review as a conference paper at ICLR 2023
Table Method comparison with GCN under decoupled input noise (upper) or label noise (lower).
Cora
Citeseer
Puhmed
Facehook 40%%
Chameleon 40%
Squirrel
inpus noise
20%
40%
20%
40%
205%
2015
20%
2055
60%
Standard
8027
.7855
7749)
8054
.7708
7583
8854
.87.9
86651
.9819
9966
.9622
9608
9933
99368
.9416
9395
9411
DropEdge
8338
.7826
8025
.7730
.6622
.9803
.9567
9432
.9126
.3588
SeuralSparse
8534
.7794
.7637
8093
77899
7468
8931
8720
.6499
9712
9691
.983
9609
9550
9348
.9699
9403
9417
PTDNet
8433
.8214
.7770
81199
-7811
76.38
.903
8776
8609
9725
9668
9493
.610
9457
9360
9469
9379
Co-teachin
8045
.7871
7550
8059
77753
7668
8951
.772
8606
9712
9717
99714
9524
9446
99447
9962
94255
9306
Peer loss
8051
.7865
.7517
8106
.7367
7653
.9177
8811
8643
9758
9703
9662
99558
9482
9412
.9362
9386
.3336
Jaceard
8200
77838
8176
77776
9784
9507
9988
GIB
8002
.8099
7741
8070
.7717
7798
8932
8808
8618
.996
.647
9650
9605
9521
9416
.9390
9-106
9397
SupCon
.3449
.8301
.0776
.7767
8867
9647
9600
.305
GRACE
.7877
.7107
6975
76155
.1511
68.0
8810
8795
8593
9015
8833
83995
8994
9007
.964
9392
93378
99633
RGIB-REP
8624
.313
8158
8299
7956
9008
8922
8687
9933
.9723
9682
9705
.6644
9480
.495
.9432
9405
RGIB- SSL
9024
8577
8421
8747
8461
.245
.9126
8889
8693
9821
9707
9668
9658
95570
.4886
9479
.4229
.9429
label noise
Cora
Citeseer
Puhmed
Facebook
Chameleon
Squirrel
20%
40%
20%
40%
20%
20%
40%
20%
40%
600
20%
60%
Standard
8281
8054
8066
.7965
77850
76599
.9050
.4139
9077U
.982
9980
99966
96666
99580
99622
.9720
9720
.9100
DropEdge
8363
.8273
.7937
.7853
9313
.9201
9673
.950
-9579
.9988
NeuralSparse
8524
.8246
.8211
.7688
.7211
.7522
.9722
9136
9089
.9881
.988
.9784
9583
9953
.9511
9933
9626
.6255
PTDNet
8460
.8214
.8138
.7968
.7765
92219
90999
.879
9783
99855
9976
96665
9626
Co-teaching
8446
.82099
8157
.7974
7877
7913
9335
9291
9319
9762
9797
96688
9642
9650
9533
9675
9641
9655
Poer loss
8325
.8036
.8069
.7991
7990
7751
9126
9101
9210
9769
9750
9734
9631
9501
9569
9636
9694
9696
Jaccard
.8289
.8064
.8148
.8011
.7887
.6889
.9988
.9355
.9096
.9022
.9255
.9758
9603
9659
.9557
.9529
.5112
.9501
GIB
.3377
81.37
8157
7986
7852
7649
9037
9114
9064
9742
9703
9771
9651
9582
9489
9641
9628
9600
SupCon
.8191
.8275
.7533
9131
9108
9617
.984
.9516
.911
GRACE
.8311
.8237
8193
7909
7630
7737
9234
9252
9255
8913
8972
8587
9053
9074
9075
9171
9174
9166
RGIB-REP
8564
.3188
8083
77866
9357
9132
9984
9889
.9785
.9738
.9737
RGIB-SSL
9314
.9234
9241
9284
9218
9250
9594
96004
.913
9857
9881
9857
9730
9752
.744
9727
9729
9726
Table 5: Comparison different schedulers. Here, SSL/REP short for BELI--SL/RGIDEE
Experiments are performed with a 4-layer GAT and c = 40% inherent edge noise.
datasct method
Cora
Citeseer SSI.
Pubmed
Facebook
Chameleon SST.
Squirrel SSI.
SSI.
REP
REP
SSI.
RFP
SSI.
RFP
RFP
RFP
constant
8398
.7927
.827
.7742
8896
.8416
.9694
.9778
.9884
.9498
.9293
.9320
linear()
8427
.7653
.1677
.7559
.8645
.8239
.9669
.9299
.434
.9369
.9316
.9265
sin(-)
.8466
.7924
8132
.7680
8637
.8275
.9685
.9944
.9477
.934
.9325
.9282
cosl:
8334
.7833
8088
.7647
.8599
.8372
.9477
9629
.9388
9444
.9251
.9033
expl
.8811
.7815
.0855
.7569
.8617
.877
.9471
.9633
.9022
.4188
.9166
.9299
positive samples
positive samples
positive samples
Table 4: Alignment comparison. Here, std. is short for standard training, and SSI/REP are short for RGTB- SSL/RGIB-RED respectively.
negative samples
negative samples
negative samples
dataset
Cora
Citeseer std. REP SSL
method
std.
REP SSL
4
clean
.616
.524 .475 445 .439 .418
8 20%
.687 _642 .543
5586
.533 ,505
40%
.655
.679
.578
6889
.623 ,533
13
60%
7322 .704
.615
.96666666 .647 .542
(a) Standard
(b) RGIB-REP (c) RGIB-SSL
Figure 5: Uniformity distribution on Citeseer with s s=
Optimization schedulers. To reduce the search cost of coefficients in objectives of RGIB-SSL and RGIB-REP, we setup a unified optimization framework that can be formed as f aLads + (1 - a) + (1 R22 Here, we attempt 5 different schedulers to tune the only hyper-parameter E 1], including (1) constant, a = c; (2) linear, a; = t, where / is the normalized time step; (3) sine, a₂ = sin(t- 7/2): (4) cosine, a, = cos(t- -/2); and (5) exponential, a; c As empirical results summarized in Table the selection of optimization schedulers greatly influences the final results. Although there is no gold scheduler that consistently performs the best, the constant and sine schedulers are generally better than others among the 5 above candidates. A further parameer study with grid search of the As in RGIB-SSL. and RGIB-REP can be found in Appendix F2
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Table 6: Ablation study for RGIB-SSL and RGIB-REP with 4-layer SAGE. Here, == 60% indicates the 60% coupled inherent noise, while the calcy represent ratios of decoupled input/label noise.
Cora ca = 60%
Chameleon fa 00%
variani
60%
Fy = 60%
60%
60%
RGIB-SSL (full)
.8996
8730
.894
.9633
.9788
.962
w/o hybrid augmentation
81.50 (5. 196.)
8604 (1.4%)))
.857 (2.6%)L) 95288
(1.3%%.)
.9466 (0.1%- 9695 (0.6%6_)
wio advesaalvllliiall
.410 (2.19:L
.705 (0.29)L
.927 (0.7%)L
99655 (0.1%)
.9322 (0.2%.) ()
.976 (0.1%))
w/o supervision (A, =
7480 (12.9%])
(7810 (10.5%)
77820 (13.0%6)
8626 (10.7% L)
8628 (1 (11.5%%
8812 (12.8%
w/o alignment (A,, 0)
8194 (4.6%)L)
.8510 (2.5%)L)
.8461 (5.95%.)
.9613 (0.5%
.9749 (0.1%%)
.9722 (0.4%-L
w/o uniformity (A, = 0)
8355 (2.8%))
.8621 (1.2%))
.8878 (1.3%))
.9652 (0.198.))
.9700 (0.4%)
.9751 (0.1%))
RGIB REP (full)
.7611
.4887
8095
.9567
.906
.9766
who edge selection , A
.7115 (1.2%)
.199 (3.3%.L)
.7800 (2.5%:)
9554(011%.)
.904 (0.1%.)
.661 (0.1%-)
wro label selection (2y Y)
7533 (1.0%
83733 (1.3%L)
.7847 (3.0%)
9484008%%
9666 (0.4%)
.994 (0.8%)
w/o topology constraint A
7355 (3. (3.3%)
.7999 (9. (9.2%)))
.7969 (1 (1.5%)))
9503(0.66..)
9658 (0.4%-.
9635 (0.45%.
w/o label constraint (Xy =))
.7881 (3.09%))
.1066 (4.4%%L)
.032 (0.7%%.)
(944((122%.)
9665 (0.4%.)
9669 (0.1%%)
5.2 ABLATION STUDY
In this part, we conduct a thorough ablation study for the RGIB framework. As shown in Table 6 each component contributes to the final performance. For RGIB-SSL, we have the following analysis:
Hybrid augmentation. RGIB-SSL benefits from the hybrid augmentation algorithm that automati- cally generates graphs of high diversity for contrast. Compared with fixed augmentation, the hybrid augmentation brings consistent improvements with 3.0% average AUC promotion on Cora.
aeeraral alignment loss. Randomly-sample pairs are with hierarchical information to be learned from, and the proposed re-weighting technique further enhances high-quality pairs and decreases low-qu low-quality counterparts. It enables to discriminate the more informative contrasting pairs and thus refines the alignment signal for optimization, bringing to up 2.1% AUC promotion.
Information constraints. Label supervision contributes the most among the three informative terms, even with label noise. Degenerating RGIB-SSL to a pure self-supervised manner without supervision (i.e., A = 0) leads to an average 11.9% AUC drop. Meanwhile, we show that three regularization terms can be jointly optimized while another two terms are also of significant values.
As for RGIB-REP, its sample selection mechanism and corresponding constraints are also essential:
Edge label selection. Two sample selection methods are near equally important for learning with the coupled inherent noise that both informative sources are required to be purified. Besides, the edge selection is more important for tackling the decoupled input noise, greater drop will come when removed. Similarly, the label selection plays dominant role in handling the label noise.
Topological label constraint. Table 6 also shows that the selection mechanism should be regularized by the related constraint; and otherwise, sub-optimal solutions will be achieved. Besides, the topological constraint is generally more sensitive than the label constraint for RGIB-REP.
6 CONCLUSION
In this work, we study the problem of link prediction with the inherent edge noise and reveal that the graph representation is severely collapsed under such a coupled 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 instantiation of RGIB, the self-supervised learning technique and data reparametrization mechanism are utilized to establish the RGIB-SSI and EGIB-REP respectively. Empirical studies 6 datasets and 3 GNNs verify the denoising effect of the proposed RGIB under different noisy scenarios. In future work, we will generalize RGIB to more scenarios in the link prediction domain, e.g., multi-hop reasoning knowledge graphs with multi-relational edges that characterize more diverse and complex patterns.

