Deep Graph Neural Networks via Posteriori-Sampling-based Node-Adaptative Residual Module
Authors: Jingbo Zhou
Yixuan Du
Ruqiong Zhang
Jun Xia
Zhizhi Yu
Zelin Zang
Di Jin
Carl Yang
Rui Zhang
Stan Z. Li
Venue: NeurIPS 2024 poster (2024)
Keywords: Graph Neural Networks
Deep Models
Primary area: graph_neural_networks
TL;DR: We propose Posteriori-Sampling-based Node-Adaptative Residual Module to enhance the performances of deep graph neural networks.

Abstract: Graph Neural Networks (GNNs), a type of neural network that can learn from graph-structured data through neighborhood information aggregation, have shown superior performance in various downstream tasks. However, as the number of layers increases, node representations becomes indistinguishable, which is known as over-smoothing. To address this issue, many residual methods have emerged. In this paper, we focus on the over-smoothing issue and related residual methods. Firstly, we revisit over-smoothing from the perspective of overlapping neighborhood subgraphs, and based on this, we explain how residual methods can alleviate over-smoothing by integrating multiple orders neighborhood subgraphs to avoid the indistinguishability of the single high-order neighborhood subgraphs. Additionally, we reveal the drawbacks of previous residual methods, such as the lack of node adaptability and severe loss of high-order neighborhood subgraph information, and propose a \textbf{Posterior-Sampling-based, Node-Adaptive Residual module (PSNR)}. We theoretically demonstrate that PSNR can alleviate the drawbacks of previous residual methods. Furthermore, extensive experiments verify the superiority of the PSNR module in fully observed node classification and missing feature scenarios. Our code
is available at \href{https://github.com/jingbo02/PSNR-GNN}{https://github.com/jingbo02/PSNR-GNN}.

Deep Graph Neural Networks via Posteriori-Sammling--aeeee Node-Adaptive Residual Module
Jingbo Zhou Yixuan 2 Ruqiong Zhang "aang":3JJun Jun Xia', Zhizhi Yu',
Zelin Zang', Di Jin Carl Yang", Rui Zhang"; Stan Z. Li Westlake University, Jiilin University, Tianjin University, 'Emory University fzhoujingbo, stan. vtam.zq.li)@aee (1)]weelllkke edu. cn
Abstract
Graph Neural Networks (GNNs), a type of neural network that can learn from graph-structured data through neighborhood information aggregation, have shown superior performance various downstream tasks. However, as the number of layers increases, node representations become indistinguishable, which is known as over-smoothing. To address this issue, many residual methods have emerged. In this paper, we focus on the over-smoothing issue and related residual methods. Firstly, we revisit over-smoothing from the perspective of overlapping neighbor- hood subgraphs, and based on this, we explain how residual methods can alleviate over-smoothing by integrating multiple orders neighborhood subgraphs to avoid the indistinguishabiliyy of the single high-order neighborhood subgraphs. Additionally, we reveal the drawbacks of previous residual methods, such as the lack of node adaptability and severe loss of high-order neighborhood subgraph information, and propose Dnsterior-Sampling-basse
Node-Adaptive Residual module (PSNR). We theoretically demonstrate that PSNR can alleviate the drawbacks of previous residual methods. Furthermore, extensive experiments verify the superiority of the PSNR module in fully observed node classification and missing feature scenarios. Our code is available at http:://hhhbbccmm///g//////////
Introduction
GNNs have emerged in recent years as the most powerful model for processing graph-structured data and have demonstrated exceptional performance across various fields, such as social networks [22].
recommender systems (6, and drug discovery 4. Through the message-ppassigg mechanism that
propagates and aggregates information of neighboring nodes, GNNs provide a general framework for learning information on graph structure. Despite the remarkable success, according to previous studies [17 32), GNNs show significant performance degradation as the number of layers increase.
One of the main reasons for this situation is over-smoothing [17 21 32 [5]. Over-smoothing refers to the phenomenon in which node representations become increasingly similar to each other as GNNs recursively aggregate more neighborhood information. This indistinguishability will inevitably degrade the performance of deep GNNs, restricting their ability to effectively model long-range dependencies among multi-hop neighbors.