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REPRODUCIBILITY STATEMENT
To ensure the reproducibility of the empirical result in this work, we have stated all the details about our experiments corresponding contents of Section and Section [ The implementation details are further elaborated in Appendix Besides, we will provide an anonymous Github repository for the source codes during the discussion phase for the reviewers of ICLR 2023.
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Appendix
Table of Contents
Broader impact
14
B Notations
14
C Theoretical justification
15
C.I Proof for Proposition 4.1
15
Proof for Proposition 4.2
15
Prool for Proposition 4.3
Proof for Proposition
Proof for Theorem 4.5
16
D A Further Empirical Study of Noise Effects
17
D.1 Full evaluation results)
17
D.2 Further investigation on the decoupled ................................
18
Decoupling imput noise and label noise
D.4
further discussion on the inherent edge noise
E
Implementation details
20
E.1
GNNs for link prediction
20
E.2
Implementation details of RGIB SSL
................................
21
E.3
Baseline implementations
22
Full empirical results
F.1
Robust methods comparison
................................
22
72
Further ablation studies the trade-off parameters
................................
22
F3
Further case studies
24
F4
Full baseline comparison with decoupled noise
28
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A BROADER IMPACT
In this work, we propose the RGIB framework for denoising the inherent 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-invariant and label-invariant graph 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.
A similar problem has been noticed in an early work (Gallagher et al. 2008), 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 (Wu et al.) 2022b), the inherent noise can be produced in the data generation process that requires manually annotating the data. As introduced in (Wu et al. 2022b), the inherent noise here refers to irreducible noises in graph structures, attributes, and labels. In short, the studied edge noise in our work in line with the generic concept of graph inherent noise in the literature, which is recognized problem in practice but also an under-explored challenge in both academic and industrial scenes.
common
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 inherent noise these common benchmarks. Thus, to fill this gap, it is necessary simulate the inherent edge noise properly to investigate the impact of noise. If the inherent edge noise exists, should be false positíve 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 inherent noise and design the simulation approach in Definition 3.T as it is more close the real-world scenarios.
Besides, to our best knowledge, we are the first to study the robustness problem of link prediction under the inherent edge noise. One of our major contributions this research problem is that reveal that the inherent noise can bring a 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 bidirectional information bottleneck that strictly treats the information source on both input side and label side is helpful in practice, especially for extremely noisy scenarios.
B NOTATIONS
We summarize the frequently used notations in Table 7
Table 7: The most frequently used notations in this paper.
notations
meanings
V [
the set nodes
8- (-(ei)) M 11/11
the set of edges
AE {0, NXN
the adjacent matrix with binary elements the node features
X X
9 G-(A,X)
the input graph of a GNN the query edges to be predicted the labels of the query edges representation of query edge eij representation of all the query edges
K Equery = (eig 11/=1
Y
hij
H
I(X;Y)
the mutual information of X and Y
I(X;YIZ)
the conditional mutual information of X and Y when observing Z
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C
THEORETICAL JUSTIFICATION
C.I PROOF FOR PROPOSITION 4.1
4.2 the uniformity loss on two augmented graphs is formed as hi t 1l, For simplicity, we can reduce it to one graph, i.e. Runif = mnn.me [e- le-1A, his Amn [.]. Next, we obtain its upper bound by the following derivation.
Proof. As introduced in Section
Rumis Σk Lij.mn E ||b]] mm te
log Runis log Bij. mn [e- |hi Amnnli
=log Eig mmnppe [e 2tt Amn -2
Biisupe log Emnope [e 2 Nmmu
N
log
20-
(5)
N
N
log (DEFE + log ZYMF
1) A (HH) log ZYMP
Here, (DD (-) is von Mises-Fisher (vMF) kernel density estimation based on N training samples, and ZMMP is the vMF normalization constant. (-) is the resubstitution entropy estimator of H = (-) (Ahmad & Lin 1976). As the uniformity loss Lunif can be approximated by entropy H (H), higher entropy H H(H) indicates lower uniformity loss, i.e., higher uniformity.
C.2 PROOF FOR PROPOSITION 4.2
Proof. Recall the hybrid graph augmentation technique is adopted RGIB-SSL for contrasting, we can approximate the expectation (H A| by 1/N E I(H;; A.l)) with N augmented graphs. Based on this, we drive the upper bound of A(Y) as follows.
N T(H; AlP) T(H; 1/N Σ (T(:;; Á.))
1=1
(1) + (HH;;
(H(HH(H)++HAA) + HHH2) ((A)) H(H, An HHH2, A2))
(V)(H(())++ H(H2) H(Aa) - H(H, H²) - H(Á,,
-1/2(I(HH; Ho) ())
=1/2(I(HH H2) +c)
(6)
Thus, lower T T(H1; H2) upper bounded as lower T(H; Ā) and T(H; A
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C.3 PROOF FOR PROPOSITION 4.3
Proof. With the marginal distribution ((Za)=Ex -P P.j. drive the upper bound of MI I(Z:; A) as:
1A)
I(ZA: A) = ZA,A [log(
)]
A(Zz)
(o((ZAlA) -L(F(ZA)I((ZZ))
EE [log(
Q (Z)
<E|KL(P (Za|A)]lQ(( (Za))]
Ptj Poylog log
+ (1 P.j) log 1- Pij RA 1-T
Σ ayE
where the KL divergence on two given distribution P(x) and (2)) is defined as (L( = Σx P(x) P())/()) Thus, we obtain the upper bound of T T(2 A) as Eqn. Similarly, the label constraint is bounded as T(Zr; Y) S Exyer Pis log Py + P.ş) log 1- T- Ry.
C.4 PROOF FOR PROPOSITION 4.4
Proof. We derive the lower bound of T T(H; Zy) as follows.
I(H; 2y) [I A)) Zy)
Pe Zr|Zza)
ZBzy.Z, 2, flog
P(Zx)
-Rzy 2, Olgg IP,. /Zy (Zy| [A) -lg(((())
Ezy. 2, 2loog P... Zr [a) + H(Zr)]
>Ezy. Z. [log w/ Zy [2a]]
Σ
Lats (far(fu ZA), Zv)
|Zy
ExjEZy
Thus, the MI T(H: Zy) be lower bounded by the classification loss, and min - ~ Zy in
RGIB-REP (Eqn. 4 is upper bounded by min A./12y Σesezy Lete AA), Zy) as Eqn. 8
C.5 PROOF FOR THEOREM 4.5
Proof. With the relaxation via parametrization in C.4 we first relax the standard RGIB-REP its upper bound as follows.
min RGIB-REI REP A X,I(H; AAI(ZA: + IZxx))
X.T(ZA; Zy) + Ax Á) Ay
As Z ~ Y, Eqn. can be reduced to min - X, T + T< + H()) where - Y C Y. Removing the final term with constant H(Y), can be further reduced min min- Å), where Aa/X,- Since minimize the -T(ZA; Y) AT Á)
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is equal maximize the I(Za:Y) AZ A), next, conduct the following derivation:
max -7(:)))
max - -M
mnax (X: ( ((A
-max
-1 -1 -
(10)
mmax
- -
= mmxx11 -Ajc 2)c- 111--1-- - (1))))(A - -
=(1 A)c
Since MI I(A;Y|ZA) 2 0 and (1ZA:A() 2 0 are always true, the optimal Z should makes -(1- (Zz AT A()) 0 to reach the optimal case. Thus, it should
satisfy I(A:Y\ZA) - and A(Z:; A(Y) simultaneously. Therefore, Za Za maxi- mizes ( -AT(ZA; which is equal minimize -T(ZA; AT(ZA; Å), i.e., the
RGIB-REP. Symmetrically, when ZA = A, Zy Z maximizes [[A; Zy) XI(Zy; Y) as (y (|)).
[(A; Zy) - AI(Zy; = (1- (); A) - (1
D A FURTHER EMPIRICAL STUDY OF NOISE EFFECTS
This section is the extension of Section 3 in the main content.
D.1 FULL EVALUATION RESULTS
Evaluation settings. In this part, we provide thorough empirical study traversing all combinations of following settings from different aspects.
3 GNN architectures: GCN, GAT, SAGE.
3 numbers of layers: 2,
4 noise types: clean, mixed noise, input noise, label noise.
3 noise ratios: 20%, 40%, 60%.
6 datasets: Cora, CiteSeer, PubMed, Facebook, Chameleon, Squirrel.
Then, summarize the entire evaluation results of three kinds of GNNs follows. As be all three common GNNs, including GCN, GAT, and SAGE, are vulnerable to the inherent edge noise.
Table full evaluation results with GCN.
Table full evaluation results with GAT.
Table 15
full evaluation results with SAGE.
Data statistics. The statistics of the datasets our experiments are shown in Table can be seen, 4 homogeneous graphs (Cora, CiteSeer, PubMed, and Facebook) are with the much higher homophily values than the other heterogeneous graphs (Chameleon and Squirrel).
Table 8: Dataset statisties.
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
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D.2 FURTHER INVESTIGATION ON THE DECOUPLED NOISE
Loss distribution We visualize the loss distribution under different scenarios to further investigate the memorization effect of GNNs. shown Figure two clusters are gradually separated apart with clean data, but such learning process can be slowed down when training with the 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 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 the decreasing trend of the corresponding losses, where the loss distribution of noisy edges is minimized and progressively moving towards that of the clean ones.
Denstt
10
opcch-100
Ratio=D4, spoch-100
opoch-100
RatiowD cpach=100
positivc
One Desee
posttvve
dean
clean noay
nogative
negative
Clears spoch+500
RatioodA, spach-500
Clean, spoch-500
Ralic=O 4 spoch+550
Dersity
10
posttive
posttive
dean
dlean
negative
Clean, epoch=1ona
Retlom0.4, epoch-1n22
Clean, spach- 1900
RahiomD epachel
Density
10
Density
10
positive nogative
posttive
deaa nozy
nagative
Seen
Trairing Loss
Trairing Loss
Training Loss
Training Loss
Training loss of positive/nagative samples with (b) Training loss of clean/noisy samples with clean data (left) and 40% label-noisy data (right).
clean data (left) and 40% input-noisy data (right).
Figure Loss distribution of the standard GCN with 40% input noise (a) and 40% label noise (b).
Visualization of edge representations As shown in Figure 7 the dimensionality reduction technique. T-SNE, is utilized here to measure the similarity of the learned edge representations h; = [/, h, with edges Gig in the test set gteat, Compared with the representation distribution of clean data (Fig- ure ((a)). which present comparatively distinct cluster boundaries, both the input noise (Figure (b)) and the label noise (Figure [cc) 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 give discriminate the true or false edges with their representations.
30
postiive
postive
20-
20
negative
20-
Compenent 2
Companent 2
-10-
-10
-10
-30
-30
-30
-30
-20
-10
10
-20
-10
20
Component
1
Component
Component
(a) Clean data
(b) Input noise
(c) Label noise
Figure 7: Visualization of the representations of predictive edges through T-SNE. The edge represen- tations are learned on the Cora dataset by a 4-layer GCN with the standard supervised learning.
D.3 DECOUPLING INPUT NOISE AND LABEL NOISE
For a further and deeper study of the coupled noise, we use the edge homophily metric to quantify the distribution of edges. Specifically, the homophily value hohooo of the edge Eij is computed as the
cosine similarity of the node feature z; and øj. i.c., hhomo = cos(z,; (z; As shown in figure g the distributions of edge homophily are nearly the same for label noise and structure noise, where the
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envelope of the two distributions almost overlapping. Here, justify that the randomly split label noise and structure noise are indeed coupled together.
noisy intuss
noisy labels clean labels
nolsy labels
Denstyy
30-
cleon imputs
Denstty
20-
Density
30
noisy inputs
20-
-0
10-
Edge homophily
Edge homophily
Edge homophily
(a) Homophily distributions of the
(b) Homophily distributions of the
(c) Homophily distributions of the
input edges
output edges
noisy edges
Figure 8: 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.c., (case input noise with high homophily and label noise with low homophily shown in Figure g and (case 2) input noise with low homophily and label noise with high homophily shown Figure 10
nolsy putu clean
15
noisy labels clean labels
nolsy labels inputs
Deralty
Denatty
10-
Density
20-
0.0
0.2
0.4
0.6
08
La
0.0
02
0.6
0.6
1.0
2.0
0.2
0.4
0.6
0.8
1.0
Edge hamaphily
Edge homophily
Edge homophily
(a) Homophily distributions of the input edges
(b) Homophily distributions of the output edges
(c) Homophily distributions of the noisy edges
Figure 9: Distributions of edge homophily for decoupled noise (case 1).
noisy inputs clean inputs
noisy labe clean labels
labels noisy inputs
7.5
Denaty
20-
Denstry
10-
10-
2.5
0.0
0.2
0.4
9.6
2.8
1.
0.0
0.2
0.6
0.8
1.0
0.0
0.8
0.8
Edge homophily
Edge homogh
Edge homophily
(a) Homophily distributions the input edges
(b) Homophily distributions of the output edges
(c) Homophily distributions of the noisy edges
Figure 10: Distributions of edge homophily for decoupled noise (case 2).
D.4 A FURTHER DISCUSSION ON THE INHERENT EDGE NOISE
Basically, the edge noise can exist in the form of false positive edges or false negative noise. Specifi- cally, the false positive edges are treated existing edges with label 1, but in fact, such edges do not exist. On the other hand, the false negative edges are treated as non-existing edges with label 0 that the predictive probabilities of such edges will be minimized.
Here, we would highlight that our work focuses on the false positive edges as is more practical and common in real-world scenarios since the data annotating procedure can produce such a kind of noise (Wu et al. 20226). Thus, if the inherent edge noise exists, it is more likely to false positive samples, while the false negative samples often intractable be collected and annotated
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in practice.
Actually, investigating the influences of the false negative samples is another line of research, such as (Yang et al., [020 Kamigaito & Hayashi 2022), which is orthogonal and complementary to our work. Learning from a clean graph can also encounter the problem of false negative samples, which 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 under-explored.
The input noise and label noise does look quite similar and coupled. From the perspective of data processing, the collected noisy edges in the training set can be randomly split into the observed graph (i.c., 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, c.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 fundamentally different.
Although both noise can bring severe degradation empirical performances, they actually act in different places from the model perspective. As the learnable weights are updated as w = w v CH,, Y), the label noise Y acting on the backward propagation can directly influence the model. By contrast, the input noise indirectly acts on the learned weights as it appears in the front end of the forward inference, i.e., H = [A, X).
Empirically, results shown in Table 3 and Table 4, the standard GNN (without any defenses) per- forms quite differently under the same proportion of input noise and label noise. Such phenomenon can inspire one understand the intrinsic denoising mechanism memorization effects of the GNN, and we would leave that as further work.
More importantly, from the perspective of defending, it could be easy and trivial to defend if these two kinds of noise are the same. However, none of the existing robust methods can effectively defend such an inherently coupled noise. As can be seen from Table 2, only marginal improvements are achieved when applying the existing robust methods to the coupled noise. While in Table and Table 4, these robust methods work effectively in handling the decoupled noise, i.e., only the structure noise or label noise exists. The reason is that, the properties of coupled noise are much more complex than the single decoupled 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.
E IMPLEMENTATION DETAILS
E GNNS FOR LINK PREDICTION
We provide detailed introduction forward propagation and backward update of GNNs in this part.
Formally, let = 1... L denote the layer index, h is the representation of the node i, MESS(.) (-) is a learnable mapping function to transformer the input feature, AGGREGATE(() captures the 1-hop information from neighborhood. ('(v) in the graph, and COMBINE(-) is final combination between neighbour features and the node itself. Then, the (-layer operation of GNNs can be formulated as m = (ABREEGATEE(MMESSSS) - 16-1,, , :u E (v)))), where the representation of node h COMBINE' (h' .m.))
After L-layer propagation, the final node representations he of each e V are obtained. Then, for each query edge egy Etrain unseen from the input graph, the logit de., is computed with the node representations h and h with the readout function, i.e., de.,, = READOUT(h, h -
Finally, the optimization objective can be defined as minimizing the binary cross-entropy loss, i.e.,
min Lets = Eey Extrasn - Yk (o )) - (1 - Y/i log [1 - o ((ee,)) where a(-) is the sigmoid function, and Vij = for positive edges while yı =0 for negative ones.
In addtion, we summarize the detailed architectures of different GNNs in the following Table 9
"To avoid abusing notations, we use the h, to stand for the final representation h in later contents.
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Table 9: Detailed architectures of different GNNs.
GNN
MESS(.) AGGREGATE(()
COMBINE(.)
READOUT(-)
GCN
m = W E'ENO Va,d; 1
h1 = o(m) + w (i
Das = 11 h
GAT
m ΣSEN() Oij W
h! a(m) W'aah h = o(m)) w
DEs h h
SAGE
m W [OOT ΣSEN(O) h/
Doss :11 he
E.2
IMPLEMENTATION DETAILS OF RGIB-SSL
As introduced in Section 4.2 the graph augmentation technique is 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
The search space is summarized in Table [00 where the candidate operations cover most augmentation approaches except for those operations modifying the number of nodes that unsuitable for the link prediction task.
In each training iteration, two augmentation operators (.) and T2(-) and their hyper-parameters 0, and 02 are randomly sampled from the search space elaborated in Algorithm. [ The two operators will be performed on the observed graph G. obtaining the two augmented graphs, namely, G TT and T 02). The edge representations are gained by h - f( enylw) h O hr,. where the node representations h? and h? are generated by the GNN model f-/lw) with learnable weights w. and so it is for h with g².
Table 10: Search space Hr of the hybrid graph augmentation.
Operator T((-)
hyper- syper-parameter 0r:
description
edge removing
Oer (0.0, 0.5)
Randomly remove the observed edges with prob. der. Masking the columns of node features with proh. 8im- Dropping the elements node features with prob. 0ya. not modify anything.
feature masking
0im E (0.0, 0.3)
feature dropping
0 (0.0, 0.3)
identiity
NA.
Algorithm 1 Hybrid graph augmentation.
Require: number of augmentation operators
1: initialize 0 + +.))
2: for 1=...72 do
3: randomly select one augmentation operator from operator space Ti(-) E Hyi
4: uniformly sample the corresponding hyper-parameter" 0, U(fr));
5: store the newly sampled operator and combine it with the existing ones U (T((1))) +
T(-):
6: end for
7: return the hybrid augmentation operators T(-)
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 as shown Figure []] Note the h. here is the edge representation. Specifically, minimize the representation similarity of the positive pairs (hl h2y)) and maximize the representation similarity of the randomly-sampled negative pairs (hl h han). where egj + Cmn-
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g² = ='=
g (G103)
Latent space H
9
Observed graph
Augmented graph
Augmented graph
00 h
doos:
h
ameg
h
= h|on]
Kj
nj
Figure 11: Illustration of the RGIB-SSL model.
E.3 BASELINE IMPLEMENTATIONS
All baselines compared in this paper are based on their own original implementations. We list their source links here.
NeuralSpansc,ntttss:::1
GCN http:////hooo//aaaaaocoo////////
GIB, http:p/s/echhhbboccoo//ssssasaat VIB, [ttp:://ehhha..co/////ss55544110
Peer loss functions.
Loss-iitioonn
SupCon, GRACE, GRACE, http://ppppgggaaao..o//aa/aaaaaa
F FULL EMPIRICAL RESULTS
In this section, we elaborate the full empirical study on inherent edge noise with various robust methods and GNN architectures.
F.1 ROBUST METHODS COMPARISON WITH CLEAN DATA
Here, we would like to figure out how the robust methods introduced in Section 5 behave when learning with clean data, i.e., no edge noise exists. As shown in Table [T] the proposed two instantiations RGIB can also boost the predicting performance when learning on clean graphs, and outperforms other baselines in most cases.
F.2 FURTHER ABLATION STUDIES THE TRADE-OFF PARAMETERS À
We conduct ablation study with the grid search of several hyper-parameters in RGIB. For simplicity, we fix the weight of the supervision signal as one, i.e., A = 1. Then, the objective of RGIB can be formed as L Lats + R1 + Rg where the information regularization terms R,/RR R2 are alignment and uniformity for RGIB-SSL, while topology constraint and label constraint for RGIB-RI -RP.. respectively. As the heatmaps illustrated in Figure 12 and Figure 13 the Å1, )a
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Table 11:
Method comparison with a 4-layer GCN trained on the clean data. The boldface numbers mean the best results, while the underlines indicate the second- -eett results.
method
Cora
Citeseer
Pubmed
Facebook
Chameleon
Squirrel
Standard
.6886
.8317
.1788
.9700
.9888
.925
DgopEdge
.6884
.8444
.944
.9669
.9000
.9629
NeuralSparse
.8715
8405
.966
.8655
9803
.6355
PTDNet
.8577
.8398
.9315
.9688
.9966
.9640
Co-teaching
.6684
.3877
9192
9771
.698
9626
Peer loss
.8313
.7742
.0855
.8911
.974
.4222
Jaccard
8413
8005
.831
9792
9703
.6100
GIB
.8882
.3277
90199
.691
9628
.9655
SupCon
.8599
8003
9131
9692
.9117
.9619
GRACE
8329
8236
9358
8953
8999
9165
RGIB-REP
.8558
8415
.4088
.9755
.792
.6800
RGIB-SSL
.2260
.1488
.9533
.845
.740
.9666
better in certain ranges. Neither too large nor too small value is not guaranteed to find good solution.
0
0.86
0.0
0.84
0.86
0.01
0.82
0.3
0.1
0.60
0.85
0.5
1.0
0.76
0.74
0.7
0.83
100
0.70
1.0
0.01
0.1
1.0
10
100
1.0
Au
Au
(a) Grid search in 40, 10
10-1
100 100, 10
(b) Grid search in f0, 0.1, 0.9.
1.0,
Figure 12: Grid search of hyper-parameters with RGIB-SSL Cora dataset (e 40%).
0.80
0.0
0.80
0
0.1
0.76
0.01
0.76
0.3
0.76
0.74
0.1
0.74
0.5
1.0
0.72
0.70
0.68
0.70
0.8
100
0.64
1.0
0.01
0.1
1.0
10
100
0.0 0.1
0.2
0.3
0.5
0.7
0.8
0.9
1.0
Av
Ay
(a) Grid search in (0.
-0 10
10"h
10
10")
(b) Grid search in 0.. 0.2
0.9, 1.03
Figure 13: Grid search of hyper- with RGIB-REP Cora dataset (c 40%).
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F.3 FURTHER CASE STUDIES
Alignment and uniformity of baseline methods. The alignment of other methods summarized in Table 12 while the uniformity is visualized in Figure and Figure T5 We have the following three observations.
First, the sampling- based methods, e.g., DropEdge, PTDNet, and NeuralSparse, can also promote alignment and uniformity due their sampling mechanisms 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 be degenerated to single points 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 a conclusion line with the previous study (Wang& & Isola 2020).
Table 12: Alignment comparison.
Cora
Citeseer
method
clean
20%
40%
60%
clean
20%
s 40%
60%
Standard
.616
.687
.695
.732
.455
.586
.689
.696
DropEdge
.606
.670
.712
.740
.463
.557
.603
.637
NeuralSparse
.561
.620
.691
.692
.455
.469
.583
.594
PTDNNet
.335
.427
.490
.522
.330
.397
.459
.499
Co-teaching
.570
.691
.670
.693
.464
.581
.683
.706
Peer loss
.424
.690
.722
.629
.529
.576
.580
.598
Jaccard
.608
.627
.658
.703
.471
.566
.674
.703
GIB
.522
.635
.652
.692
.439
.524
.591
.623
SupCon
.060
.045
.033
.024
.086
.077
.060
.065
GRACE
.466
.488
.522
.556
.454
.551
.589
.609
RGIB-REP
.544
.642
.679
.704
.439
.533
.623
.647
RGIB-SSI.
.455
.543
.578
.615
.418
.505
.533
.542
Learning curves of RGIB. We draw the learning curves of RGIB with constant schedulers in Figure 16 Figure 17 Figure 18 and Figure 19 We normalize the values each plotted line to (0, 1) for better visualization.
For RGIB-SSL, the uniformity term, i., (H),, converges quickly and remains low after 200 epochs. Similarly, the alignment term, i.e, [(H; H2) also converges in the early stages and keeps stable in the rest. At the same time, the supervised signal, i.e, [(H; Y) gradually and steadily decreases as the training time moves forward. The learning processes generally stable across different datasets.
As for RGIB- REP, observe that the topology [ZZ 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 from 0 to 60%, these two regularizations react adaptively the noisy data A, Y. Such phenomenon shows that RGIB-I -RP with these two information constraints works as an effective information bottleneck to filter out the noisy signals.
more noisy case. That is, when noise ratio increases
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positive sampless
pozitive samples
poskive sampliss
positive samples
12
12.
32-
22-
76
70
95
36
66
16
64
128
64
126
32
96
126
regative
22
20
128
228
94
128
24
148
standard
(b) Dropedge
(c) Neural Sparse
(d) PTDNet
positive sa mpmlle
positive samples
passive samples
posil ove samplees
32
12
52-
16-
36-
94-
36
120
320
126
126
32-
32-
46
54-
05-
56-
12
64
56
122
3
32
64
720
12
64
120
22
61
66
130
co-teaching
(f)
peer loss
(g) Jaccard
(h) GIB
positive malpses
positive samples
poskive samples
positiessmppae
12-
44-
56-
122
122
120
nioga Live samplss
negative samples
reeaavve samples
regative sampli
32
12
50
56-
05-
50
52
64
76
125
32
+4
95
:25
22
64
1ce
22
56
128
(i) SupCon
GRACE
(k) RGGB-RRE
(I) RGIB-SSS
Figure 14: Uniformity on Cora with 40% inherent noise.
25

---

Under review as a conference paper at ICLR 2023
positive sa samplees
pozitive samples
poskive samplics
positive samples
12
22
32-
22-
70
70
95
36
66
16
64
128
64
126
96
126
regative
22
20
228
94
128
24
148
standard
(b) Dropedge
(c) Neural Sparse
(d) PTDNet
positive sa memlle
positive samples
passive samples
posil ove samplees
32
12
52-
16-
36-
94-
36
120
320
126
126
32-
32-
46
54-
05-
50-
12
64
56
122
3
32
64
720
12
64
120
22
61
66
130
co-teaching
(f)
peer loss
(g) Jaccard
(h) GIB
positive mampses
sostiiessamples
poskive samples
positiessmppae
12
44-
06-
122
122
120
nioga Live sampls kss
negative samples
reeaavve samples
regative sampli
32
12
56
05-
50
52
64
76
125
32
:25
22
64
Lce
22
56
128
(i) SupCon
GRACE
(k) RRGI-RRE
(I) RGIB-SSS
Figure 15: Uniformity on Citeseer with = 40% inherent noise.
26

---

Under review as a conference paper at ICLR 2023
0.-
24.
28.
38-
Igrment reprartatinn
0.4
of Symbiyy arbastoo
24-
wnferrrtty regud riasiin
Fa requbeitution
0.0
34
200
20-
000
(a) clean
(b) E 20%
(c) s = =00
(d) 6 60%
Figure 16: Learning curves of RGIB-SSI. on Cora dataset.
1.0
Lbel conubrant
Lbbl camutrale:
constrint
0.2
5.0
3.6
topeboy
0.2
22
02-
02-
220
400
000
500
1000
202
400
500
502
3000
200
400
500
500
:000
200
400
+32
500
1000
Eouch
Eparch
Eproch
(a) clean
(b) e 20%
(c) == 40%
(d) 60%
Figure 17: Learning curves of RGIB-REP on Cora dataset.
0.2
02-
02-
gresent rey And De0i
58g Tpan regi beiovton
50g Toaee beloninn
Lyrreer (og cniDattee
0.4
bart raga ataction
0.0
Faozh
(a) clean
(b) 20%
(c) 40%
(d) 60%
Figure 18: Learning curves of RGIB-SSL on Citeseer dataset.
03-
coserwer
stal convrriie
0.-
2.4
0.6
0.2
22
22
-oodl byy Car 500001
puperiiee Ices
32
220
600
1000
203
600
833
3000
200
100
833
1000
200
432
800
1003
Eooch
Esoch
Epoch
(a) clean
(b) == =20%
(c) = 40%
(d) 6= 60%
Figure 19: Learning curves of RGIB-REP on Citeseer dataset.
27

---

Under review as a conference paper at ICLR 2023
F.4 FULL BASELINE COMPARISON WITH DECOUPLED NOISE
The entire evaluation with 10 baselines and two proposed methods are conducted keeping the same settings in D.I
Results on each dataset are summarized as follows.
Table 16
Table
Table
18
full results of GCN/GAT/SAGE on Cora dataset.
Table 99
Table
20
Table
full results of GCNNGAAT/SAGE on CiteSeer dataset.
Table 22
Table 23
Table 24
full results of GCNIGAT//AGG on PubMed dataset.
Table 25
Table 26
Table 27
full results of GCN/GAT//AGG Facebook dataset.
Table 28
Table 29
Table
30
full results of GCNGGAT/SAGE Chameleon dataset.
Table 31
Table
32
Table
33
full results of GCNGGAT/SAGE on Squirrel dataset.
Table 13: Full results of GCN with edge noise.
kyerr
elean
20%
40%
60%
20%
input
40%
50%
20%
40%
60%
Com
9183
8870 0108
0145
79992_0190
8047
9020=
8733
8686 0102
0213
119702205577
8027 0580
7856-t 0526
7490 0498
X281
0133
8054=
0213
8060:
0172
82562 0494
7940:20033
774292
0226
71777.0007
5035
0370
77933 61
77566:1 0270
8249
0110
7925
0137
7901
0145
Citsseer
8968 0092
8651
1.004
8355-1.0011 1.554
8254
$767:1 0097 86151-0090
8585 0085
8834
0155
3619
0101
3674:1.012 0122
8317
7864
.7380
3054--
7753-4-0225
7850-
0174
88611 01663
.7110::.222
7720:1-0101
721124:0470
7900=
.7618:5.0229
Pubmed
1737
9419
(271 0007
9590
9468 0012
0337
0637
9597::
9178
8870
88748220011
8641
2.044
18759:
8651
9030=
9039
0070=:
.9081 Lt: 0055
0856
8751
86366600036
8355 0025
87422 0029
8662 0041
9050
0083
9112- 0159
9063:2
0033
Facebook
988771 00000
.9880 0007
.966611.000 0007
9943-1.0010
9878-1.000 0008 9852-1 0006
9834-1-0011
9892-.000
0006
9888
9888:1.00 0007
9870 0009
9829 0000
95200 0424
9638.1.000 0402
9819:1 0015 9668-1 0147
9622-1..113 01.51
9882
0007
9880
0007
9886-: 0006
.9899
979884-00113
01338
+1179
9764-1-0034
0009
9409-1-0160
9863
9865-
0012
9916-=.0010
Chameleun
98233: 0027
9753 0013
0015
96961.
0002
9657
0009
9784:1 0017 976211 0016
9754
0023
0775_0011
9769
0018
9755: 0036
1997
0011
9496-2 0190
9274
96081
9431:1 n261
9368-
91622
.9752 0036
9662:
0042
9511
9286 0067
9656
0450
0229
9752= 0027
9766
0035
9745:
Squirel
.9761
9610---0000
9722-4.000
.9725 20011
9432-1.0006
990611
(001
193611.0025
9416:1.0042
9395 0011
110101.0000
7720-00016
2220-00016
9710:1.000
-9694 0028
3484-5-0049
94229- 0008
10099
9489-6-0057
.408 0021
0002
9688- 0028
9675-
0027
9656:: 0034
28

---

Under review as a conference paper at ICLR 2023
Table 14: Full results of GAT with edge noise.
kyyr
elean
20%
mixed acise 40%
60%
20%
input nobse 40%
60%
20%
lubel noise
40%
60%
Com
0026 0070
8786-
0155
8439
8286- 0120
5014
0070
1525 0097
8800 0077
8991 = 0088
8841
IM19
-880=_0000
8783= 01KE
8323 +
0181
8000 4212
7841 0095
2616
0107
34544 0118
8377H 0146
8530 0121
8357
0082
8281
8650
.7032 0452
0127
7801
8376
0069
Chiseer
8911 0090
8596120125
8358-200177
8271.0121 36891 0096 8526. 0130 85124:0074
8762 0076
8650=
0102
8648
0166
.2775--0248
.2518--0183
0092
7798
8299 = 0098
7807
.2773..0270
.7139:1.02 0251
7970.1.131 7860.1
0107
.77111.0126
7963
0129
7883
0162
7801
7801-0100 0161
Puhmeed
9406
9173 0008
8954
8884
9155: 0024 9176-t 9100: 0000
93066-0038
9271
0232-0034
89600=20000 m066
8610
0045
8454
8539
0048
8668 0040
8547:t 0050)
8476
0037
8817=
8172-
866--000 0062
8631 0072
8315
0059
3176
80402
0092
8374
0036 8801 0321
5067.1
0306
8480 0088
0414
8313=0077 0071
Facebook
9876 0008
9869
0005
9836-1.000
0008
9836 0007 9878.1
0006
9861:1
0006
9858.1 0008
9872
0006
9864--0000
0009
9857
0008
9875
2012
99220+-000
0014
5955-4
0011
97734
0046
9875
0010
9974-
1:0025
9577
0006
9851
9860
9770
9725
0017
9650-2.0018
9625
0018
9767.1 0026 9747: 9759.t
0018
9746-0023
9743=0011
9721
96522-003
9605
9741
0028
9738
0027
99442 0005
9699
0029
9573-2.0066
9482
0064
9644 0053
.9543- 0075
94744 0074
9122 0043
9688
0055
9698
0065
Squirel
9940-1.001 0000001110000
9680
0007
9635 00017
.988
9702-1
0008
9690:1
0010
9639:1.000
0011
9719=00018
0018
9701=0011
9686-000
0012
99200 0023
9581
1:
0046
9456-1.0061
9335 0062
9992:1 0047
9455:1 0075
94151
0061
9682:
0000
9690
0028
9686_ 0021
1.97888.000 0007
.507
.930-1-0164
.925 0089
9487 0065
-419:--0000
.9255
0073
.9520
0070
9507-.0102
Table 15: Full results of SAGE with edge noise.
kayer
elean
20%
mixed moise 40%
60%
30%
input noise
40%
60%
20%
libel noise
40%
60%
Com
9045 |006
18733- 0101
8820--.0177
8409 0086
5006 0074
3857-t 0093
89174 0063
8868
0070
8695
0106
8564-10000 0091
8664 6009
8225 +
0009
.783
0093
.7595
(229
3007 0110
1437-- 0099
8387- 0183
8309 =
0090
8046
0074
7920
0191
8426
7787
0423
0222
.005-10421
-707-
0235
0130=00344
Chesserr
8648 0998
8438
844120013
82792
5644 0109 3608:: 0064
86472 0161
8560= 0120
8533=
8412=0122
29141101011
26961.0213
.75391.0119
81711
8226.1
0203
806R=0116
-7705-0172
8390 01K7
7708 0168
7223
.72011 0236
30861
0145
7997:1
0183
7877:1 01:4
7903=
7690- 0201
7564=0099
8095
9136--0002
90944.005
0005
0025
9578-t
0023
8817=00144
8793=00045
8446 m058
8827 (027)400006
8063+-0001
.86194.003
8715
0080
2901-
0082
.0033
0067
8386
0085
.3300-0104
3228
0092
8360= 0224
8314220088
(105
8333
52344
0335
8838c: 0105
85664:
0199
8242
0104
8161=
0071
8190=0129 0129
Facebook
0882 0008
9858 0008
0827.20008
0788 2012
9881:1
0008 986111 0007
996t:1
0008
9867
0006
0840
0006
9825= 0014
9859 0013
9821-1.0015
1.025
.998 1198-1.000
9819.1
0015
9815.1 0024
98155
0025
9866
0015
9817-0000
9809-00009
961611.0991
951711
0228
9761=
9768_00045
Chamcleon
9786 0000
9663 0093
96015--00088
9536 0028
97544t 0019 97344 976624
0018
9700==0011
9670=-0026 9603=0099
9645
3580+5.005
.905
0010
5921- 0017 57214
9721-
0016
99299 0022
9606-1-0046
.9511 +-008
.95704
0005
9679-1.001 0051 9908- 0011
.9440-1
0059
9673-
9641
00038
9625-00022
Squirel
9745 0015
9680-1
0015
9626-1.0015
9570-1.0000
9737.1
0010 9736:1
9731
0011
9692=0005
9643
0012
96066-0017
0689 0052
9584
0107
95774-0076
95411.0001
9637.1 0092
96302:
0079
960711
0090
1663__002)
9612- 00449
9612_00020
9682-1 0045
0555-1.0065
9528-1.0000
9961
9592-1 0053
9600:1 0036
9551 0012
9574
9540=_0207
9583=.0025
29

---

Under review as a conference paper at ICLR 2023
Table 16: Full results on Cora dataset with GCN.
GCN
layers
method
20%
mixsed noise
40%
00%
20%
Inper nolse 40%
60%
20%
label nouse
40%
00%
Standard
8870:1.0108
8:1011.0145
7959:..011 0190
9091 12 0085
.941311
0065
.8905 1: 0099
.9020
0011
8793:2.009
.872::1.0133
Droglidge
87734-0128
8805 0175
9016- 0006
3092--
.6635 +:
9016
8791
2060- 1095
SearaiSparse
5369-1.0000 0096
-215-=01555
0158
9118-
39003
00.13
.849 9086
8999
0009
8797:900073
5543-4
PTDNet
8853110127
84511.0068
8180:10100
0106
9095
0108
8926.1 0106
8825 9096
9000 0080
8801 1 0067
.8702-1.0477
Co teachang
8887- 0202
.8499+ .119
8026-
0257
9084
0132
3969-- 0107
8901
+ 0101
90021
+ 0200
3929
01.26
3099 0115
L=2
Peer
.867
8912-1.0190
0115
.8472 0237
8-611
7970-1
7964:
0162
9107
9016:1
8806
0138
9015
10941.0110
1 0021
8856
8874
.0133
0093
S7211 0135
2813 0291
GBB
8857--10250 1226
5057 6300
9126- 0065
29733+-0141
0148
8955 0100
9003 + 0136
2828 .0135
2807---010 01446
SupCon
8827 5827:1.0125
.::1 0157
8125 0119
9006
0074
3577:1 0073
5846 6: 0124
8876 1: 0082
8668 H: 0088
.721
0092
GRACEE
8588.-00155
8220 0204
7987=
8102+
8048
0092
2987
3396-
1074
RGIB-
8915 0091
.816++01330
-258--0077
9137
0060
2959-4 0088
.8911 +: 0100
9066
9066+00100
8958
0020
3935- 0087
RGIB-SSL
.2722 0091
10044
5892-1.0000 0107
0049 9312
9108
1 0063
.9014
0007
9473
1 0054
9427
942715.0051
9280
0084
Standand
8111110013
74191.0035
6970 0377
8027
0580
7786611 0526
..490 0498
R281 +: 0133
8054
0213
5060:1.000
0172
Droglidge
8017
.7423+ .0335
7303
0235
8338
- 0132
77226 + 0377
.7544
0425
8363
+ 0110
8273
0105
-1.48 0141
SearalSparse
8190:1.010
.33881.0399
.339
7293 0393
8534
1 0250
7794.1 0285
7657
1. 0529
K524
11 0094
8226110011
8211 0106
PTDNet
8047-1.040
.7599
0216
8133
82144 0122
.770 0624
8460 0128
8214
813811015
0151
Co-teaching
8199-1-0026
.2479+-.0377
00255
8055
- 0609
2751 0064
22530 0550
8446- 00219
8209 H 00881
2157-- 0246
T=4
Peer loss
Juccad
8185-1.0022 0026
0211-100218
7498:1.0418
7.58811.0388
7018:10013 0413
7024:
8051
8200
1 0664
0772
7838:1
7566
1 0623
0858
.7517
7617
0099
K325
K289
1 0177
1 0201
8036 0227
8064
- 0229
S1481 0227
5069 1. 0193
GIB
$198.4 0331
1458
2002
0607
3099-1
0566
.7241
+ 0050
8337
+ 0123
8137
8157- 0270
SupCon
8240:1.011 01:47
78191.0221
7490--0033
8309
0 0124
X301
1 0218
R025
5. 0210
8491
1: 0120
82755..111
X256:1.0100 0108
GRACE
7872+ 0207
6940
0248
6929 0040
7977= 0211
7107
0188
.6975
+ 0114
8331
+ 0106
M23T 0252
8193+ 0246
RGIB AGIB-REP
83134
0098
7966
0110
7591--01-2
0112
8624
X313-4
.8158
0199
8854
0149
8318
2018880015
8297-1.053
RGIB-SSL
0072
0000000000000
9021
3577.1
.8121 0156
9314
9224
924111 0019
Standind
7040:1.0133
7429 0226
7177=
8035=
7973:1
7546 0270
K249 0112
7925
7901 0145
Dioe
7041
8130
6009 0208
7900=
0560
74RR 0187
.170 0187
8358
+ 0148
8187 0076
TW90: 0513
NeuralSpurse
790111.000
73244
7018:10033
8082
0418
7432.11 0086
.3333 0243
8415 1 0167
81601.0181
5081 0101110015
PTDNet
0166
7601 0225
7167=
0158 8129
7873:1 0199
7570
0511
K348 0144
0558 7933
7899 0466
teaching
.920-1-0132
.7387+.0244
0244
3151 0262
8079-00499
0170
.7623
+ 0310
8296
1 0065
8020
7976:1 0375
I-6
Peer loss
lucard
8015:1.011
8000.1.0150 01.59
2966.1.0300
7211 0250
0286
8199_0036
0361
504
8001110016
-1 0204
7607
7600
1: 0272
0364
8409 1: 0116
K277 0138
.79991..033
8005 0729
0289
7918:1 0240
7004 0230
GIB
796111.010
.7544
23000
8036
0445
5067
.7639
1. 0402
8347
1. 0111
8016:1.0156
79477 0147
SupCun
8090-1.02-2
.73651
0227
6920-0015
0251
7845:1
7734
8273
0002
8161
8157-1
GRACE
7576:1 01-48
71872
6860 0213
7693
0161
7171 0223
6896 2 0272
8309 0347
8134 0310
3102:1 0234
RGIB-REP
8100 01137
.::39.1.0221
0221
0010:10012
8282
0123
7857
0142
7623
1 0144
8365
1: 0163
8247
82:0011011 0119
RGIB-SSL
862311-0126
.80001.0240
0240
7357 0342
8632 0187
7878.1
0368
7310 0483
9184
0070
9120220108
21 0081
Table 17: Full results on Cora dataset with GAT.
GAT
layers
method
2006
mixed nouse
input noise 40%
60%
20%
label nouse
40%
60%
Stankrdd
$786.1.01555
8489:1.0131
82266.1 0120
9014:2.0070
$825.1 0097
8880 0077
8991 1: 0088
S841 0097
8830:1.010 0101
Dropkidge
.841-4-0114
-2794-9112
0144
3720-4-000
5517-4-0196
NeuralSpurse
1820-1.0134
8447110011
32.881.0112
90511..007
0007
3384.1.00
0121
8828
0067
8982 1: 0060
8855
0110
5761:1.0129
PIDNet
X799-t-0152
8487
2314: 0075
9023-2 0105
8838-
0116
8827
+ 0090
9019 + 0086
N772 0082
S726- 0060
Co leactcinng
8883-4-0148
.8571
8338-+.0160 0169
9069
0134
3396-1
9289
0221
.1000 0358
3969
0243
loss
8367-1.0139
.862
0206
8343-1.01
9026
0135
01.69
8073
9057
1 0212
881811.019
8918
Jaccard
8809-4-0190 0180
0164
833442.0122
0124
9066
0142
3823-+
0081
90005
+ 0215
9010
6092
8919
01.86
8929
0114
GIB
8826-1 8826-1-0199
8864
+ 0218
3375 9- 0294
9260
0330
9092-4
0061
9162
6- 0213
5000
4 0128
8925
0 0195
8915 0098
SupCon
8709 1-019
8962
1 0021
8300 0132
8957
0115
8827
0131
8805
1116
8881
1 0065
.73011.015
9652 0144
GRACE
8286-402224
7764
0129
7328-+.021
7615- 0979
7309
0297
+ 0100
2805
0125
S807
RGIB-REP
8759-1.0132
8374
8239-1.0134
9006-
3835--0079 0079
8798
4 0013
8993
0086
A833
0115
5810 0125
RGIB-SSL
9142 0092
8858
87774
9234 0053
0067 8973
8806
2 0144
0053 9389
9347
0058
9311 0057
Standurd
8323-4-188
8005 0212
7841
0095
86162.
0107
8454 0118
8377
1 0146
8530
0121
8357
0082
8381 =1 0114
Droplidge
4237-4-0157
.7782
00772
0072
35154.0083
0050
0144 8508
.208
0090
1 0015
8516 1 0100
8373
17735-.00 00142
8374 0157
8309-1.0154
.7544
7769.1.01
0135
8575
8450
0110
1 0148
0123
$3951.0122
83481
PTDNet
83641:0147
8045 0096
7890:1
0081
8669
0130
8445
8851
1 0146
850T
0113
8399
0099
8170-1 0117
Co teaching
829411-0222
.8001 H: 03300
7895 0177
8696 0196
8344
0163
8823
1 0150
8534
11 0144
8374
4. .0313
$541 0169
L-4
Peer loss
83441-0276
.8067 12 0.254
7933
0142
8520
0202
835311
0145
8376
1. 6
8533
1 0021
5827 0075
S111 0203
Jaccard
8319-400277
2932-. 0107
3634-
01.3
SS49 1112
84064-000 0144
8537
+ 1220
24024-011
-014
-354+0015
GIB
8352.1.0067
8811 00244
7945
0279
8860-1.000
0099
3579
0404
8501
2 0000
8434-1
53-0--.013 0130
SupCon
8324110102
8033 -. 0099
77761.01 01
86201
0162
S441 0098
8337
1 0386
8514
0108
8381 0092
831811012
0121
GRACE
.700 4-44
.6711
+
6656-1
0578
3707
0267
31544
6966
0291
8021
0230
RGIB-REP
8274-100153
.7882
3552-4.00 0657
8652
8370
8110
81541.0014
8180
0181
8332
$259 0166
RGIB-SSL
8760:1 0112
8669
11001
8304:1.01 0176
8865 0125
8553
0127
8349
1 0030
9163
1 0080
9075
0087
0087 9006
Standand
8067.4 0102
.7514
0389
7037:1: 0452
84142
0127
7
0537
2678
4 0641
8376-1
0087
8154
0009
8074
0169
Droplidge
8051:1.0111
.7375
7110011 0388
0132
7514.1
0615
7248
1 0511
8199
1 0135
8312 00663
8112:1 0079
NeuraSpursee
8169-1-01300 17266:0016
0596 7149-1
84632.0183
7997
0149
7273
1. 0992
B100
0113
8257
0123
$149
0224
PTDNer
8007:1 .7460 0711
71452
82532
7799
8464
8254
0116
Co-teaching
805911.0312
175761.0038
.3070-1.455 0455
7968-10000
05
2717
1 069
8549
1 0298
8204
1.0884
8115 0389
L-6
Peer loss
Iaceard
8138-1-0336
8155-1 0400
.7537
.757211.411 0411
71232 0454
707001.052 0521
8455-10214
8495
0142
7918
70472
0547
0661
7679
7767
1 0737
0771
8403
8374 0060
1 0229
8318 0197
83451 0067
$049 0224
$108 0180
GIB
$188-1.0056
1.509
201111.0112
8152-1.030
$009.1
.7923
0718
0137
.22011.0100
5020-100222
SupCom
7356:1 0629
6644
0607 61151 0455
7535 0856
7102 0655
6241
1: 0433
8083
1 0568
8040
0384
7869 0392
GRACE
$748-1-0659
.5499
5611
3675
07788
5537
6 8607
5632
H 0583
5588
-: 0692
6176- 0961
RGIB-REP
8148-1.0155
.78331.0179
6812-1 0264
8106-1.113
0129
8001
0178
7433
1 0663
8366
L 0107
8274
1. 0151
8192-1.11 0179
RGIB-SSL
861311.0107
8194
7858.1
86572.0115
$2131
8045
9049
8960
3985
30