Several methods have recently been proposed to alleviate over-smoothing in deep GNNs, According (26], these methods fall into three categories: Normalization and Regularization 34 33) Change
of GNN dynamics [5), and Residual connections 32 [6]. Among all of them, the residual- based
Equual contribution.
t Corresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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method is inspired by the success of residual neural networks (ResNets) 9 in computer vision. This type of method introduces a residual connection to the GNNs architecture. For example, JKNet [2
learns node representations by aggregating the outputs of all previous layers at the last layer. DenseGCN [6] concatenates the results of the current layer and previous layers as the node representations of this layer. APPNP [15] uses the initial residual connection to retain the initial feature information with probability a, and utilizes information aggregated at the current layer with probability 1 GCNII [3] shares a similar framework with APPNP, and it further introduces an identical mapping to avoid overfitting.
In this paper, we study the over- smoothing issue in GNNs, with a particular emphasis on residual methods. First, we revisit the over-smoothing phenomenon of GNNs from the new perspective of overlapping neighborhood subgraphs and explain the essential reason why the residual method can alleviate the over-smoothing. In essence, these methods mainly use multiple neighborhood subgraph aggregations to alleviate the indistinguishabilityy of the single neighborhood subgraph aggregation, thereby improving the performance of the model. On this basis, we find that these residual methods often lack node adaptivity in utilizing multi-order neighborhood subgraph information, and at the same time, they still struggle to mitigate information loss when dealing with high-order neighborhood subgraphs, which hinders further improvement in the performance of deep GNNs. Although some residual methods, such as DenseGNN, can avoid these drawbacks, they tend to introduce more parameters at deeper layers. This can lead to significant memory consumption and is prone to gradient explosion, limiting the scalability of the methods.
Node-Adaptative Residual Module (PSNR). More specifically, this module introduces a graph encoder learn the posterior distribution of the required residual coefficients for each node in different layers with minor overhead. And then, we can obtain the specific fine-graing graigra node-adaptive residual coefficients by sampling from the distribution. The contributions of this paper are follows:
Considering these limitations, we propose a Posteriori-Sane Saaterrorr-Sampllng-bbeee
Perspective: We revisit the over-smoothing issue from a novel perspective of high-order neighborhood subgraph coincidences and explain why the residual methods can alleviate Through this lens, we reveal several significant drawbacks of prior residual methods that limit the performance and scalability of GNNs.
Method: We propose PSNR, lightweight and model- aggnstig module mitigate the draw- backs of previous residual methods and provide theoretical justification for its advantages.
Experiments: Extensive experiments verify that the PSNR module can effectively mitigate oversmoothing and further improve the performance of GNNs, especially in the case of missing feature that require deep GNNs.
2
Related Work
2.1 Notations
connected undirected graph is represented by & = (V,E), where V (11, U2. . UN] is the set of N nodes and & C V x V is the edge set. The node features are represented in the matrix E N x where d represents the length of the feature. Let A E
denotes the adjacency matrix and Aij = only if an edge exists between nodes 9 and By. N the diagonal degree matrix, where each element d; represents the number of edges connected to node v;. À A - = represent the adjacency matrix and degree matrix with self-loop, respectively.
2.2 Graph Neural Networks
A GNN layer updates the representation of each node via aggregating itself and its neighbors' representations. Specifically, a layer's 's output H' consists of new representations h of each node
computed as:
(1) where hy indicates the new representation of node ;; and fo(.) denotes the update function. The difference between different GNNs lies in the update function fo(:) and the AGGREGATE(
h = fe (h, AGGREGATE (fh,) | 21 E V, (v,, uy) E E)))
function, which are also key to the performance of GNNs. Graph Convolutional Network (GCN) 14

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Table 1: Common residual connections for GNNs.