---

Under review as a conference paper at ICLR 2023
Table 18: Full results on Cora dataset with SAGE.
SAGE
layers
method
mised moise
20%
inyur moise
40%
60%
20%
lahel noise
Suaddan
47014-0101
8520
8469
9006
0074
5857::
89174:
0063
8968
0070
8695 0106
8564: 0991
DiogEdge
834420000
8600+-008
8478=0095
9013=0000 0090
5026-t
0086
90050+ 8084
8985
0079
8725-20000
5699
NearalSparee
88214-0123
8570+-00000
=491=.0155
9020-
0090
25300
0087
89449 0098
8885
4 0111
M681
10110
2638-20000
FTDNer
886022010 0109
85%-2-0101
8474 009
9040
0101
80254
0085
89474
0070
8002
+ 0006
8721
+ 0068
85862 0075
Co-teaching
8794
8569+_0141
8557=0000
8910=
0099
3891
8955
0123
4 0216
0291
86324
L=2
Peer loss
Iaccard
88174 0007
88284 0170
88134-0115
8480=-0055 0058
8474 0081
9042
9080
0099
0209
2073 0210
3366:1
01
8897
8960
0045
0064
8922 0112
8876
0111
8695-
8768
0251
0183
8739:1 01:49
5596:1 0113
GiB
8766 0196
86794.0125
8546= 0246
9089
0252
5004:
0098
8998
+ 0544
8958
0000
8727
0101
X6144 0125
SupCon
88441-0117
85071
8499 0076
8974
8004:
8943
8916
0004
8742
0092
56011 0093
GRACE
8123 0113
7978+
8000
7944=0017
8087
5010
0160
7963 0120
8250
8228
5361 0102
RGIB-REP
8748 0094
x380==0102
9016=
8876: 0075
89144220093
8863 007%
8628
0088
2449 0017
RGIR-SSL
91024 0074
.8967200114
099320070
9196 0072
9059:1
0090
0084 9062
0073 9278
9174-2
0089
91632 0076
Stundand
8225
.783340009
7595= 0259
8607=
0110
8437:
0999
8387
0183
8309
0000
8046
0074
7920 0191
DrogEdge
0144
29444..033
7842=0049
8734
3685:
8622
0146
8466 0111
81992.011
X125
0059
NearniSparse
0157 29900+.0108
7573=00128
8703
0108
3549---013
01:45
.3594+:00
0061
8418
+ 0148
8025
0148
2069- 0106
PTDNer
01:0:20149
7147
7900 0177
8626
5607: 0090
8650 0154
8435
0012
8125
0117
7061 0231
Co-teaching
0137 7815
7658=0544
8660= 0168
X533- 0149
8424
0196
8550 0545
8117=
01157
2088+ 0275
L=d
Peer loss
8200+-0007
.7863-
0174
0294
X334-
.8451
0024
8363
+ 0142
11244-.0191
-9344-00331
laccard
0151
2850+-0115
7602-0048
8619
0255
3465 0227
84% 0233
8378
0006
8046 00094
7976- 0288
GBB
8286:2
79474-0130
3729 0578
8844
0280
2457
8227
85744
0115
7947- 0229
SupCon
82052 0141
78094.0176
7383 6218
8568
0115
$450:
0153
8445 0187
8476
0106
8150
0170
7041 0129
GRACE
6342.4 0745
64244
6711=
6465=
61724
64%6 0544
6434
0384
63762
0251
6438 0449
RGIB-REP REP
8274+.011
78224.0143
7092-0202
8634-00121
8420
0144
8367
0:44
MONT
0187
7991
RGIR-SSI.
8837 0065
88782 0116
8613
0148
8960 0109
55174
19
8825 0113
9150
0018
9041
= 0075
0072 0023
Standard
0423
74204
0251
7180
0248
8256
0222
7947
0561
8005
0421
8158
0168
7707 0255
7660:
0134
Drogkidge
8055 0228
23398 0560
7176
0389
8262 0153
8193-
0679
8089 + (220
8340 0161
7993
0091
77897
D144
NeuralSparse
39534-0177
.1378+-.180
0233--0238
0020
52344
.7800+
82144 0107
27988
H 0033
21622
0160
PTDNer
7999:20051 26644-.0199
7352= 0202
8311=00145
33674
0078
8110+=00100
8222 0121
78255 00778
7744-
Co-teaching
7877-4-0477
-2545+..8512
72122=0532
8306= 0256
3991 0695
8000 0433
X524 0256
.7720 01265
77687- 0266
L=6
Peer loss
3781 .7445
7277
8300 0254
3020
0624
5043 0449
K309
+ 0149
.7734
H 0338
76622 0262
0437
.2393-4.0145
7277=
3073-
8017 04944
8148 0186
.7707
+ .124
0204
GIB
7814+-0993
244334.0442
7339=0037
8366--0194
3106- 0689
.8040+0000 0617
8172 0255
.7006 0226
7689-
0190
SupCon
.7599110055 0056
201911.0285
6673:10017
8219-
0389
3618
1 0666
7159:1.0717
8242 1: 01559
7880--0155
.786611.0
0148
GRACE
4806-t-010 0455
0248
014
6536- 0365
66144 0394
7239 0231
2035
0160
7014
D111
RGIB REP
8149--00166
27157+500725
2099-00473
0473
8391
3149
77227
8358
0100
22974
0140
2046-
00355
RGIB-SSI.
8662- 0130
8470
0108
8746 0091
36341 0099
8603 2 0156
8983 0089
89302 0108
5040:
0076
Table 19: Full results on Citeseer dataset with GCN.
CCN
layers
method
201
mixed moise
60%
20%
inpus moise 40%
60%
20%
label noise
Standard
18661 1.651+
8855-+.0154
8254- 0016
8767= 0097
561
8858- 0085
88344
0155
8629 0104
3674-t 0122
Dropkidge
8613---0112
8112=00158
0117
3557-4-000
.84334-000
0164
8802
3695-2.133
0133
X689-1.01 0143
NearalSparse
.8085
.8.224.0138
8239- 0069
3801
0085
8634-1.00
0009
.8614:1.00 0014
8527
1 0094
8753--0111
.5720-1.0150
00156
PTDNer
8646+ 0155
2204+-0011
8223 0142
8805
0995
3647 0112
8571
0116
8813 0074
8734 0126
5508 0697
Co-teaching
8089-4
.8166 0215
8278-
3029-1
.8699
8833
81144
0081
2847-10094
0094
L=2
Peer loss
Jaccard
80822 0164
87384 0159
8874+-0182
8308-
5292=
01.7
8783
8725
0080
8090
35745 0186
3065 0155
8606 015%%
8638
0131
8849
8009
0144
.231
8718
8653
0100
0276
5308:1 0212
X680 0154
GBB
8825 1 0119
3476- 0331
8306 0156
8819
0381
20000 0318
8688 + 0136
8914 0253
3681
00128
3707-4.00
SupCon
8344
0101
0153
8140 0121
8501
3335
0186
8436
0128
8337
506020011 0131
GRACE
8450+
0123
.2225+00100
7808=0008
8529
0137
77677
0171
897
807000100
8646
0122
3699- 0156
REPB-REP
8885 0088
5117=-0113
8751
0074
2637--0012
0122
8600
0119
8795
0101
8638
86388.00175
3044- 0095
RGIR-SSL.
91994 0001
.8957
8759= 0087
9271
0855
5018: 0090
8942 0145
9495
0 0059
0515 0049
04802
0062
Stundard
0144
23300
0201
7085
0146
8054=
0240
3708 0325
77583
+ 0258
77065
+ 0124
7850
0174
7669- 0240
Droplidge
76354-0000
.3393- .170
3094-
.7573
0016
.797
0015
2853-
36332
0142
NearalSparse
7765 1.0123
01666
7148 0237
8093
7509
0160
.7468
1: 0289
7968
10119
7721
1.10
.152-1 0209
PTDNer
.7955-0031
77425+-.1200
7783==0130
9119
- 0105
7811 0191
.7658 0017
2206 + 0014
.7765-2
.0124
76222-1 0215
Co teaching
7333 7533-4-0181
.7238
0245
7131=.0157 0157
8059
- 0263
3753-- 0408
.7668
+ 0212
.9914 0031
.7877 (0291
7913- 1943
Peer loss
0215
.7345+
7104=
0242
8106
37672
.6633
0515
7991
0734
.7991
(228
.751 0245
Juccard
7473+-0010
23324
8204
7107=0072
8176-
0285
3776-
0471
.7725
0432
8001
01187
.2887-
.19%
7689-1
GIB
.509-1-0030 .036
0210
9020- 0298
5717-4-0112
.7788
9 8411
7996
0120
7882-4.0186
0275
SupCon
7554+ 0196
7458
7299 0218
8016= 0999
7767 0111
7655
- 0164
8024 0108
7993
0012
7707 0166
GRACE
76324
722424
0219
6844-
7615-
J151
.880+
0232
7509
.2630-20014
37372 0907
RGIB-
3575
35194.0181
2222-00227
$299--0134
0134
3996
1996--00330
.7771
0178
.083
+-0152
.7846
0034
7945- 0203
RGIB-SS1.
80944 0108
84272-0174
8137 0170
8747 0144
3461
+ 0109
8245
.0142
9204 0085
9218 0099
9250-2 0071
Standard
4087
.2100+.0222
7106 0168
12200-
0161
3400
0401
.7212 04700
25000 0011
.7241
00157
76688 0229
Droglidge
7683 06634
..1919.01599
7006-10000 0504
7645
1 0641
5442
0160
.7209
H: 0012
8007
4: 00152
7925
1- 0044
3656
:: 0196
SeuralSparse
76411-0180
.731 2.011
7036-10585
7913
0152
7476
0279
.7391
1.0255
.793 1 0391
.7739
0188
7668
0256
PIDNet
30614.0153
3072- 0385
7982--012
0034
-
00114
.7464+
01888
.7913 0122
.7742+
00555
7382++ 0151
Co teaching
739511.0168
.71915.0099
7188 0248
7813-- 0185
.350
0475
.7310 .496
8004 0017
.7885
02275
.7701
1 0245
I-6
Peer loss
7396. 10661..0080
7415-4-0444
.71024 0030
7105-1.0255
7135 0111
7166:
0155
75121
74459
0488
.750
.7309
1 0009
.44/3
7915 0169
79225
+ 0176
.7768
.7825
9 01255
0299
76995 1. 0239
17:10- 0298
7370-1.0531
.7226 0268
0234
2435
.722 1 0697
.7276
1 .054
7945 1:71
.7803
0221
7681
0251
SupCon
.714:2.0122
74154-8166
7205=
1907=
7543 0109
.7670 00183
.745
+ 0231
.6659
.014
7573 0180
GRACE
8995 0222
6901
1 0210
6946- 0152
7061
- 0260
2007
0247
6815
0109
7752
+ 0220
.7339
2 0151
3773- 0351
RGIB- REP
7725 0077
.429
0202
0110
7781
0211
7559
1
77415 0211
7900
0252
079711.018]
.786
0194
RGIB-SSL
84174-0010
279555.0148
7133
8379-00065
0065
3006-
0173 -7793
0964
9062 0066
5001 0098
31

---

Under review as a conference paper at ICLR 2023
Table 20: Full results on Citeseer dataset with GAT.
layer
method
20%6
mised 40%
00%
20%
input 40%
60%
20%
label moise
40%
0076
Suadand
8886-4 0125
8307:20110
0121
8689
6096
8536-t
30
8512.40114
8762-4
8650
5648:2
0166
Dropfidge
8366:1-0133
8100:10000 0098
8750=2.0079
8496:1
0101
8.122..013 01221
8820-1 0086
8679 0112
5673:1.01 0114
SouralSpurae
8573 0101
83224 0092
8743- 0117
8577::
0067
8580 0135
8536-1 0080
8734-2 0076
9657:2 0089
PTDNet
8802-1.0107
4.311401137
8157..0077
8560:1
8874
8784
8669110112
Co-teaching
8628-1 0220
8366.2.0122 0022
3199
$720220022
8521 01.39
88100 0224
8924
0122
8888-1 0365
8919:: 0305
L-2
Peer loss
Jaccard
8677 0125
8615-1-0197
.8799
8877 0170
82352.013 0120
82234.0024
0124
8721
88412.007
8539 0173
8556:1.011
8498
8559
+ 0316
0309
8543
8878
+ 0185
4 0143
8676.1 0195
86552 0288
3661
5611 0258
0256
GIB
8610:1.0030
8324:
87391..014
8741
SupCon
8495 0100
81382
81552 0099
8611 8086
8454
0111
8303
0172
+ 0137
8499:2017
8179 0185
GRACE
8092110221
2.64
748791.0278
80141.0370
7628:1 02:00
7433
1 0245
5758
1 0146
8768:1.0073
$651.1.011 0172
RGIB-REP
8545 0108
8310
81371 0091
87362.0107
8366:1
0097
8503
4 0159
8778 0093
86962 0081
86142 0084
RGIB-SSL
9106-1.0102
.8829 0055
3677:4.000
91724.0072
8909
0086
.8785 4:01
0071 9419
9410
9410:+00000
Suaddad
0026-t0017
7518:20019
0185
0191
5043
0105
7912
8174+
7998
7934 0156
DropEdge
8063 0079
.6244 0211
7434:1.13 0124
8131
7977
0178
7814
0060
6267
0148
81032.0077
8057 0148
NesralSpurae
.7584- 0142
.736 = 0172
7550 0129
82832
0130
3088
0088
7911 + 0174
8259 0119
81 8138 0092
7086 0109
PIDNet
9000--0113
.7734
7597 0185
8254
0105
81322 0089
7950 0143
5137 0243
8082 0194
5036 01.99
Co-teahing
8016:1.084
2005
7521
83135.0173
8109
7547 0350
Led
Peer loss
Jaccard
806811-0222
80641-0178
77771
7807:200255
75442 0191
75171.0186
0145 83462
#258:1.0124
8108
8063
0122
0138
7901
7945
+
1 0073
8206 1 0168
8160
0339
80361.0170 0176
8015
7025 0207
7099 0215
GTB
8170:1 02.30
01541
7645 0247
84222 0365
81124
0212
7972
1 0305
8197.4 0749
8080 0155
8010 0177
SupCon
794011
0125
7478220145
7972
7852 0201
GRACE
.7119 0133
66111.0035
64/91.0579
2216110026]
5947
0660
6000
7773
1 1010
773911.047 0475
7882 0328
RGIE-REP
7991 +: 0107
.7743
0164
7418
81552.0156 0156
7006
0157
7373
- 0008
8108
+ 0118
7946 0180
7035 0131
RGIB-SSI.
8520 0145
8800
80292
0098
8992 0120
8251
0132
8145
0110
5084
0001
9101 0076
9102
0117
Standard
-7877+00117
.277
71339
0251
0134
7
0107
7741
+ 0126
7963
+ 0129
7883
0102
7801
+ 0161
DoopFidge
.768 0088
7116: 7497
0119
7854 0230
7640
7840
7806 018
NeuralSpurse
.7004 0099
7460:20170
7247:2
8047=00101
7647
7248 + 0096
8057
7855=-017
7880= 0148
FTDNet
.78554-0193
7286-2.225
0257
7927=.0287
7822+ 0132
.7579
S000
0085
7977
7890 0145
Co-texching
7819:10041 .789920035
7236-2 0292
79642-0189
7809 0181
7740 0185
7033 0406
7918 0348
7079 0245
Peer loss
7846-4-0214
11599
7257=2025)
79792,0172
7955
77786
7865
7912 0148
Juocard
7902:1-0117
.7365_.0330
7157:1.020 0248
80561.0224
3008:1
0226
7733
0245
7964 0226
7936
0255
7847:1 0218
GTB
7818-2-0230
.7378 0785
713722 0416
8161
7908 0183
7763. 0176
8007 0155
7955 0166
7794 0044
SupCon
.7370 0124
9160=
6670)
0442
7667
0402
7729 0356
6999
2 0597
7810 0219
77522
0119
7591
0362
GRACE
5068-1.0128
2004
5108:1.033
0319
4956.1
5181
52881
5068
RGIE-REP
78174:0129
7062
0681
7254:2 0188
7883
8160
7789
0168
7620
0176
7981
+
7711 0487
7817 0164
RGIB-SSI.
8275 0148
7999 0136
768122014) 0140
8261 0096
3024 0087
77006
8855 + 0103
8918
8040 0119
Table 21: Full results on Citeseer dataset with SAGE.
SAGE
byers
method
mixed noise
Input noise 40%
label nouse
20%
40%
600%
20%
60%
20%
40%
60%
Standard
8138-1-0143
1.001
8644
16109
3608
1: 0064
8847 .0161
8560
1: 0120
8533:8.0112
8412-1.0122
DropEdge
865111-0118
8593-1.0129
8503:1 0125
8834 0159
8791 01088
8806
8809
1: 0085
8834 0064
8810::101100
NearalSparse
9668-4.0070 0079
84754.0009
-437= 0069
8746-
0102
3674-4
.8722
0183
8670
0120
8645-
0036
PTDNet
0620:1-0125
1888.1..013
8477-10077 0078
8740
0094
3675 01.00
.800 1. 0124
8657 1: 0128
157611.0177
3830-1.0073 0073
Co-teaching
8306 00443
8:13.1.0179
8323 0144
8652
0133
5606.1
0067
8650
0247
8757 0269
86601..00
0101
8392.1 0063
L=2
Peer loss
Juccard
8525
85141 0208
4525-4-0157 0157
.8/921::044
.8440 0120
8319
8291
8661
8691
3305
3693
88715 0152
.8335
+ 0151
8563
00166:10133 0120
+ 0278
8629
8835
0107
0210
S413-1.015
2363
0241
GaB
8574-4-0192 0092
.577
-323
0125
8693 0186
26314 0310
87799 8400
8000 0197
X576
0182
2403 0157
SupCin
8344 0000
.8241 0013
8168
0111
8485
00355-000 0133
3493-1 0147
8454 0161
8457
+ 0109
8394 6 0096
X323-4.011
GRACE
8283
0295
13192
8253
0208
8361
0190
8273.1
.831
1 0134
81M 0193
8422
1011
3100.1 0233
RGIB-REI
85144-0120
2213
0118
35374--011
0105
8864
0153
8592
8533-.015
0154
2455-2.0125
RGIB-SSL
.000 10004
88944.00199
3916--00788
9045
0076
29415
+ 0097
.892
1 .041
9143
4 0082
9075
6. 0000
5087-1.0100
Standard
7914110101
7686
23399
8171
0173
8226.1
0141
8157 0200
8OSS
1 0116
.7891
11. 11165
.7705 0172
Dropfidge
7889
0110
7678 0206
8300
0130
5007 0953
8271 1 0115
8058
1 0118
7899
0173
7826 0282
NeuralSparse
79341-0115
0015
8331 123100000
8242
.8324
0014
8111
1 0146
2904
8 .029
.7747
0242
PTDNet
7972-1.015
.78041..020
7563- 0230
8299-1011
$25911 0119
.3344
R121
1
.79221.0000
7845 0166
Co-teaching
7978-1.0188 0189
7679
0230
7557=
8226
0172
$2971 0202
R256
2: 0263
R253 + 0203
8083-2 0218
7836:1 0347
Led
Peer loss
Juccard
7913-1.0116
790111-0108
.7710+
7691
0269
0030
7626 0165
0236
8197
8359
0264
3321 0-055
S2M11 0189
8239
85901 0064
+: 10215
5135
8167
1 0119
4: 0121
7940 0194
7972
0157
.7411 0227
7855
4: 0195
GIB
7739 70111
0254
7691
8190
83061
0275
R151
1
7947
0266
SupCon
.780-1101100
2672-1.0145
0025
8191
0197
1081 0138
7882 00137
8.021
.5783-1.017
0172
GRACE
6196:1 0253
6104
6308
0323
6286
2 0287
6365
0223
6198
0157
6495
1: 0202
6431
0210
61111 0191
RGIB REP
7854 0123
.7003+
3562- 0200
81514
8131+-022
0021
+ 0118
.7863-4-013
.140
.7799 0212
RGIB-SSL
0157
8/1421.0147
8052:1.0127
0118 8525
.8564
1:0144
8867
1 0008
.1161 8866
0124
Standand
.77881.0166
.7223
7204:1 0236
8086
7997.1
78772
00145
7907
1 0219
7690
756411 0096
Drog
7756:1 0143
748522
0104
7290= 0761
8070
0125
5060:1 0165
8026 0195
7961
+ 0174
.769 2 0149
7514:::
0212
NeuralSparse
.757-1.0175
.764
1.64-1.000 0205
7206-10112
8039 0164
5008
1 0157
6016
1: 0223
7002
1: 00189
7762.1.0177
7588 0160
PTDNet
7844-1-0116
77525--0270
0270
7435
8070 0130
$1281
0108
50135
7008 1 0063
77830 0182
7634 0210
Co-leaching
7797.1
.7248.1 0643
7368
8087 01%
5016-1
7864
1011 0904
76711
0232
L-6
Peer less
Iucard
7800:1 0257
7796-1.0158
7211-1.0622
7267
7248:100390 0330
7294110233
8135=-0025 0225
81682
S012.1 0206
$08811
.79141.020
799001
0200
7000 0074
8005
3. 2009
.7749 0269
277111.0316
.6672-1 0098
0617110147
GIB
78234 0158
735722
0808
7247= 0332
8180= 0388
5073 0241
K049 037A
706T 0231
77100 0367
7576-2 0097
SupCun
7619:1.0176
.71931.0426
0010-10231
7861
7800
.75291.0019
7775
.771511.0206
771171 0235
GRACE
5608-1.011 0071
67672.0441
0494
6591
0344
6660:1
0166
6670 0292
6966
1 0067
667811 0682
6679 0476
RGIB-RB REP
7766-4.0070
7:79--.0150
7427=.0127
7981
1. 0104
3929-1-015
01.49
.7844 0228
7959
H 00127
.7865
0.163
3695 H: 0169
RGIB-SSL
8372-1.0169
.22611.0189
8184:.0135
8481
18111.0075
8252.1
0129
8795
1.012
8802
0100
87.191.0176
32

---

Under review as a conference paper at ICLR 2023
Table 22: Full results on Pubmed dataset with GCN.
QCN
layer
method
20%6
mixed
40%
0406
20%
inpu: 40%
60%
20%
40%
600
Stondand
9477440000
9271+-0077
9141=2004 0001
9590=
0012
9468-t
9357
0016
9646
0004
9637
0021
0597::
0022
DrogEdge
939411.0025
.9555-.000 0027
0033
9467
0022
9302.1 00221
9116:: 00022
9554 0026
9558220002
0027
9519:1.00 0019
SearalSpurae
94792.000 0021
02512.00139
9120 0029
9558 0019
93154: 0033
9269
0014
9654
001
95254 0022
9588:2
0017
PTDNet
.9644
9111:10032
- 0022
9320
9272.1.007
9027
9651
961611.002
9584:1.000
Co-leaching
9502-1 0085
933551.010
9160: 0696
9510
0098
9131 0054
9255 1: 0113
9676
1: 0018
95602.
0127
9608:1 0098
L-2
Peer loss
Juccard
9496.1 0009
9000:1 0034
9925
9339 0044
03087
9235
9140 0138
0061
9554
9558 0068
0019
9327 0067
9307 0019
.250 .2199
9283 8000
9615 0175
9550
0002
9532 0006
0521 0178
95911
0545:1 0144
0054
GIB
9509-1 0205
9268.1
9131
0006
9599
9100:1
9:28
9559
9573
9597
SupCon
9345:: 0020
92572
2017
9118:
0031
9583
0015
03452 0000
9214 0023
9625
0021
9522 0018
0506: 0010
GRACE
93411 0032
.319-1.0027
0027
9154:
06:9
9109
0046
932111-0032 0032
9225
1 0073
9459
1 0004
3516:1.0027
9511:1 0022
RGIB-REP
9537.1.00 0013
9368
0018
9270:
0056
9579 0022
9467:1 0023
9365
0016
9696
0003
9690 0023
9671 0015
RGIB-SSL
0022
.471
0002
9201
.9305
+ 0098
.9121
0025
9719
9724 0015
9011
0012
Standand
8870
8748
-641=0041
0854=_0051
87594
8661
0040
9000
0062
9099
0022
9070:
0062
DropEdge
87111:0149
84822.
0045
83542 0962
8682
0158
8456:1
0036
8376
0046
9313
0071
9201 + 009]
9240:1 0077
NearalSpurae
800810000 0080
8733-2
8610 0609
8911 0090
0001
8649
0044
9277 0108
9136
0117
0089 0084
PTDNer
8872 0071
8753-
8623=
8903=
0087
S776 0078
86P 0055
9219 0112
90P9 0104
5033 0101
Co-teaching
8760
8638-
0000
8931= 0045
5790.1
8606 0063
9315
9119 0324
L-4
Peer
Jucard
8872.1.006
8061 0130
8803 0060
8815
0009
8512
0057
0136
-017
8987
0221
5811
.76411 0099
01
8639 0073
8643200122
9136 0116
9098 1: 0110
9135
9101 - 0046
0116
03102 0095
9096.1.012
GIB
8899 02.39
8729 0205
85442 159
8932
0756
8508:
0083
8618 0317
9017 + 0089
91144 0065
9064: 0059
SupCon
0061
8718
8525=0110
1807 0000
87194
8558
9131
9108
0095
91622
GRACE
8912-1 0034
874911.008
0962
8810
0034
8795:1 0099
8501-1.000 0010
923 1: 0088
$252.1
0052
9155 0043
RGIB-REP
901711 0044
8834 0082
8652= 0018
9008
0033
8817 0054
8687 0056
935T 0008
0343:2
0062
0132 0062
RGIR-SSI.
9225 0125
8918 0065
8697 0555
9126 0046
55894
0062
8693
2 8049
9504 0036
9604
0028
06132 0023
Stundard
8870++00056
8731
8855
0025
8742+
0029
8652
[044
90500
0083
9112
0159
5063 0636
DrogFidge
8670110009
8421
0044
8342- 8058
8623=
5407
8328
9140 0014
9102
0082
0972
NearelSpurse
8814+
85% 8079
-603=-0900
8792=
0072
3005
0031
8612 8046
9140 0065
30 0047
0080: 0077
PTDNer
1007
8610
00-0
8791
0040
3570
0032
8619 0040)
9123 0004
9086
0077
91052
0012
Co-teaching
8850:1-0126
8698
8568 (150
8773= 0825
87671
0113
8691 + 00995
9101 0372
0311 0200
0204 0162
Peer loss
8120:2.0125
8633-2.077
8534=
8734
8775:t
8668
9155
91722
0089
0251=20107
Juccrd
8852-1.0092
8658 0085
8555 0055
8711 0031
379411 0126
8670 0200
9045 0117
91182
0108
9065:1 0059
GIR
8738 0140
8605=0199 0193
8718= 0070
5767
10
8623 0114
9081 1 0131
91362
0147
01 0106
SupCon
0796
8627+
8533=
0045
8705
0108
87334
0026
8643 0047
9231 0081
0294
0075
9218 0992
GRACE
5778
8664
0051
8675:1
8591
9296 0064
9190
0052
9110:1 0049
RCIR-REP
8846:2 0071
8715 8035
8640=2004
0643
8818=
0854
87164:
0021
8047 8646
9161
+ 0081
91 16 0075
01:9:1 0095
RGIB-SSI.
8915 0062
.8577
0633-0006
8891 1 0057
8732 0034
8659 0005
9450 0042
9488 0065
948422 0027
Table 23: Full results on Pubmed dataset with GAT.
GAT
layers
method
20%6
mixed noise
40%
20%
injut nolse 40%
60%
20%
lahel noise
40%
0006
Standard
.9173-400028
.8984 H 00030
3884-4.0033
9255--
-0124
99176-6-00 00035
.9102
00000
.3306 +- 00088
.971
9271-000030
-9232--0000 0034
Droplidge
.102
0022
8970
.033
$837-1.000
0004
9208 00.3
9177-1
.9087
9374
.909
1.000
9967:1.0019
NeariSSpare
9106-t
0021
8952
00151
39400:2.000
0029
9179-2.003
00129
5/210=b
0052
9141 4: 0033
9510 0009
9297- 0002
9252-1 0021
PIDNet
.919 + 2015
8943
+ 0033
3844
0037
9068
0027
9210-
0026
9943 6: 0025
9311 + 0001
9280 0034
922674
00024
Co-traching
9163-4
0044
.994
0050
0100
9058
0089
9180:t
0009
9136-
9556
9364
9376- 0249
L=2
Peer loss
Jascard
.9188 1 0057
9156-4
0115
.8900
8965
100000000000 0094
3083
3823--.00
38331
0086
0041
9042
0092
9181-4.0175
9171
0175
0089
.928
9173
+ 0070
0214
9400
9304 8 0009
92964
9355 0127
0084
9219- 0229
9282-1.0190 01009
GBB
9136- 0061
8966
- 001 10
3981
0115
9052
0313
9208-
0097
9383
4 8206
9306 0009
9291 0965
9277 0089
Suption
5002-4 0086
8881
20057
886334
0027
9026-
0025
5080-1-000
0053
9200
9123
5009-
GRACE
.220
2220-1.0811
.782
9893
3914:4.700
0767
8273
0671
8053-4-0000
7992
0695
8792
00:4
3996
0-057
8946-
0211
RGIB-REP
9190:4
0025
9054
= 2017
3039-2.00
0056
9250
0053
9164
0024
9099
0002
9511
+ 0036
9506 0005
9276: 0022
RGIB-SSL
9223 4 0022
9054
- 00052
3900 0027
0021 9183
5001
0023
8977
9: 00029
5400
+ 0007
9407 0022
5878
0018
Standard
8610-1.000
1010
8833
1: 0012
5.000
8547-1
0000
8476
H: 0037
8817
4 0012
8772
0010
3096
: 0062
DropEdge
8800-1
0034
8862
= 0028
83251.
0094
8605
0041
8557.1
0057
.854
1: 9009
8852
1 0082
8851
11. 0055
SS71
0034
NeuralSparse
.891
8805
30125
8402--.000 0025
3012
8042
8572-4
0048
8459
0005
8906
0009
3800
+-0070
0075
PTDNet
16141-0008
.8568 1: .0026
3372-1.0338
2610
.033
3541:1 0000
18532
1: 2007
5909
1 0052
8855
5. 00044
5385
1. 0064
Co-teuching
1688-1-0125
88500 0065
8411:11000
0081
0066 8666
8556:1
0103
8968
1: 0118
S791
1 0107
8996
0035
5968 0132
Ls1
Peer loss
Juscard
8830-1.0089
8613-4-0040
.802
.8233 + 00105
0082
8265-1.01
83422+.011
0000
01:36
0195 8677
8652-
1
3501-
0143
5065
8885
.8711
1: 11
9
5860
8351
+ 0110
1 0078
10000
0008
3550-1
$750-1.011
5535- 0069
0119
GIB
8616: 0096
8834
- 01443
X570--.011
0116
8914
0189
S486 0009
1842
4 0122
8823
+ 0111
:300 0076
SuptCon
85344-0066
8292
- 0122
2048
1. 0064
8625-
- 0065
3447-
0009
83500
0144
S708
4 0057
3696-
00002
9629
- 1005
GRACE
8355 1
8202
L. 00387
8016
0909
8139
0344
3078
0649
7822
1 0856
7959
1 0839
81851.070
8008
1. 0796
RGIB-REP
8656-+
8443--
3339-1.000
0033
8674+
0055
8528-
0006
8000
0084
8925-
8833- 0062
RGIB-SSL
8891 48077
.8651 + 0006
2480-1
0066
8878-
0039
3650-4-000
0044
8489
+: 50055
9276
+ 2042
.277
.002
9264- 0004
Standard
8315 0059
88116 0073
3040-1.0000
0092
33744.00036
$201
0021
8057
1 0006
8150 0086
8414-1.011
0011
8113-1 0071
Dropfidgee
8468-1 0407
8236 00550
5004:1.00
0071
8655
0037
8303
0044
7015
1: 0334
8501 1 0058
85681 0058
8523-1
0381
NeuralSparse
8426-1-0025
.8234 H 00032
3098
0067
8323
0342
$258-1
0090
.797
9622
0091 .8633
8577-1 0075
PTDNet
8381-1.0019
.8177
50311.003 0260
8508
0044
$2511
0281
0014
8635
8198110025
$152-1.000 0065
Co-leaching
8360:1 01
8111
0012
2081 0113
8301 0101
8246:1
0398
8071
1 0512
S729 0080
86471 0107
8424 0079
L=6
Peer loss
Juscard
8354-1-0128
8333-1-01400
.8186 0011
8819
: 0073
0082 S0111.
00004
8 0221
8416
$108
.112
8277
8340:1
0421
0059
8816
8833
1. 0497
6: 0336
5501
8528
4- 0032
1 0100
8507 1.010
1: 0015
3310-1 0089
20011100108
GBB
8266 8316-4-0125
00555
3064--0241
8189-.00136
8206+
1045
0420
8551
+ 0149
-495--0012
2361--0072
SupCon
8218-1.022 0238-1-0236
.7763 1: .0306
7113-1.0211
08821
3025
0286
.7555
4 0297
8543
4 0122
3390-1.0300
A3-00- 0071
GRACE
7773-1-0737
.729
L. 0344
6821
7015 1340
6953
0805
6643
1 1009
S212
1 0262
51551.0455
5543-1 0519
RGBI-REP
8324-- 0039
8162
00SS
7954-4.0000 0068
8248-
0070
S145--
0088
.7892
8518 +
8496-
0083
5492-
RGIB-SSL
8738 1:051
8843
0033
8124:1.008
8709
.023
.432-1
0046
.8187
8.000
9208
0013
.9250 0029
9935-110031
33