Residual Connection
Corresponding GCN
Formula
Res
ResGCN 1G
H. H +o [D LAD iH, Wa-1)
InitialRes
APPNP 15
H. = (1 a) D AD H +aH
Dense
DCN
AGG H....... HR-1)
JK
JKNet
32
Hutpuut AGG(H, H..1)
is a classical massage- paassanggg GNNs follows layer-wise propagation rule:
Ha+1 = (D - H.,
(2)
where H3 is the feature matrix of the k-th layer, W₂ is layer-specific learnable weight matrix, (+)) denotes activation function.
2.3 Residual connection
Several works have used residual connection to alleviate the over-smoothing issue. Common residual connections for GNNs are summarized in Table [ where H₃, represents the output of the k-th layer, W4 a learnable weight matrix for the k-th layer, serves a hyperparameter denoting the residual coefficient, and (-) denotes an activation function. Additionally, in DenseGNN, AGG represents function with the concatenation of outputs from all previous layers as the input to the current layer, while in JKNet, AGG refers to the aggregation of all previous representations through concatenation, max-pooling, or LSTM-attention only at the final layer. Details can found Appendix B
Why does the residual method alleviate over-smoothing?
In this section, we revisit over-smoothing from the perspective of overlapping high-order neigh- borhood subgraphs. Based on this, we elucidate the role of various residual methods alleviating over-smoothing and identify their shortcomings.
3.1 Revisit ovor-smgg from the perspective of neighborhood subgraphs overlapping
GAT-2
GAT-4
GAT-8
GAT-16
GAT-32
GAT-64
Cora
Citeseer
Pubmed
0.30
SMV
0.25
0.2
0.20
0.1
2
21
22
2
24
2
21
27
2
24
20
22
2
2
2
27
Degroe
Degree
Degroe
Figure 1: SMV for node groups of different degrees. More results are shown in Appendix C
For message-passinn GNNs without the residual connection, the information domain of each node after k-layer aggregation is a corresponding k-order neighborhood subgraph. Intuitively, the size of its k-order neighborhood subgraph grows exponentially as A increases, leading to a significant increase in the overlap between the k-order neighborhood subgraphs of different nodes. As result, the aggregation result of different nodes on their respective k-order neighborhood subgraphs becomes indistinguishable. This explanation can be partially validated from the perspective of node degrees. Considering nodes with high degrees tend to have larger neighborhood subgraph overlap, correlation between neighborhood subgraph overlap and oversmoothiong can be validated nodes with higher degree exhibit more pronounced over-smoothing. To verify this point, we conduct experiments on three graph datasets: Cora, Citeseer, and Pubmed. Initially, nodes are grouped based on their degrees, with nodes having degrees falling within the range of [2 2(+1) assigned to the

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i-th group. Subsequently, we perform aggregation with different layers of GCN and GAT and then calculate the degree of smoothing of the node representations within each group separately. The metric proposed in [9 is used to measure the smoothness of the node representations within each group, namely smoothness metric value (SMV), which calculates the average distances between the nodes within the group:
1 SMV(X)= N(N 1)
EX.,. X,.),
(3)
i#j
where :),,) denotes the normalized Euclidean distance between two vectors:
D(x,y) =
x x
(4)
From the definition, a smaller value of SMV indicates a greater similarity in node representations. We show the result GAT in Figure More results can be found in Appendix C It can be observed that the groups of nodes with higher degree tend to be more similar to each other within the group in different layers. This finding supports our claim.
3.2
The role of residual method in alleviating over-smoothing
After verifying the conclusion that neighborhood subgraph overlap leads to over-smoothing, a natural idea to alleviate the overlap of the single neighborhood subgraph by utilizing multi- order neighborhood subgraph aggregations. In the following section, we show that the previous k-layer residual-bass GNNs essentially represent different forms of utilizing neighborhood subgraph aggregations from 0 to k orders.
Table 2: Utilization of neighborhood subgraphs by various residual methods.
Model
Closed/ttrrative form formulas
ResGCN
H En DNHH
APPNP
H₂ (1 - a)* N + a Σ (-1 (1-A)
JKNel
HA AGGINH. N
DenseGCN
H. AC 1.... Ho)
In the rest of this paper, take GCN, classical residual-free message-passiin GNN, example. Assuming that H is non-negative, the ELU function and the weight matrix can be ignored for simplicity. Combined with the formula of GCN in Eq.