---

Under review as a conference paper at ICLR 2023
Table 24: Full results on Pubmed dataset with SAGE.
SAGE
layers
method
mised moise
60%
20%
inyur moise 40%
60%
20%
lahel noise
Suaddan
191344 0032
9094- 0015
903520004
9295=
9378::
9410::
0013
88174
0014
87932 0043
5757.2
041
DiogEdge
91424
0026
9188+.8007
0034
9179-
9178:+
0013
9165
0013
9021 +
9005-20011 0010
5011-
NearalSparse
-918-- 0030
913774.0051
9094-2003 0634
9271=0001 0019
9209 00026
42554 0017
9003
+ 5002
8925 00050
2364:
0052
FTDNer
91144 0026
91382
0030
9104 0637
9267= 0020
92114: 0027
9229
0034
9050 0008
8941 0140
53681 0013
Co-teaching
91644
-160+
0027
9092=
0132
9258:
9228
8827
9054
0236
3356-20011
L=2
Peer loss
Iaccard
9108 0123
910011 0104
917442
2167
0094
9109
9044: 0087
0621
9236 0098
9208
0079
9238:1
9259: 0175
0051
9308
9307 0076
0141
8861 4 0052
9005
0118
8785
8835-2
0066
0092
53421 0048
59091 0058
GBB
91733 0049
9125
0211
9096 0620
0152 9338
92944: 0125
9299
0194
5880
0125
8865
0122
39551 0058
SupCon
0026
9148
9106:
0035
9105:1
9250
9033
0055
8929
5558
GRACE
83174 0067
8891
8336=-0060
0068
8288
0071
83442
0053 8305
8330 0046
8344
0078
8138- 0075
RGIB-REP
91922
0022
9136
0027
9119- 0657
9228
0051
9322
0031
95511
90055
0003
9015
0001
5000 0043
RGIR-SSL
927664 0052
926554
0030
9313 0631
9273
0055
9305:
0040
9354 0025
9304
0017
9294
- 0049
9374
0066
Standand
0027
8619
0073
8715=
0080
8001
0082
9033
0057
8386 0085
8330
0104
8368
0092
DrogEdge
X067
8740
8816-
0077
9072
9016::
9087
0019
9036
0014
5048 0059
NearniSparse
0037
3825-
8046
8822-00051
8982-0000 0967
9146-1-000
0022
-190++0000 0041
.847
+ 0005
8535
0091
8472-
PTDNer
85422 0045
8883
0019
8846 0052
8885 0055
9057: 0055
9127 0015
8464 0105
8458:
0066
2408 0065
Co-teaching
0104
8752+
0050
8667 0122
8727= 0079
39711 01:59
.901
+ 0058
8001 + 0240
3498
0368
X5144 0988
L=d
Peer loss
8683
.817
0100
3065+-0117
8726=
3997-4
.9055
+ 0019
8550 0151
8514-2.01
0015
8413+
laccaed
8646-
.8725+00144
0140
0125
9017:: 0205
01155=01144
8378 0137
8385-
01 18
5276- 0685
GBB
0836-20003
8824
0101
2918 0095
.921 00245
8479
0103
0415
X330==0000
0120
SupCon
8768 0086
8687
0100
8518= 9190
8907=
0854
8957
0104
8579 0113
8065 0346
8811
0052
58101
0044
GRACE
71424
6741
0484
6446
70122
0516
6498
0806
6647
0535
6540:
0424
RGIB-REP
8746+
0049
8708
8758=-0071
0004
2904
0048
.841
8504
8466
0070
3533-
0118
RGIR-SSI.
93151 0036
9237 0035
0033 9314
0046 9276
9510:
0039
9286-1
00155
0000 9461
0057 0448
0021 0410
Standard
83144 0081
81054.0279
x333
0089
8224
0535
8538
0105
8566 0199
X342
0104
X161
+ 0071
X090
0129
Drogkidge
8781+
3665--.000
00776
8556-
0081
1900
085
8721
0122
8628
+ 0152
8500
+ 0000
8698
0074
8426-
0096
NeuralSparse
86114-0101
858911.008
8482-.0182
0182
5694
0083
3838-1
.889
9 0057
8368 + 0014
8344
0000
8353
0081
PTDNer
8425+ 0073
8811+
8067
8525-
0066
8430=
0107
3627+ 0115
-8759+00008
8232 0076
8195
0017
5198= 0060
Co-teaching
x349--0167
8265
- 0985
2007---0144
8612
X299 0552
X243-
X143- 1409
L=6
Peer loss
8339-1.0162
$111++0025
8335=0008:
0317=-0:11
3610-1.0135
8605 +: 0023
K277 + 0224
8260
0197
.201
0316
0115 .180++
8347= 0124
8298
33644
866T 0594
8260 0138
8217
+ 0150
X1400 0192
GIB
8309--0000
6236-0206
8226--0385
0355
3527-4-000
8647 0144
8310+-0142
M237
0125
X141+ 0165
SupCon
8288 1.089
.8276.1.0311
8174:1.0070
8617-
2443-1 0528
8365 5: 0350
8745 1: 0001
8632
.066
0076
GRACE
0133
7151+ 0368
6737= 0522
7102 0296
7392+
0257
6751
+ 0444
7120 0166
70042 0588
77330 0286
RGIB REP
8493
0027
84424-
0083
8373-
8439
3527
+
85366
8275 + 0218
8148 -00M
X211 0145
RGIB-SSL.
8915:: 0060
.8971 0051
8056-20044 0046
8948 0079
9036:1
0042
8995 2 8040
9313 0042
0911
0036
0794: 0053
Table 25: Full results on Facebook dataset with GCN.
OCN
layers
method
20%
mixed moise
60%
20%
inpus moise 40%
60%
20%
label noise
Standard
1.880- 0007
9986 BM0T
9843
0010
9878
0008
5852:t
0006
.9834
0011
9892 0006
9888 0008
5888-1 D007
Dropkidge
9866
0007
9845
9821=
00009
9764-
9535-+
.9705 + 8012
9783
9879-80000
5881-1 0007
NearalSparse
.95774 0007
98611
10008
98837+0000
0009
9772-
0007
9632
0007
.9752
.010
9788
0006
9885 00007
9688-1 0005
PTDNer
98766 0008
9962
0008
9838=
0008
9776 0007
9548-
0008
9728 + 0011
9859 0006
0786
0009
5886-2 0006
Co-teaching
9871 10052
991224
8104
9876-
.9200
99166
0226
9873
0121
5825 1009
L=2
Peer loss
Jaccard
9759-4 00045
9755:: 0009
9737
9796
0015
9895
9707 0006
0696
97.9
9797 0074
0098
9907
9855
0051
0040
9720
97533 0150
8019
9729 0000
9763 + 0001
9773
0730
01 10
0014
9729-2
9774:
0042
0034
GBB
97722- 10009
9733
0.022
9720
0158
9796
0245
99200
0173
9707 + 0011
9724 2017
9714
1404
59117 0086
SupCon
0008
9748
2014
9665
9534-
9783
9713 0000
9569
0160
0924
GRACE
8883-
0120
3811
8412=
0224
8950
3792-
0119
88550
8221
8864
0007
8826
0051
.784:1
0104
RGIB-REP REP
9881
0007
9900
5646--0009
9880-
0007
9934-1
0005
.984
+ 0011
9895
+ 0007
9891-.0000
5994-1
0008
RGIR-SSI.
9845:2
0005
98800 0005
9765 0014
9840 0011
9808::
0012
9757 00099
9802
0005
9859 0010
3866:2 0007
Stundard
9829- 0020
9520-
0424
9438-
0402
9819
0015
9668 0147
.662 +: 00154|
9882
+ 0007
0880
0007
5886
0006
Dropkidge
.0228
9662
0096
9473-
0016
.9531
+0012
.9733
9771
0008
99766+
0009
NeuralSparse
9625
0020
.9638
0009
0007
9712
9691
.95K31
1: 0071
9781 1 0007
9781
000000008
973411 0008
PTDNer
972254 0017
9485=009--
9725=
0014
5003 0089
.9233 - .0257
98799 0003
0880
0008
57133 0008
Co teaching
9620-4-0113
95226
0439
9480 0050
9712 0084
9007
2- 0141
99144
0046
.762 + .000
9797
0167
96388
- 0154
Peer loss
9807
9536 0500
9450
0457
0758=
9622
0116
970P 0085
9750
0128
9734 0137
Juccard
97944- 00011
9579++9471
9422--0452
9784--0127
0127
5702-+ 0295
.6638
+ 0540
9702
+ 0016
9725
0004
5959- 0025
GIB
.773- 0182
9608--00544
9796
- 0260
9947-4 0200
.6550
0033
9742
.0033
.9003
00023
.971 0000
SupCon
9538 0067
9508+
9297=
5647
0888
95174
0113
9401
+ 0135
9647 0164
0567
(114
0553 0133
GRACE
8899-4 0100
8865- 0126
3315- 0108
.895
0115
0006 9884
8972
00146
.887 0125
RGIB-
9632
0026
99700+
9519--0063
9833--000
0014
9723
.9622
0087
9883
0007
0007
RGIB-SSI.
9829-t 0023
9711 0002
9643-2000
9821 = 0019
9707
0021
9668
- 00300
9857 0010
0881
=:: 00088
5857:2 0007
Stundard
9798-40013
96099-01138
5968---0179 1179
9764
0034
9002
0096
.9099
0010
9963
+ 0013
.9855
+: .0122
5877- 0010
Droglidge
97744 0030
9635
00063
9500=1.0100
95400
- 0034
9579-1
1 0083
.411 6: 0054
9556
1 0014
9642
8 0006
9555
- 0012
SeuralSparse
9774110020
9596
0067
9435 2110
9609
2 0029
9496
1 0073
9990
0007
9561
1: 2012
9562
0011
9465
0016
PIDNet
9767-4
96606+
01331
9449--0173
0614
9568
+ 0064
-416--0044
-045
9500 0011
9460-
0014
5560- 0009
Co teaching
9638-1.000 0001
996651.0235
0261
9520-
0051
1 0140
.9889
1. .0288
9474
1 .068
9958
1.143
9561
0169
I-6
Peer loss
969911.0045
955711.0141
9362
0221
1
9554
0053
949111
0135
9645
0034
9496
1 0063
1581
11. 0200
9650:1 0021
00022
99533--
00137
94425-
39116-1
0115
.923014
0010
95228
+
99255
1 .0075
54123 1057
9649-1.0143 01:45
.6002
0199
9116
9511
0333
95521
1 0261
.9555
1: 0046
9556
1 .099
9930-_.000 10000
9578 0052
SupCon
9462-4 (050)
9577
0013
9460=
0075
9555
.9253
0010
9411
+ .088
0550-1
.0087
95
10:10 0122
GRACE
9009-1-0188
3363-
8259-1.001
0019
9071
- 0077
3582-
0228
92744
+ 6029
8942
+ 00533
9022
2 0013
3977-1 0100
RGIB-RIE REP
9774
0001
9680
0097
9501
0134
0024
9619
0069
.923
0083
9968
0008
9863
0010
99672-1.00 0011
RGIB-SSL
0023 9751
9641
9295-
0122
9763-
0634
9623-
0040
.945
+ 0084
9845
+ 0008
9843
0012
5847:1 0011
34

---

Under review as a conference paper at ICLR 2023
Table 26; Full results on Facebook dataset with GAT.
layer
method
20%
mised 40%
0,%%
20%
input 40%
60%
20%
label moise
40%
0076
Suadand
9869:4 0005
9836:2 0007
9878:
0006
9861-:
0006
9858-4
9877.4:
9864 0000
9857:2
0008
Dropfidge
9645--00007
19532...000
95171.0009
96522_0007
9636:1
0067
.933 1: 8008
9755 2:55:1100 0007
9743:2 0010
9739 0012
NeuralSpurae
9661 + 0007
9651
4000
96.88 0006
9765- 0008
96524 0007
9744 2: 0006
9663:1 0010
96572: 0006
9656:2 0007
PTDNet
.9551
1.000
9735 0007
9766..
9650:1:00
9757-1.0008
Co-teaching
9723 0006
.6855
0054
9762:1.000
0639
9632.2.
0057
9720:1 0001
974
1: 0046
9700
0:000
9789 0230
9792-1 0288
L-2
Peer loss.
Jaccard
.971 1. 0038
9760 0033
9644 = 0000
.966
1. 100057
9736-1.00
9725 0076
0078
9556
9699 0020
0064
9758.1
0730:1 0104
9708
9780
4 0103
0012
9720
9657 0194
1- 0100
97.89
9807
1: 0064
0207
9701 0087
971 0078
GIB
9679-1.
96162
9709-1
9738 0019
9799
SupCon
97184: 0028
9646 0085
0087 9689:
9753 8008
9737 0009
9629
977011
0018
94562 0038
9419 0029
GRACE
8377-1-000 0009
.871611.026
02663
0264
8343..0341
8502.1 0271
8457
1: 8407
5768
1 0457
0310
$791 0319
RGIB-REP
9965 0: 0006
99854
0009
9837. 0006
98724
0007
9855:
0007
9655
1. 0008
9671
1: 0007
9861 0009
9659 0006
RGIB-SSL
.988---0011
.9664
99174.002 0021
97644..011
0012
99744--0015
.9734
0021
9778 4.008
9784
0009
9791
0009
Suadard
98575-0001
0850
9820
0014
9855
0011
9827H
0019
+
9673
9874
0010
9874
0005
DropEdge
983911 0009
.991 2: 0033
9767:1
0024
9836
0009
9772
0022
+ 0000
9657.1 0005
9845
0009
9644 0007
NesralSpurae
9854 0009
.81122 0019
9768
0027
9839
0012
0700
0028
9736-
0000
9677 0009
9869
0000
9868 0007
PIDNet
9843 0039
9830=
9745 0076
9844
0012
9787
0039
97524
0017
96724
0005
9868
0010
9872 0007
Co-teahing
9789:1.0019
9777 0086
97171.011 0111
0703-
0050
97181
9783
9645
0750
9646 0014
Led
Peer loss
Jaccard
9757-1-0079
9944 0079
.9663 0067
9690220105
9687: 0046
9663:1.015
9610
9607
8068
9616:1 0184
8610
0064
9783
9655
1 015
0073
9677 0053
9689
0025
9638
9625 0009
0000
9691
9601 0036
= 0203
GTB
9609 01.34
9747 0077
9709 0204
9721
0297
9741
0199
9740
0111
9747 0018
97681 0093
0799 0016
SupCon
9709-1
0464-20254
91342
0309
98242
9745-
9580
94822
GRACE
7803 1: 0649
77477
7618:1 1058
82591.0660
7588
0963
22
133
844011
0370
8468
0337
8122-1 0403
RGIE-REP
9655 0014
9839
0016
978311 0034
9838
0019
9806- 0013
9765
1 0037
9673.1
0008
9871
12 0008
9667 0007
RGIB-SSI.
9751 + 0033
9007
96022
0043
9767 0030
9683
0003
9603
2 0002
9755 0078
9772
0032
9762 0047
Standard
9805 0025
9677 0314
9804
9738
0044
9710
0036
9854+
0015
9860
0007
9867 0012
DooeFidge
96434
0054
0274
0042 9736
0589
9606
9735
9736
9736 0009
NeuralSpuree
97464 0077
9667=
9560:2 0122
9740
9668 0056
9652 0043
9735
0017
0742220011
9733 0030
PTDNet
97699--0059
99679=00077
9582-.0200 0296
9754=.0012
9688 0045
9645
0019
9738
0041
9741
9756 0012
Co-texching
9745 0034
96962
9625 0408
97472
0084
0833
0108
9738
2 0015
9623
0724
96762
0072
0731 0215
Peer loss
9000s:0014
97162_0125
9776-1
9766
0074
9671
9605
9768 0063
Juocard
9789:1: 0084
9651
9620:1. 0337
9673:2
0007
9809:1
0110
9731
1
9657.1 0064
9607
0004
9751 0030
GTB
9839 0037
9777
0422
058421 0174
9868
0200
9760:1
0135
9701
+ 0349
9648
+ 0054
9640 0946
9774 0029
SupCon
9428 0194
919T
87792 6347
0171
9459
0978
9312 8544
9465
+ 0173
9484
0194
9023 0333
GRACE
72341. 0687
6075.2 1089
7415
1242
7256
70624.104 1042
RGIE-REP
9779 0031
9711
0043
9433:
0392
9763 8041
9666
0087
9614
0057
9847 0017
9848
0013
9856 0089
RGIB-SSI.
3653 0107
9491 5 0105
9254-2 0206
96762.005 0057
9502 0087
9384 2 0067
9757 0065
9675
0052
9713 0105
Table 27: Full results on Facebook dataset with SAGE.
SAGE
byees
method
mixxed noise
input noise 40%
label nouse
20%
40%
600%
20%
60%
20%
40%
600
Standard
19888 0.008
9820-1.0008
9788
11 0012
9881
2: 0008
.962
1: 0007
.982
1: 8008
.967
1: 5006
98801500006
982251..00 0015
DropEdge
9945-1 0008
98071.0011
9765
0011
9766 2. 0007
99749:1
1:00110000 0011
.952
0008
9761
1: 0008
.93912.000 2010
971911.000
NearalSparse
977334
0008
.66994.9011
9683-
9675
9000-
.9639-
8011
.667
9640
9627-4-000
PTDNEt
.755
0007
.922311.0099
9779-1000 0012
9775
0007
9761 0008
.954
8007
9766 1: 0007
9743
0007
9726:1.000 0014
Co-teaching
9759-1.0100 10101
99722.1.000
0004
9728
1228:100 0100
9702
0050
9742.11 0023
97391
008
9764 0125
9768 0205
9700:1 0282
L=2
Peer loss
97763 0082
.9720 0072
9990-1
0103
0005
977814
.9787
.9721
+ 0019
9942-4.0
-91934 0031
Juccard
9744711.000 0018
.922011
8009
9021-
0096
9729-
97600 01.39
.9522
0102
9733 1:
9706 100059
9716:1.00 0098
GaB
.5788- 0089
9765=
09220
9724
01.99
57077 0287
.9778
4 8006
9750 2012
9715-
0067
5977- 0050
SupCin
9482 0166
9268
10S5---0000 0046
9141-
01399
9793
0043
9782-4
0072
9796
+ 0077
9279
+ 2018
9252
0074
9155- 0041
GRACE
911311-0110
.8007
0295
9077
0069
9012
0113
8885.1
0177
.031
1 0104
9053
1 0126
.950
5067
0137
RGIB-RED
9800++ 0009
98344
2012
9878-
9963-
0007
.983
+ 2011
9875
0006
9860-20000
5855-
RGIB-SSL
9840-1-0000
0007
99299
9811 0006
9838
- 0007
.921
1 0008
.9823
.004
.641 9 0008
.9839
00007
9641-1.00 0010
Standard
99224110015
0015
.783-1.0025
9698 00400
9849
0015
981511
0024
.915
0005
9844
0015
9817.1.000 0007
9608
0009
Dropfidge
9618.1 0026
9764-1 0034
9722
0022
9625
0015
9601
0043
9600
0003
9649
0008
9627
0017
9615
0012
NearalSparse
9811- 0001
.9766
9740
9636
0023
9:00
96155
15 0016
9548
1
9628 9 0014
.9133
0025
PTDNet
9520 0012
.97571.000
9738 0030
9623-
0041
9409-
0001
9807
1 2002
9652
1 0007
9632
0009
9621-1 0024
Co-teaching
9660:1 0009
965331.
0075
9659
0014
9600= 0098
96141
0091
9626
0079
9654 0253
9706
0136
9628:1 0263
Led
Peer loss
Juccard
9668 1. 1.0105
9621 0019
0019
.9676-1.0054
.97151.0005
00056
0004
9916
9773-1 006K
0630
9681
9682
001000000 0079
0145
9675.1
9611
0014
0091
9782
9600
1: 0012
0056
9671
9725
1 0107
:: .204
9609
97261
1. 00888
0063
9660 0002
1900519.2270
GIB
9774-10000
9975
9733-
0054
9708-
9616-1
9679
0171
9765 010|
9714
2018
0107
SupCon
9169-1.0144
1.9911
9001
9326 0208
9118.11.000
9188 0019
9295
1: 0011
90225
1 0148
GRACE
8683 1. 0292
8335
0696
8545
0415
8585
04224
8721 0261
8413 0564
8108
1 0213
8827.1
0375
8376.1 0330
RGIB REP
99228
0016
990444 0020
9950-
9843
.908
0000
9857
+
9833
00025
54300+ 0019
RGIB-SSL
9817 0016
.97971.0014
0009 9768
9828
9798
.9977
2013
.009 .921
.920
0011
9626.1 0010
Standand
975111.0055
9603
0365
9616:
0091
9715
9672-1 0057
.5177
0228
9761 1.0136
9788
0045
9749.1 0151
Drog
9657:1 0009
96761 8088
9578-2014
0149
9734
0050
9616:1 26
9665 0073
9631
+ 0013
9606
0046
0651:1 0145
NeuralSparse
060011.0075
9577.1.022
0222
9672-1
0051
9647
0048
9691.1 0156
9673
.006
9632
1 0017
9616
9615 0051
PTDNet
0029 9668
993331
0142
9617
0111
9774 0028
969311 0063
9642 0055
9629
1 0015
97361
0010
9708 0037
Co-leaching
9644-1.0116
966991
9658 0195
9730
971811
9583
0750
9742 0118
9744
9770.1 0282
L-6
Peer less
Iucard
9750:1.0119 0119
0611-1.0141
.600-1..339
0666-1.0453
9681=10085
9686 0004
9613-_0167
9666 0261
9918.1 0053
961 0066
.52411.
9683 0583
0223
9655
9634 0170
1. 0020
8769
9784
1: 0018
0110
9650-1 0256
97811 0178
GIB
9739-2 0197
96562
0564
97492
9743= 0351
97174 0034
9615 0291
960T 0153
0050 9779-2
07412 0222
SupCun
.85911.0139
0130
8947
8910.110101
.9096
0215
91411.0114
916611.0144
9135 0122
GRACE
8202:1
0165
772551
7901
0018
8234-1 0547
7960 0915
8061
1: 0367
8262:1
0306
8103:1 0351
RGIB-RB REP
97700
0040
9557-4-0119
9374=.0149
9749
0046
9683
0072
9551 0093
9K38
+ 0000
.922
H .0015
9614 0029
RGIB-SSL
9773 0032
.920-1.0031
9678.1.033)
9781 0027
9729.1..000
0023
.692 0050
97741.0011
9786-1.001
2012
0017
35