2 the o-order neighborhood subgraph aggregation can be formulated as ~ where N = D AD- To show more intuitively how different residual models utilize multi-order neighborhood subgraph aggregation N* H, we rewrite their formula in Table Details of the derivation of the closed-form formula in this part are given in Appendix D As can be observed, the output of GCN's residual-based variants contains multi-order matrix products that represent different order neighborhood subgraph aggregations from k. There are two main ways to exploit them: (1) Summation, such as ResGCN and APPNP. Such methods employ linear summation over the aggregation of different order neighborhood subgraphs; (2) Aggregation functions, such as DenseGNN and JKNet. Such methods make direct and explicit exploitation of different order neighborhood subgraph aggregations through operations such as concatenation.
However, the utilization of multi-order neighborhood subgraph aggregations in these methods presents the following issues: Firstly, the summation methods all use a fixed coefficient sum the neighborhood subgraph aggregations. Consequently, these methods inherently presume that the information from the neighborhood subgraph of the same order is equally important for different nodes, which lacks node adaptivity. Secondly, for ResGNN, APPNP, and JKNet, when the number of layers increases, the output of these methods still involves many high-order matrix products that are ovor-smoo This leads to severe information loss when aggregating high-order neighborhood subgraphs, which in turn degrades model performance at deeper layers. Although DenseGNN seems to alleviate the above issues to some extent, the recursive use of all previous neighborhood subgraph aggregation would introduce more parameters as the model deepens. This increases memory consumption and raises the risk of gradient explosion at deeper layers.

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4 The Proposed Method PSNR
4.1 Methodology
To solve the above issues, we propose a node-adaptive and lightweight residual module named PSNR. The motivation is learn the adaptive residual coefficients for each node, thereby achiev- ing fine-grained and node-level neighborhood subgraph aggregation to improve the performance of GNNs. However, directly learning these coefficients through backpropagation presents signif- icant challenges. The primary challenge lies in the lack of transferability of learned coefficients. Specifically, in tasks such semi-supervised node classification, nodes in the test and validation sets often cannot propagate information to the training nodes through multiple message- message-passing steps. Therefore, we cannot learn effective coefficients for these nodes during the training phase. As remedy, we regard node-level residual coefficients as hidden parameters. Our strategy involves estimating their posterior distribution P(7k A, HH, k). In most cases, since the training, validation, and test sets orig- inate from the same distribution, the posterior distri- bution learned from the training data possesses trans- ferability to the validation and testing nodes. We can assume the posterior distribution to be Gaussian:
H
k 0 H' k-1
Encoder
GNN Layer
Posterior Encading
The ~ N (N((A, HE, k), o H&,
4 Residual Connection
A graph encoder can be used to model this distribu- tion. This encoder leverages graph topology and node information as inputs. Subsequently, to enable back- propagation, we employ the reparameterization trick. This technique enables represent the sampling pro- cess from the aforementioned distribution as follows: Figure 2: The framework of PSNR Module.
Sampling
PSNR Module
HR
T = H(A, k) + (A, k), N00,0).
Furthermore, we parameterize (( H3. k) and o(A, k)) as an arbitrary GNN layer. While the posterior distributions of residual coefficients vary across different layers, we do not parameterize specific encoder for each layer. Instead, we employ the same graph encoder and use the positional embedding generated from the layer number differentiate the posterior distributions of residual coefficients for various layers. Consequently, the PSNR module can be formulated as follows:
H h-1 GraphConv (H&-1)
H2 H1 + o(diag(m--)) (H1 H3-1),, 7p-1 N (pR-1, ok-1?