---

Under review as a conference paper at ICLR 2023
Table 28: Full results on Chameleon dataset with GCN.
QCN
layer
method
20%6
mixed noise
40%
0476
20%
inpu: 40%
60%
20%
label
40%
000
Stondand
195344 0025
9696+.0002
9657=
0029
9784-
0017
9763:4
9754 0023
9775
0018
9769 0018
9785: 0636
DrogEdge
9716:1.0030
066011.0025
9579 0029
9646
0026
9602.1 0024
9661 1: 0023
9656 1: 0018
9664 12. 0029
965111.0031
SearalSpurae
9-012
0697:-0010
9633= 0030
9682 0026
9646:
0023
9627
0022
9679 001
9669:2 0022
965122 00799
PTDNet
.662.1.000 0032
9637:
9679-1000
9619:1
.9627 5005
9690
9674
0.022
96-1110003 0033
Co-leaching
967911 0059
9641 0086
9676::0088 0088
9767 0096
971 0084
07091:0010
97001 0218
97291.000
0015
97451 0272
L-2
Peer loss
Juccard
96811 0103
9647.1 0057
96571..018
96672200106
9660-10033
9658 0059
0030
9611
9781 0043
0190
.799:1 0005
0784:1 0014
.777
9759 8005
.028
9711 1 0010
9763 0159
.954
9695 0000
009]
9150:1 0086
9716:1 0036
GIB
9649.1 0052
9634
9200
97144:
97861
9674
9660
20000
9778:1 0089
SupCon
9625:1 0024
9677
00774
0632
9757=
0018
97184 0023
9697 2 0014
9749 0013
0074 9755
0750:1 0630
GRACE
9145.1.0105
.898.1
3935
9881
5081 0121
9005 0062
J9931 0012
9171
1 0012
9181
0054
9196.1 0058
RGIB-REP
97541 0026
96981
9652: 0027
0797 0025
9757 0016
9744 0028
9604
0023
9798
0016
9782:1 0028
RGIB-SSL
997944 0023
9655 0025
9595-
9709
9709
.9644-
0024
0025 .9638
9736 10000
9797 .0022
9979: 0034
Standand
9616-40003
939-
9274
0008=00038
9433
0761
0271
9686
0000
0580
0021
9362:
0035
DropEdge
9568- 0044
95481 0058
9407 0050
9567=-0039
9433:1
0055
9453
0088
9580 0000
9579
0033
9578:1 0033
NearalSpurae
99992 0026
0013
9402 0657
9609 0027
9540
0178
9348
0169
9583 0017
0583
0050
9571
0011
PTDNer
50077
95144-0016
9424 0997
9610= 0M224
94574
0194
9500 0000
9585
+ 0002
0576
00150
0666: 0635
Co-teaching
9516110199
9483=
9524- 0123
94461
9447
0256
9642
9650
0058
9513:1 0050
L-4
Peer
Jucard
95432 0090
9500.1.006
0193
0558
0267=-0370
9344:
9507
0053
0656
94361 0286
948741
0780
9364 0355
9417
9631
9401
1: 0051
+ 0161
9659 1. 0020
9501
- 0055
0569: 0227
9557.1
GIB
95541 0006
9561
0: 2.092
9121 0267
0267
9605
0060
9531 0231
94162
051
9651
1 0061
0587:2 0100
94891 0029
SupCon
9467 0045
9516:
9544
0580
0051
0477 0038
GRACE
8978.1.018 01:45
89671.0000
8949
0030
9007.1
0045
8.64 0013
9053 1: 0110
9074
0136
90751.00 0113
RGIB-REP
97211 0035
9621
12 0016
9510 0652
9706
0027
9604 0052
9480
0053
9785 0001
9797
0017
9785 0032
RGIR.SSI.
3665 0022
099740003
9500=
0650
9688-
0025
9570
0047
94%6 0002
9730
0017
07522
0050
07442 0632
Standard
9662-+00042
9511
9286
0000
9656-
0045
9450
0177
9276
0229
9752
0027
9766
0085
9745
0640
DropFidge
0:12
9484 0053
9608
9481
9331
9537 0012
9559
0547 0024
NearelSpurse
9644 0035
9550
-066
0588=00055
9468: 0038
9387 0053
9525 + 0033
0541=
0051
0535: 0637
PTDNer
.065
9384= 0088
9637= 0024
9553 0051
.991
00344
9537 0007
9559
0012
9541:2 0632
Co-teaching
968411 0058
0567-2.0130
9119=
9505 0121
9534:
0222
9271 2 0308
9679 0198
05ST
0112
0557 0782
Peer loss
0128-10010
9551
9308=
3441
9358
9575 0164
9667
0250
059122 0236
Juscrd
9754:1 0063
954911.0088
2202:00104
9637
0035
9495:1 0213
9457110330
9500 0040
9550
0128
9433:1 0066
GIR
9600:1 0003
066112201346
9129 0738
9544_ 0212
0547:1 0395
9456 0219
9619 mm
0905
0112
0516:2 0124
SupCon
9537+ 0054
9452+
9352= 0846
9534
8968
94544 0068
9375
(023
9618 0036
9627=
0052
0557= 0067
GRACE
90041
8005
0030
9025
89631
8952
9154
91662
9151 0082
RCIR-REP
0078 9666:2
9580
9467=000
0067
9662=
0027
9406
0041
9453 8049
9740
0013
9776
0021
0751: 0011
RGIB-SSI.
9633 0032
94162 0036
9177 0082
9627 0037
9482 0037
9276 0093
9712 0017
9742 0025
9726-2 0632
Table 29: Full results on Chameleon dataset with GAT.
GAT
layers
methed
20%
mixed noise
40%
injut nolse 40%
60%
20%
lahel noise
40%
Standard
.9725 4-
-9650+-0000
96625-4
19267
00125
55147-4
(742-4-00220
00/20
.9759 H- 00018
.9146
+ 0023
9943-+-000 2017-
99111--0000 0041
Droplidge
9916.1.002
.9630
9556311.00
0023
9747
0022
9752
9926-1.000
9698
NeariSSprre
.53334 0029
9667
00125
9615-2.0022
0028
5762-2.002
0012
57148-t
0028
9744
4: 0015
9702
+ 0002
9742 0055
5904-1: 0042
PTDNet
.52361 -022
.667 + 9004
9620
0027
9768
0023
97255
0021
9739 6: 0022
9070
0021
9631 1: 0022
9611
0058
Co-traching
96174
9629
0047
9626-2 0051
9624
0084
5645-t
9647
9684
9610
9688
0092
L=2
Juscard
loss
.9755
.9622 9663
.9665 - 0062
00001
9683--
9602-
96524.012
0130
5621
99699-1.000
0112
0051
.9616 0123
9611
0078
9508
96.38
4 0006
9612 0047
96.24
00774
9603- 0134
9620- 0089
GBB
9628 0018
9639 0115
9625
0208
9681 1 0091
9643-
0257
9601
+ 0285
9644
0119
9688 0095
9631
- 0115
Suption
30009---0000
9611
0195
9607
0032
97524
5021-4
0026
9701
9639-
GRACE
J908
77829
0494
3396-4.000
0693
79924
0371
-960-4
0071
.7913
0418
7908
0261
.7985
0322
.789111000 03:00
RGIB-REP
5933-4
0051
9668
-. 0001
9629-
0026
5769-
0018
57522 0016
9753
2018
9706
0001
9753 0029
9732- 0001
RGIB-SSL
.835
0-037
9593
3005
95.36-1.000
0041
0002 9640
5587
0048
9945
+ 0005
900P + 0015
0022 9728
59119
00066
Standard
9921
1-035
9652
H: 0002
9605
L. 0031
9741
8 0028
9686
00000
9674
L. 0007
9740
0007
9738
E 0027
9712
0047
DropEdge
9666-1.0031
.9572
= 0034
9469
0043
9659
0028
9598
0021
.9996
1. 0010
9749
1 0024
.747
0016
9718
0031
NeuralSparse
9687-4-0027
9903
9536-+
0033
9699-
9600
1036
950
+ 0017
97745-
+ 0002
9756-5-0122
9027 0035
PTDNet
9907-1-0041
.96401:000
15009100034
9708
0027
9602
0025
9548
3: 0055
9754
11 0005
.9611 1.. 1.00
9709
1. 0011
Co Co-texching
9726-1 0096
.9488 0030
9639.1 0088
9551
1 0037
9503
0027
9556
1: 0003
9691
1 0009
9664 7565
9622
0242
Ls1
Peer loss
Juscard
9776-1.0089
35145-4-00
.992
.9714 0008
0098
9068-
9068+.0103
2013-100112
0103
9729
9637
0041
0024
5002--
9977-1
0211
008N
2012
9557
9049
+-
1 0078
9551
.0996
9973
555T- 0068
00137
GIB
.000:t 01448
9646 00641
9641-10089
0089
9629-2.019
0197
56422
0189
6047
96646- 6044
9624-
00105
9609
- 0064
Suption
004444-00333
9537
0106
9381
0381-1.012 0125
9675
- 0028
9631
0082
3955
1 0008
3594
+- 0131
994224.013
0013
5236
-1 00022
GRACE
5429 1. 1.665
.085
1133
6273
6178
0940
5839.1
1022
.6133
1. 1544
5534
1 0139
.61551..00 1055
6333
0818
RGIB-REP
3704++00035
9645
2012
9591-.003
6036
56544
0046
9572.4
0047
9741
0040
9745-
9718 0039
RGIB-SSL
090744-0034
.932
- 0034
9289-1.0000
00059
9526-1.00
0037
9451
1 0045
9300 .0083
9619
+ 0003
.9655
0042
9643
0038
Standard
9659-1.0029
.9533
1: 0006
9482-1.005
966811.0033
9543-1
0075
9874
0000
9722
0013
9688-1.000
0055
9698
:: 0065
Droptidgee
9461 4002
.9295
12 0063
9062:1 0216
9469
2.99 00.99
9293-1
0087
9180
1 0063
9698
1 0006
9689
0046
9694 0037
NeuralSparse
.953 00039
.9391
+ 0057
0.055
9473
0149
9038-1
0043
.9250
9688
0007
.9005
0026
9096
0063
PTDNet
9508 0308-1.0075
.959
0222
9255
9535 0002
942411
0053
2003
97211
.9177
9697-1 0016
Co-leaching
9609-1 0033
.9277 .019
950111.0141
9709 0129
9569-1
0115
9483
0155
9671 0085
9698 0260
9K11
=: 0176
L=6
Peer loss
Juccard
9773611-0116
9680-1.005 0057
.9570
.900 - 0055
9485 1. 0076
95101100106
9659 1659-10000 0000
98151.0185
9563-1.
9555
0-355--000 0057
01
9968
9459
1: 0006
1. 0100
5900
97501 0124
11 0126
.732 0093
9701 1: 0196
9962
9972 0122
2 0078
GBB
9682-4-0138
9616
9612-.0165
0776
5610-
0922
96244
9557
+ 0071
9514--0000
-077
9985--001
0138
SupCon
9981 148111.0135
.9300=.0171
9015
9455 0267
9490
0157
9114
0229
9411
4 6006
93370
00174
9211-100 0144
GRACE
5681 1 07785
.107 - 0257
11471100661
1599--0076
5046
0124
5167
1. 0502
5850
0827
5884
$518 0104
RGBI-REP
9612-4-0000
9410+-0099
9258-4.0166
0169
9531
0054
5417-4-00
0058
.913
+
9747
5707 0056
RGIB-SSL
9367-1.0058
.933
0016
9199.1.0000 .053
9418
0085
9226611 9226:1.0056
.9205
0000
9576
0066
.962 0079
9557 0083
36

---

Under review as a conference paper at ICLR 2023
Table 30: Full results on Chameleon dataset with SAGE.
SAGE
layers
method
mised moise
20%
inyur moise
40%
60%
20%
lahel noise
Suaddan
96811-0006
96054 0013
9536
0026
9754
0019
9714::
9762:4
0018
9700
0019
9670 0026
0601:2 0049
DiogEdge
9004+ 0025
96144-0002
9556=-002 0022
9677=
9669:t
9673
9630
0016
9678=
9625- 0047
NearalSparse
.969---0031
0031
00029
9650== 0029
9751= 0025
9751
57335 0002
9945 8001
9708
+ 0011
9661 0.051
9006-2 0051
FTDNer
96742 0035
9994-2 0012
9538 0019
9752 0022
96374 0033
9651 2 0029
9601
2 0016
96582
0005
05984 0042
Co-teaching
0040
95494
9543=
9453=
0425
94094:
0009
0595
0099
9525 0909
L=2
Peer loss
Iaccard
9677.1 0074
9638:1 0096
9981
9650220012
0006
9812:
9589 0656
0021
9503
9541
0007
0091
9561 0075
0540:2 0190
9517 0015
05571:0100
9907 0010
9625
4: 0146
0522
95062
0155
0027
9587:1 0227
054512
0118
GBB
9660 0006
9593-4
0214
9530= 0621
9671
0007
00000000000
9659
0275
9537 0055
0525-
0044
0586 0068
SupCon
963641
9666
9509
9585:
9503
9567 0004
0530
0027
0516:1 0037
GRACE
82154 0332
8327=
1372=00245
8369
0146
2251 0441
8295
8610
8583
5641:
RGIB-REP
9099 0028
96266
0034
9765-
0022
9741+
0018
.9700
00144
9933
0017
9715 0025
06700 0037
RGIR-SSL
9740:1 0030
.9009.00027
9686 0025
9752 0020
9737
0031
.741
0035
9789
0016
.791
- 0028
9781 0029
Standand
96452-0035
9505=
0010
9221
0017
9721
0021
9721 + 00166
9678
+ 0006
9641
+ 0028
0605
0051
DrogEdge
96444
9585-
8M12
9516 0024
9652=
9662
0019
0021
9663
0000
9614
0638
NeuraiSparse
9649--00022
93764+.000
0015
9495
-619
0003
9002
0022
-9620+-0000 8001
9675
+ 0017
96424
0051
9003- 0046
PTDNer
9651 0036
9574
20144
95062
9624 0027
9622 0015
9650 00135
9687
0004
9646
0025
0596:1
00556
Co-teaching
97722 00500
9665-
0040
95822
5083=
0083
9033+
0101
.613 0007
9617
- 0255
9638
0070
9637 0066
L=d
Peer loss
9542
9663-4
0044
9496-
9610-
0110
9667
.9601
9657
96794
0012
0661+1.010
laccaed
943- 0076
9485-+
0069
2653--0115 0115
9683+
0218
9697 - 0157
9606 0043
0517 00448
0681= 0088
GBB
9557
9038-0017
9644
0214
961 0200
95599 8005
9576 00026
9464 0080
5303 0065
SupCon
94942 01300
9351
0001
93632
0960
9592
0122
95384
0122
9517
0125
9478
+ 0009
0501 + 0126
04492 0999
GRACE
8040 0364
85424
8415
8266-
8302=
8642
0533
7462
77972
0306
7W92 6341
RGIB-REP REP
90664
9582+
0520
-010
97194
0028
9722
0012
9000
9696
0023
9671 0058
RGIR-SSI.
9775 0027
9687.
0078
9655 0000
0021 9740
0707
0040
9721 + 0044
9748
0003
0051 9762
0040 0754:1
Standard
0006--00066
9541 00s5
9470
9679
0051
9669
0001
.640
(009
9673
0006
0641
+ 0058
9625 0042
Drogkidge
9532+ 0028
9487
0063
5592
0828
9582+
0035
9580
0299
9500
+ 0014
9561
+ 01150
5953- 0024
NeuralSparse
9525 1 0043
.9333
0066
9480-
9597
9509-1
.9530
0057
.589
+ 0024
95524
0025
.95
15 0034
PTDNer
95364 0042
9557+.0015
9483--0977
9706=
0035
9581
9459
0077
9580 0004
0550
0022
9429- 0045
Co-teaching
.500 00700
9355=-0029
9492- 0031
9457-+-000
-007
.920 +: 0015
9445
+ 0009
9365
H 0015
94977-1-011 0115
L=6
Peer loss
9953 H 0006
.9:09
10066
0015
5697 0084
9656-1.011 0110
.9622
+: 00366
.921 + 0210
9595
9559
H 0047
9416-+.0113
9535-+88116
9535=-0111
9610=
9368:
9463
9518 0057
9421
+ 00052
9433 0137
GIB
9009---0133
95794-9000
9568=-007
9691=0031 0315
9098---000
9992
0279
9633
9366
00774
9947--011 0015
SupCon
9310:1.0107
9178-1.011
9291
0114
9244 0271
.981 1: 0231
9342 0022
9375
0015
9013-1 0087
GRACE
0779
.775
7718= 0880
7275
= 0909
79144 0755
810P 8K55
7754 0258
7629
0508
7686- D437
RGIB REP
99636
0002
9955
2M12
9455-
0012
5641
+
9660 +
9683
+ 0013
9681
0033
5664-
0042
RGIB-SSI.
9665: 0030
9601 0045
9563-20033 0636
9662 0052
96641
0040
9610 0032
9609 0022
97162
0033
0008
0011
Table 31: Full results on Squirrel dataset with GCN.
OCN
layers
method
201
mixed moise
60%
20%
inpus moise 40%
60%
20%
label noise
Standard
1,621 0011
9519- BM20
9444 08124
9610==
0012
5490
0091
9401 +: 0096
97744
0013
9931 1231::200 .011
9712-
0011
Droplidge
9571-+-0015
9480-+-001
9270==0022
0022
0011
9426-+
9939 8006
9623
0015
9612-.0000
0000
9604-
0014
NearalSparse
0011-1-0019
9528-+.002
9409
0022
9621
0017
9495
1 0016
.9544 +: .002
9741
9 0013
9724-4.0015
9922-1.0010
0010
PTDNer
966124 0018
98133
0036
9424-
0624
9619
0016
9488:t 0635
9363 - 0002
9744 0011
0723
0015
9718 0009
Co-teaching
9907
.9535
2044
9465-
9525-4
9454
9666
0304
965111
L=2
Peer loss
Jaccard
.61334 0027
9600:t 0015
9608
9568
8057
06166
9609-
0697
0014
9636-
9622 0159
0107
9491 0125
95944t 0111
94%
9397 2 0119
8091
9528 2 0017
9694
9: 0089
9628
01
9401 0010
0676:1 01.49
GBB
9622-4 01955
94778
0094
9489
0159
5294
0260
94611
0091
9432 + 0058
9457
+ 0099
5462
0017
9492-1 0060
SupCon
9494-1 0013
9439
8009
9458-
9421=
95491
9489
9444
0011
9423
0015
9118 0909
GRACE
9292-4
0011
9316
2010
0015
9290-
9317:4
0010
9527
9054 0018
9058
0015
9052:1
0018
RGIB-REP
9614-1-0019
0019
99544+.00066
0035
9611
9611
0021
9512- 0092
9992 0043
9761 + 0013
9753-+.0011
5945--.000 0015
RGIR-SSI.
28711200011
9476 0029
9402 0020
9534 0016
9486-t
0027
9951
2
9752 0023
0749:2 0019
9738 0023
Stundard
9433-4
0036
94006
0011
9386 0025
9416 0042
9398
0011
9411
+: 00400
9720 0016
9720
0016
10:00:1000 0021
Dropkidge
9439-4 0063
.937-
0014
9376-1
.9358
+ 00033
9008
9603
0003
0016
NearalSparse
949111.000
0065
.90991
8022
9297-
0033
9169
9400
0028
.9177
0062
9633
9626
0016
0016
PTDNer
9485 0056
055266200054
9804= 8048
9469
- 0048
5400 0050)
.9579 80026
9633 + 2018
9625
0012
56226
0012
Co teaching
9461 H- 0059
.982
0054
9274 0051
9462
0101
9415 + 0092
.306 + 0053
9675 + 0292
9641
00115
9655 - 0264
Peer loss
9417
99355
9286
0044
9503
9386
9936
9696
9694
0003
9696 0006
Juccard
9443-2 0047
9327
9244--095
9288--011
0177
9335-
0146
.9200
0055
9529
+ 0029
95124
0088
5501=100100
GIB
9472---0127 0027
493294.0155
0202
9390--000
0045
99006 0196
.939
+ 0024
9641 + 0005
9628
9628-500032
9601
0012
SupCon
34734 0010
9348+
0017
9901
0372
0010
9343- 0087
9505
- 0003
95
9516 0042
0595
0011
9511 0040
GRACE
91994
0008
9330
2007
9563- 0016
9592-
9958-
0019
9174
0014
9166-1 2013
RGIB
9509-4 0036
.955
0077
9434--0009
9895-0000
0053
9422-4
0059
.9805
0000
9735
0013
.933
0015
9937-4.000
RGIB-SS1.
9499 0040
9426 0019
9425=2000
9479
0051
9429
0017
9429
.0034
9727 0015
.729
0019
9726-2 0011
Stundard
9484---0049
99299+-000
9408--000
00039
9489
- 0057
5408-
+ 0021
.9366-
.0022
.903 + 0002
9665-
.012
9636-1
0034
Droglidge
9400-1.003
.9375
..025
9249910011
9474
0 0057
9409
1 0030
.9665
+: .002
9669
1 .0022
9652
8 0031
9638
H 0038
SeuralSpurse
9508 30081100019
9-12911.001 0044
9391 0029
9532
0060
9412.11
0011
.388
1. 0006
9881
1 0020
9647
2. 0018
9671 0019
PIDNet
.941++.000
9514--000
0044
9432-4
0068
.9800
-600 00023
9667
00055
9649-4 0025
Co teaching
9480-1-0144
9954-11.0074
1076
9538--
0051
9424
0018
.9379
20122
.672
.019
9694
9651-1.011 0195
I-6
Peer loss
Juccard
9519-4-0138
9539-1.005
99459 0100
9+12011.0127
9298--0015
$405: (960
0015
5965
9485
0070
01.39
0012 941661
.9228
9662
.0105
0001
9684 0152
+
.687
96.3
+ .0022
1. 0188
9673- 1077
9621 0189
9447-1.0103
.9159 80126
9110
9131
0269
9413
0201
.9366
1: 0228
.633
1 2001
.61999.011
0.1114
9612-1.00
0004
SupCon
9590: 0023
.0019
9580 0024
9519:
0018
9464
.0026
9633
+ 0051
0524 00448
5419-1.00
0012
GRACE
5399-4 0007
9372
0006
9248-10011 7017
9397
0012
9330-1
0009
9349
+ 2017
9213
+ 2007
9218
0013
5211-
0014
RGIB- REP
9556
0028
999221.000
0090
9463 0063
9570
0018
9496
1 0060
.420
1. 0056
9577
1 0029
9676
1. 0051
9662
0018
RGIB-SSL
9462-4
0029
99771
0412
9509
0032
9474-
0017
9449
0000
9642 2012
9659
0017
9647:1 0010
37