(5)
Uk, σ GraphEncoder,,. (H₁ H', + LayerEmb(())).
where GraphConv(-)) is the k-th layer of any backbone GNN, H₃ denotes the node representation matrix of the k-th layer, and H'2 represents the output matrix of the k-th layer. The first equation corresponds to the aggregation operation of the backbone GNN at the k-th layer. Tis represents the node-level residual coefficient the k-th layer, where the element 7i corresponds to the residual coefficient at the i-th node. In addition, 7) follows a -igeeniimnsional Gaussian distribution: N(pk, ox ²)),, and each time before each residual calculation, the distribution is first sampled to obtain the exact residual coefficients. μa and σı represent the mean and standard deviation of the distribution, respectively. diag(1)) represents diagonal matrix transformed from 7/k- where the i-th diagonal element is precisely the i-th element of 7)a- o6-) represents the sigmoid function, which constrains the residual coefficient to (0, 1). GraphEncoder, (-) is a posterior encoder, which can be any graph convolution layer that is independent of the backbone GNN. LayerEmb(() represents the positional embedding (29] with layer number k as input. 7 is a learnable coefficient serving as the layer embedding factor: The analysis of the residual coefficients can be found in Appendix H
Also, it is noteworthy that PSNR introduces randomness by sampling residual coefficients during both training and testing phases, thereby adding learnable random perturbations. This is different to other methods including DropEdge [24], GRAND [ and DropMessage 77) that only incorporate randomness during the training phase and primarily introduce perturbations to node features and
graph structure. We provide the theoretical analysis for this design Section Additionally, we clarify the difference between PSNR and other subgraph-based methods in Appendix F
5

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4.2 Theoretical justifications on the advantages of the PSNR module
In this section, we theoretically show that the PSNR module achieves finer mner-graieed and node adaptive neighborhood subgraph aggregations while avoiding the loss of high-order subgraph information. Firstly, combining with E the matrix form of the iterative formula for PSNR-GCO is:
H = H A (H - D 1/2 AD A Hx-1
For simplicity, we use Λ to denote ding((n:) Subsequently, based on this recursive form of formula, we derive the closed-fore expression of PSNR-GCN as:
R-1 H II,, M, II N. +M) M
k-1k-1
where Ñ; - and M, -A (I+) formula can be found in Appendix E The first two terms consist of cumulative product terms of different orders, similar to the form N thus approximating a new version of neighborhood subgraph aggregation. Additionally, the formula involves the aggregation of all neighborhood subgraphs from to A orders. This ensures that our method, like other residual methods, can comprchensively utilize multi-order neighborhood subgraph aggregations to enhance performance. Furthermore, since A& is diagonal matrix computed by a learnable posterior encoder with graph structure, node feature and layer number input, the neighborhood subgraph aggregation of PSNR- GCN is fine-grained and node-adaptive. This sets it apart from methods like ResGCN and APPNP.
H1.
The detailed derivation of this
Beyond that, we can also demonstrate that the PSNR module can reduce the information loss of high-order subgraph aggregation, thereby further improving the performance of GNNs deeper layers. Specifically, we aim to prove that as the order k increases, the smoothing rate of the cumulative product terms in PSNR is slower than that the matrix power terms other methods. Since we need to analyse the smoothing rate, which involves analyzing the asymptotic behavior of cumulative product terms or matrix power terms as the order increases, we can generalize the problem setup. Therefore, only need to prove the following proposition to support claim.
Proposition 1 Let = (S, = A,Nij E No), where A5 represents a random diagonal matrix with each diagonal element A3,85 satisfying 0 < Aja < L, and let X be any feature matrix. Then, as the order k increases, the product of elements in the set and the matrix X. denoted as S.X, converges to runk-one matrix with identicul rows slower than X GCN N
x (A) = -I=
Since the diagonal elements of the diagonal matrix A are all results of the sigmoid function, we can assume that they have lower bound >> 0. Therefore, in this setting, each element S; in the set S is a row-stochastic matrix satisfying the following property.