---

Under review as a conference paper at ICLR 2023
Table 32: Full results on Squirrel dataset with GAT.
layer
method
20%
mixed noise
40%
0476
20%
inpu:
40%
60%
20%
label moise
40%
0076
Stondand
9600:: 0007
96354-0017
9588-000 0024
9702-
0008
9000:
9659922000 00144
9719
0018
9701 = 0017
9686:2
0012
DrogEdde
9619:1.0014
95671100000
9505 0022
9627 0008
9593:1 0014
.955 1: 0011
9684 0015
9662 0022
9661 0018
NearalSpurae
9656-2 0009
960722.0012
9533 0019
9662
0011
9639:1
0010
9584
0017
9709
0012
9697:2
0014
9685:2 0009
PTDNet
0012
9546:10022 0024
9668
0010
9630:1
9600311000
9021
96.011:0000 0015
9698
0012
968911.0016
Co-leaching
9566:1 0034
19662.1.0066
0026:00026
9505
0045
9509.1 0092
95091: 0016
9591 1: 0293
98802.001
0174
9360:1 0089
L-2
Peer loss
Juccard
9715 0098
9720:1 0097
9726220005
066111.0053
9682 0106
96/8:10094
0202 9807
9706
9768.1 0011
971 0064
9704 0107
9645
9637
9681
4 0121
1- 0081
9668: 1. 0.004
96352 0069
9627.1 0002
9681: 0116
GIB
0055
9610:
0000=00001
9965:100000
9638
0123
9660
9611
967911
SupCon
9471 0019
93862
0023
9353
0926
9539 0032
9463 0026
9411 0014
9639 0028
9631 0019
9636 0021
GRACE
7300:1 1193
7533-1.1135
70152.1383
7.94
1101
8903
1807
7052 1659
5109
1: 0382
.7780 1044
7608 0620
RGIB-REP
9675.1.00 0014
.631
00311200012
2012
9587 023
9692
0007
9660:1
0013
9653
0016
9735
0012
9719
0013
9715: 0010
RGIB-SSL
9453 0020
99421
9285-
0002
9418
0022
9394-
0033
.932
9002
0007 9726
0037 9705
9695-+0004
Standand
19581
94%
8063
9335=00002
0592
0047
9455:
9415
0061
9682
0060
9690
0028
0686 0021
DropEdge
0541
93401
0046
9234 0959
9476
0054
9354::
0067
9292 0079
9539
0047
9513
0054
95 0110::000 0037
NearalSpurae
0044
9383-
0046
9120
0044
9401
0071
9337
0045
9884
0044
9559
0001
0555
0052
0556-1 0022
PTDNer
94164 0054
9510+
0059
9302
9447= M051
93934
0053
95515
0131
95744
+ 0099
US5T= 0022
9552 0042
Co-teaching
9504110009
90241.0130
9029-
9821=
0074
9345:1
0126
9371
0191
95761
9592
9551 0087
L-4
Peer
Jucard
95301 0071
956111.0133
9:123110160
9506
9296. 2110
0370=00--0
9500
9578 0185
0965
9411 0086
9583.1 0256
9558 0000
24142
9548 0178
9572 0074
9572 00K3
9500 0088
9593 0134
9554 0089
GIB
93514 0080
02711:0000
9098 077
0548
0156
9504:
0240
9550 0348
95442 0023
0549 0073
9524 0060
SupCon
9997
9322
9266 7440
9340
9314
9574
9557
0065
9556 0049
GRACE
5710:1.1110
5658.1
6203
5684
1202
5771 0945
602) 1. 1200
5291 1: 0694
5097.1.011
0139
5029
0072
RGIB-REP
9541 0059
9432
9116: 0963
9526
0637
9401 0072
9389 0015
9681
1 0078
967222
0029
9660 0048
RGIR-SSI.
93472 0038
0331
9260= 0652
9321 M222
93274
0020
9270 0003
9902 0031
9651
0014
0641 0021
Standard
9957++0000
93094.0144
9254= 0089
9487=
0065
9419:
0041
9255
+ 0003
9585
+ 0097
0520
0070
9507
0162
DropFidge
004710009
9181 9279
9768=
9763:1
9175
94124
0002
9431
0076
9444 0090
NearelSpurse
9999 0088
9279+-0167
9224=
0965
93424 0023
9213
0115
9468
0118
9474
0102
9504 0051
PTDNer
9213=
9460= 0048
9340 0052
92733 0094
9482
0088
0541
9453 0077
Co-teaching
9219-10059
9229 0329--.021
9392= 0093
9357:1 0098
9261 2 0100
9406
0383
0445
0272
0490 0784
Peer loss
9:30:2-0136
9916220220
9149=0147
9170
9017
9230 0221
0341
91872 0925
Juccrd
9577:1.0122
9374
0157
0111
940611 0111
930911.0165
9303 0153
$3902
0151
91142 0223
GIR
9204-1
9475
0127
9484
0211
9487 01
9438 0307
9421 1 0133
9484
0101
0486-2 0006
SupCon
0209-20000
9174+
9107 0052
9278=
0957
92234
0054
9210 0008
9140 0072
0175
0088
9182-2 0081
GRACE
5000-1.0099
54271
6004-.1210
60131
5342
5064 0145
5017
5074 0190
RCIR-REP
9404-: 0067
9248
0018=0051
9437=
0084
97991:
0070
9238 8063
9473 0071
0073 9486
9397 0045
RGIB-SSI.
9286 0033
9245
9165=0055
9276 0023
9745 0015
9257 0037
9903 - 0043
0586 0027
9499
:: 0067
Table 33: Full results on Squirrel dataset with SAGE.
SAGE
byyes
method
mixed noise
Input noise 40%
label nouse
20%
40%
600%
20%
60%
20%
40%
600
Standard
9660:110015
.9626 10015
997001..011 0016
9737
0010
9726
1: 0010
.9721
6: 0011
9692 1: 0015
9643-3.0012
0006-1.0017
DropEdge
9657-1-001
9602:1.0015
9535
0020
9622
0010
9625.1
0009
.909
00000
9593 1 0009
.95001 0009
9593-1.000 0007
NearalSparse
9572-4 0008
.9177+.0012
9562-
9534--000
9534-4
.9521
8016
9589
9534
6009
PTDNEt
.578
0008
9513.1.0013
9559-
0021
9630-
0010
950111 0007
.95411.000
0010
9596
1: 0014
9529
0.00
9596:1.000 0019
Co-teaching
9526-1. 0031
956011.00
0071
9526
0629
9403 0050
9583:1 0061
9585
0101
9533
1 0002
0533
0306
9539.1 0026
L=2
Peer loss
Juccar
9552 00079
9658
.961911.000
.9507
0050
0068
9598
9630-
0020
0066
9512
9600
9534-4
961911.07
0173
.9117
.9599
1: 0171
+ 2013
9525
9620
+ 0005
1: 0105
9608
.955
1 00178
L. 0068
9600 0014
10:4 0187
GaB
.90661 0079
96600-
9661=
0060
9632
0260
9517:4 0213
9556
+ 0010
9565
+ 8009
0595-2
0002
0111
SupCin
99225 H. 0015
9941++0001
9409
0000
5585
- 0061
9:55
6 01011
9933
+ 0002
9533
+ 03066
9539
-. 00026
9400
0021
GRACE
8543 0076
.8572.1.000
0094
8528:
0116
9918
8563
0068
3601
1 0087
.8528
1 0015
8477
1: 0119
8459
0000
8152-10011
RGIB-RED
9701 0010
96474+9004
-597--0088
9757
000
9747-4
0007
.9557
+ 2011
9735
9707 0007
5682- 0016
RGIB-SSL
9536-1-0015
0016
19800
9448-
0000
9734
0014
.9799
- 0015
.9803
+: .0044
9765 9 0001
.9022
9691-1 0005
Standard
.984-110107
9577.1.0076
9541 002]
9637
0092
9600
0079
.607 1: 0000
9663
1 0000
9612-1.000
0049
9612-1 0020
Dropfidge
0518-1.0030
0577.2.002)
9515
0033
9656
0051
261 0093
9573
0111
9683
0000
962811 0014
9608
0012
NearalSparse
9601-1 0083
.9570 .040
9513-1 0000
9618
0066
.959-11
1:
.9605 1 0061
.652
1 0005
.632
.0033
9605
0021
PTDNet
962111.000 0050
.9511
.009
9536-1 0033
9667-
0045
9668.1
96691
0052
9675
1 0003
9635
1607 0026
Co-teaching
9518:1 0181
9468.1
0093
9479
0024
9525
0086
9577.1
01.59
9528
0137
9625
+ 0044
9610 0220
9675:1 0097
Led
Peer loss
Juccadd
9632 H. .0196
96222-1-0135
.6622 0008
960551.0155
9587 0100
9602-00088
9877
01:5
0084
9626
9630011
1: 0120
0253
95101
9643
6: 00445
027%
9505
9637
4: 0001
1 0006
6508 1. 0003
9617-1.0170
9599
9691 1: 0016
0039
GIB
9537.1
9538 0275
9531= 417
9603
0881
9610:1
0115
9580
0310
9508
9523
0064
0559
SupCon
9478 0012
9412.1.0099
9342-10033
0029
9267
0045
9368
1 0028
.9499
0006
9375
1: .004
9935
$335-1.003
9017-1.002 0021
GRACE
$859-1.001 0115
.871
8609
0212
8824
0113
5305
1 0261
8653
0178
K386
1: 0130
846611.010
0103
8347.1.00 0180
RGIB REP
00088
9958--
0035
9536-
9682-
0029
5967-4-00
9584
00714
9689
+ 0014
9664
6- .0027
0022
RGIB-SSL
0002 9496
.9663-1.004
0047
9422-
0064
9531
.999 1
9019 .9551
0000 9622
9603
0087
9601
Standand
9555
0065
9958811
0038
9461
0054
9592
9600:1
.951
100022
9574
01192
9540
$5401.0207
9583.1 0023
Drog
9544::
0069
9411
9463=
0641
9566
0070
9473:1 01
9519 0053
9464 0023
9407:2:
0084
0196:1 0025
NeuralSparse
9563
1066
.91551.0000
9457
005
9591
1 0046
9410011 0052
9415 1: 0001
9371 1 0005
9341
0001
90199.1 0033
PTDNet
9565-1-0060
0060
952541.0050
0050
9463 0856
9593
0045
95841
0044
9992 00188
9584 0003
93421:
0018
92.6.1 0018
Co-leaching
9421110071 0071
9396-1.013
9251 0130
0585-
041111
9388
9144 0111
0204 9496
0983
L-6
Peer less
Iucard
9473 0108
9535-1 0058
1.0000
9305-1.0057
9133-10008-
9225_
9339-
9000
0053
0659
9312.1 0058
9386-1 0084
.9331 1. 0009
9453 0044
9296
9348 0200
1 0211
0210011 0285
3.755..115
9135 0035
9L71-1 0029
GIB
9437-1 0155
9323-2-0155
9284 0162
9402 0230
3406-1
31
9533
0335
3407
2 0360
0274 0364
91942 0679
SupCun
9392-11 00:19
.9300
9221=10136
0266__0022
931331
0102
.9309
1.
9261
1 0001
9202
1.00
91961
GRACE
8418:1 0983
8870
0623
8732--0151 0151
0151
8126.1
1348
88161
0148
8383 1 0007
8370
0295
3487:1 0121
RGIB-RB REP
9558
H. 0047
.9:74
9455
0052
9583
0044
9566
1 0035
9947
.000
9662
+ 0057
9646
6 0021
9634-1 0000
RGIB-SSL
9437
0026
.9391
9266:1.0022
0012
9881
0064
9519.1..00
0012
.969 1.
9578
1. 0001
.9133..0000
0076
9601110001
38


Reviews:
[{"review_id": "i_5tf_9y8NZ", "reviewer": "Reviewer_sr13", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper focuses on the edge noisy scenario in the link prediction task and proposes a method named Robust Graph Information Bottleneck. The self-supervised learning technique and data reparametrization mechanism are utilized to instantiate RGIB. ", "summary_of_the_review": "The research problem and the idea are good. The analysis, experiments, and writing should be improved.", "strength_and_weaknesses": "Strength:\n1. The coupled edge noise in the link prediction task is a good research problem and this paper proposed a reasonable solution. \n2. The motivation is clear and the idea is novel. \n3. The experimental results show that the proposed method is effective. \n\nWeakness:\n1. The inherent edge noise is defined as the additive edge noise, which seems to be not very unified since much noise for link prediction in the real world comes from missing relations. \n2. The understanding part in section 3.2 is not very formal and rigorous. It’s unclear whether the quantitive and qualitative analysis by alignment and uniformity is specific to edge noise. \n3. The design of the RGIB objective function is based on intuition, and its rigor needs to be further explained. \n4. The relationship between the two instantiations, their respective advantages, limitations, and scope of application are not well explained. \n5. Most of the baselines are published in 2020 and the latest one is published in 2021. Some SOTA methods should be included for comparison. \n6. In the IB principle, the trade-off parameter is important, and needs more analysis. \n\nOther questions and suggestions:\n1. Some related works are missing. There is a recent work[1] that incorporates the IB to purify the graph structure, which should be included in the related work or included in the baselines. [2] is also a method for graph denoising from the perspective of information theory. \n2. There are many graph adversarial attack methods that perturb structures. Since this paper focuses on the robustness of link prediction, how does the proposed RGIB perform under adversarial attacks?\n3. As the scalability of graph algorithms has become increasingly important in recent years, I would like to see the performance of this method on larger datasets and its time consumption analysis. \n4. I’m wondering how the distribution of H and the MI terms in the objective function change during the training process, which is helpful to understand the rationale of this method. \n5. Are the alignment and uniformity specific measurements for edge noise in link prediction? Can other denoising methods also promote alignment and uniformity? \n\n[1] Graph Structure Learning with Variational Information Bottleneck. AAAI 2022.  \n[2] Principle of Relevant Information for Graph Sparsification. UAI 2022. \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "aB-FDcw_X6", "reviewer": "Reviewer_zAFv", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This work find that the inherent edge noise can perturb the graph topology and labels, which may reduce link prediction performance. Thus, the authors propose an information bottleneck guided method, namely RGIB. RGIB achieves robustness representation of graphs. The experiments show effectiveness of RGIB. This paper is very well written and the theoretical proof is very detailed.\n", "summary_of_the_review": "I thought the article was well written. The experimental part is slightly lacking and I hope the authors can answer the questions being asked. \n", "strength_and_weaknesses": "Pros: The paper is well written, and despite the considerable complexity of the method, its presentation is relatively easy to follow. The task of interest is well-defined, and has clearly been effectively solved on the datasets considered. The experiments provided in this paper are extensive and quite comprehensive. Specifically, the authors verified the effect and superiority of their proposed RGIB method in detail from different angles. The experimental results are also convincing. \n\nCons:\nI am mainly concerned with the following issues.\n\n(1). The baselines do not appear to be new enough. Almost all of the RGIB baselines in the experimental section are from 2020. Can the authors include baselines from 2021 and 2022?\n\n(2). The work mainly focuses on robustness. Is there a metric to evaluate robustness of these methods?\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "DjnwafItP8", "reviewer": "Reviewer_m5JE", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "A method to handle graph-edge noise is introduced based on an information-bottleneck theory. It is shown that adding edge noise to real data will increase the link prediction error. The introduced method is shown to reduce this error.", "summary_of_the_review": "\nSadly this paper is too unclear to follow and understand. As well as addressing the above pointed out issues, I recommend focusing on explaining more clearly the edge data generation steps, the contribution added starting from the GIB work, and explaining why these additions improve on the existing work. I look forward to see the revised version as this is a very intersting research area.\n\n[After Review Period]\n\nThe paper has a very nice contribution with respect to the motivation [identify and remove noise from graphs for link prediction] and the intuition behind the idea [extend GIB to separate the label and feature noise], and the paper is clearly structured. Many of the concerns have been addressed and modified in the paper and so I feel that the paper is closer to being ready for publication. It is still unclear how the claim that the public datasets either do not have noise, or that this noise is always FPs. Another reviewer commented that typically the noise in graphs are missing edges, which contradicts the authors comment. The experiments on the public dataset without added noise also show improvements with their method, implying that there must be some noise. This leaves a number of unanswered questions about the noise in the data. Maybe the authors can inspect samples of the edges being updated by RGIB in the clean datasets to see if qualitatively they can start to investigate these questions. Also, could the synthetically generate data to analyse the behaviour of their method? Due to this and the number of initially found issues I believe this is still below the acceptance level, although it has improved during the review period.", "strength_and_weaknesses": "\nAddressing noisy edges in a graph is a sensible and practical direction of research, as it is not often considered, is likely to be present in real data and will reduce model performance error. This paper demonstrates that errors increase for the link prediciotn task with synthetically added edges to public datasets.\n\nHowever, it is many parts of this paper are unclear, how noise is added, what a graph illustration is trying to show, what the x and y values represetn, what values in a table represent, how well this approach works on the original data without synthetically added edges and more. This paper needs to explain more clearly many parts of the work.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "8SgdcfREt7", "reviewer": "Reviewer_gCKh", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes a robust graph information bottleneck (RGIB) for link prediction. This paper extends the previous graph information bottleneck (GIB) in a non-trivial manner by considering the fact that if the graph structure is assumed to be noisy, the labels in link prediction (i.e., hold-out links) are also noisy, and therefore the MI maximization term in the original GIB objective can still introducing noise in learned latent representations.", "summary_of_the_review": "A good paper that theoretically extends the GIB framework to handle label noise problems in link predictions.", "strength_and_weaknesses": "This paper has several strong points that I appreciate very much.\n\n1. The authors address a novel problem of label noise in the GIB framework.\n\n2. Two adverse consequences of the existence of label noise in the original GIB framework, i.e., poorer alignment and worse uniformity, are clearly identified with good visualization.\n\n3. The authors proposed a self-supervised learning-based extension to GIB, where the alignment and uniformity issues seem to be addressed with good theoretical support.\n\nMeanwhile, I  also have some questions for the authors.\n\n1. Since the target links are randomly selected from the existing links, isn't the structure loss and label noise the same noise rather than two different noises that are coupled together? Although I understand that the use of coupled noise makes it easier for the theoretical analysis for the discussion of GIB's issues and RGIB's advantages, I still think coupled noise may not be the best-fit description for the case.\n\n2. In addition, there seem to be insufficient discussions regarding the limitation of the methods. I wonder whether the strength of the RGIB will hold if the \"coupling\" between label noise and structure noise is weak. Moreover, I'm also curious whether the RGIB framework can generalize to the same problem setting of the original RGIB for node classification, where there exists noise in the node labels?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}]

Author rebuttal:

Decision: Reject