Property For row-stochastic matrix St, there exists an 0 satisfying the following conditions: 6 S Sk.ij < 1, (i. 2. Sk.ij 0, if (i,j) E
We can refer to the conclusion in matrix satisfying Property 11 Leveraging mathematical tools such as joint spectral radius from the perspective of dynamical systems, this work proves that the convergence rate of GAT is bounded below by GCN. Due to all the matrix S, from S also satisfies Property [ this conclusion can be borrowed to prove Proposition]] T thereby demonstrating that the PSNR module can alleviate the loss of high-order neighborhood subgraph information. Furthermore, this conclusion also explains the introduction of randomness during both the training and testing phases. This random perturbation further reduces the smoothing rate, thereby enhancing the model's 's performance.
describes the attention matrix GAT as row-stochastic
4.3 Complexity analysis
We take PSNR-GCN as an example to provide a complexity analysis of the PSNR module. The time complexity of vanilla GCN layer mainly comes from the matrix multiplication of N and H. hence its complexity On"'d). And the main computational parts of PSNR module are the
(6) computation of mean and standard deviation, sampling of pk scalar multiplication and matrix

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addition, which correspond to complexity of Onn)d), O(n). O(nd), and O(nd), respectively. Thus the time complexity of the PSNR module is O(n²d) and the time complexity of a GCN layer equipped the PSNR module is O("²d). d). As for space complexity, PSNR module needs to store the computed mean and variance for each node, i.e., O(2n), which can be approximately considered as O()). Section 5.6 compares memory consumption with other residual methods on large graphs.
5 Experiment
In this section, we assess the performance of the PSNR module in comparison to other methods and answer the following research questions (RQ): RQ1. How well does the PSNR alleviate oversmoothing? RQ2. How does PSNR perform compared to other baseline when used with different backbone? RQ3, Can PSNR enable deeper networks to perform better under the missing feature scenario? RQ4. How scalable is the PSNR on large graph datasets?
5.1 Datasets
We conducted experiments on ten real-world datasets, including three citation network datasets, iee. Cora, Citeseer, Pubmed (27], two web network datasets, i.e., Chameleon and Squirrel (25),
co-author/co-purccaae network datasets, i.e., Coauthor-CS (28). Amazon-Photo (28) and three larger
datasets, i.e., Flickr [20). Coauthor-Physies (28 and Oghn-arxiv [1]. More details of these datasets
and data-splitting procedures can be found in Appendix G
5.2 Baselines and experimental settings
We consider two fundamental GNNs, GCN 14 and GAT 30). For GCN, we test the performance of PSNR-GCN and its residual variant models, including ResGCN [16), DenseGCN [16). GCNII 3 JKNet (32] For GAT, we directly equip it with the following residual module: Res-GAT, Init Dense-GAt JK-GAT and PSNR-GAT. And we adopt the GraphSAGE layer as the GraphEncoder of the PSNR module. The impact of different graph encoders on the experiments can be found in Appendix ! In addition, we compare three recent representative methods belonging to different categories aimed at alleviating oversmoothing issues; DropMessage 7 for the drop category, DeProp (8] for the norm category, and Half-Hop M for graph data processing. For the missing feature setting. we also conduct comparisons with several classical oversmoothing mitigation techniques, including BatchNorm [T], PairNorm 33 DGN [34,, Decorr [12, DropEdge [24]. For
all baselines, the linear layers in the models are initialized with a standard normal distribution, and
the convolutional layers are initialized with Xavier initialization. The Adam optimizer [3] used for training. Experimental results are obtained from the server with four core Intel(R) Xeon(R) Platinum 8358 CPUs @ 2.60GHZ, one NVIDIA A100 GPU (80G), and models and datasets used in this paper are implemented using the Deep Graph Library (DGL) and Pytorch Geometric (PyG). Further details on the specific parameter settings can be found in Appendix G
5.3 Effectiveness in mitigating over-smoothing (RQ1)
In this section, we aim to assess whether the PSNR module can mitigate the oversmoothing phe- nomenon in deep GNNs. We select representative datasets Cora, Amazon-Photo, and Chameleon. Using GCN the backbone network, we compare our method against several residual methods: ResGCN, GCNII, JKNet, and DenseGNN. We set the number of layers to 2, 4, 8, 16, 32, 64 and test on ten random splits, with the average accuracy serving as the final result. The experimental results are depicted in Figure 3 Consistent with the analysis in the main text, most methods can alleviate over-smoothing, but deeper layers, such as 64 layer, over-smoothing still occurs. In contrast, compared to other residual methods, PSNR maintains stable performance even at deeper layers, demonstrating remarkable effectiveness in mitigating smoothig in deep GNNs. This is attributed to PSNR module can effectively reduce the loss of high-order neighborhood subgraph information. It is noteworthy that among the various residual methods, another initial residual method GCNII also alleviate over-smoothing well. However, the subsequent experiment will reveal that although the performance of GCNII remains relatively stable with varying layers, it leads to a decrease in overall performance.
7

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Vanilla
Dense
PSNR
Cora
Amazon Photo
Chameleon
70
90-
Accuracy (16)
70
50
70
50
40
60
30
2'
2a
2
21
27
2
21
2
25
2
2
2'
2
2
Layers
Layers
Layers
Figure 3: Different residual methods' effectiveness in mitigating over-smoothing.
Table 3: Summary of classification accuracy (%) results on real-world datasets. The best results in bold, and the second best results are underlined.
Method
Citeseer
Photo
Chameleon
Squirrel
Aug.
GCN
78.56.11.57
66.00.1..37
90.1910.83
91.22.10.73
67.7010.17
47,43
6.67
ResGCN
80.11:10.98
66.40:2.29
90.86:10.99
91.31:0.85
70.70#1.54
52.43=1722
3.67
GCNII
77.40=#1.43
66.69=1.95
90.34+0.78
91.46+0.84
69.69+1.36
46.13=1.50 5,5
5.00
DenseGCN
74.9442.46
62.5442..56
90.28-+1.06
90.38-+1.02
67.82:1.27
49.57-1.58
7.17
JKNet
79.42=#1.48
06.4922.332
90.94+0.89
90.85=1.19
18.26+1.115
49.87=1.44
5.17
Holf-Hop-GON
80.74++ 1,80
67.76-42.19
1.83
19.32++1.17
60.07:-2.34
42.78-1.011
6.33
DropMesaage GCN
1..00
67.66.1 1.84
90.64 10.64.2.34
91.40.10.90
65.91
45.71 1.76
4.83
DeProp-CCN
67.24:12.02
90.98:1:2.04
91.38.1..33
5.17
PSNR PSNR-GCN GCN
8..01
68.06:12.12
91,23 1.00
91,51 10.69
71.51 1.90
54.95 1.73
1,00
GAT
19,21+1.31
67.12:41.55
89.43++10 62
89 09.6440.84
68.04+1.333
47.93=1.99
6.00
Res-GAT
79.72+1.94
67. 17-19-+1.9
89.92-+0.86
91.40++0.74
72.66-0094
55.98-2.12
2.83
Init-Res-GAT
79.67=1.711
66.84=22552
89.61+1.022 1.02
91.53++1006
89+11.81
51.29-1.42
4.17
JK-GAT
80.04++1661
65.83++2.11
90, 10-+0.94
00.82++1.24
67.98+111
50.43-144
4.50
89.72.10.
88.92
7.00
DropMessage-Gia
80.36:11.92
67.82-42.04
90.67:1668
90.9810.90
63.23:12223
45.23=11355
4.33
DeProp GAT
76.00112.08
61.16.1.55
87.34 1.42
89.7611.52
Half- Hopl-H
1..2411.69
66.74:1198
89.66:11445
89.92.10.77
62. 86:1:222
4..
6.83
PSNR-GAT
80.47::1.62
68.01=2.144
1..38:1.21
91.64:0.61
7..24::1.69
6..85=1611
1.33
5.4 Fully observed feature setting (RQ2)
The PSNR module can effectively address the performance degradation of GNNs at deeper layers, but can it further enhance the overall performance of the model? In this section, comprehensively evaluate the PSNR module across a wider range of datasets. Under the fully observed feature setting, set the number of layers to 2, 8, 16, 32, 64, and conduct experiments on random splits for each dataset, taking the average accuracy as the final result for each layer. The results of all layers of the model can be found Appendix AA] To evaluate the overall performance, we record the best results of each model in all layers for each dataset in Table 3
As can be observed from the Table PSNR outperforms all baselines in most cases. For example, compared to the vanilla model, PSNR improves the test accuracy on the Squirrel dataset by 7.52% and 12.92%, respectively. Compared with the vanilla GCN and GAT, the proposed PSNR can significantly improve the performance under the fully observed feature setting.
5.5 Missing feature setting (RQ3)
When do we need the deeper GNN? Real-world data often contain missing features. In that scenario, previous research [33] has shown that deep GNNs can help improve performance. For the nodes with missing features, due to the lack of information, they need a deeper network to gather more neighborhood information, thereby achieving better node representations. However, deep GNNs face the issue of performance degradation. In this section, examine if PSNR module can improve the performance of GNNs in the context of missing feature.
Consistent with 33 18 12], we evaluate the performance of GNNs on three datasets, Cora, Citeseer, and Pubmed, and remove their node features from validation and test sets. Under this setting, the test nodes need more propagation layers to reach the training nodes. We reuse the metrics that already reported in [12 for None, BatchNorm [II], PairNorm [3), DGN R4, DeCorr [12], and DropEdge (24. For all residual-baat based models, the results obtained by varying the number of layers in

---

Table Test accuracy (%) on missing feature setting. The best results are in bold and the second best results are underlined.
GCN
GAT
Cora
Citeseer
Pubmed
Cora
Citeseer
Pubmed
Module
Acc
#K
Acc
#K
Acc
#K
Acc
#K
Acc
#K
Acc
#K
None
57.3
3
44.0
6
36.4
4
50.1
2
40.8
4
38.5
BatchNorm
71.8
20
45.1
25
70.4
30
72.7
5
48.7
5
60.7
PainNurm
65.6
20
63.1
68.8
50.3
63.2
DGN
76.3
20
50.2
30
72.0
75.8
54.5
5
72.3
DeCor
73.8
20
49.1
30
73.3
15
72.8
15
46.5
6
72.4
15
DropEdge
67.0
6
44.2
8
69.3
67.2
48.2
6
67.2
Res
76.8
8
60.4
10
76.6
76.5
60.5
6
76.9
Init-Res
65.1
15
70.4
77.1
60.6
8
76.8
Dense
66.2
4
51.5
2
74.1
68.5
10
52.7
2
75.1
10
JK
75.5
30
60.4
8
76.9
77.0
10
60.3
4
76.8
DropMessage
75.5
10
61.0
6
74.6
76.5
61.1
8
76.6
DePrup
71.4
76.1
68.04
48.3
2
75.8
Half-Hop
73.7
8
59.48
6
76.5
76.0
20
59.6
4
76.9
PSNR
77.3
20
61.1
15
77.0
30
77.9
8
61.9
15
77.3
10
{2, 4,4, 6, 10, 15, 20, 30} and running five times for each number of layers. We select the layer #K that achieves the best performance and report its average accuracy. The results are reported in Table By examining the results in Table 4 under the missing feature setting, the optimal number of layers achieve the best performance is significantly higher than in the fully observed feature setting, demonstrating the importance of deep GNNs. And PSNR outperform other baselines in all cases through alleviating over- smoothing more effectively. Specially, on the Pubmed dataset, PSNR boost the accuracy of GCN and GAT by 40.6% and 38.8%, respectively.
5.6 Performance on large graphs (RQ4)
To validate the scalability of PSNR, conducted additional experiments on three larger graph Flickr and Ogbn-arxiv, to further validate the effectiveness and datasets i.e., Coauthor- Specifically, scalability of our method. report the performance of GCN and various residual methods on three datasets and the memory consumption on the largest dataset, Ogbn-arxiv. The experimental results are presented in Table 5 from which we observe that PSNR-GCN scales well and achieves the best results across all three large datasets. Meanwhile, in terms of memory consumption, PSNR is slightly higher than GCN and ResGCN, comparable to GCNII with the same initial residuals, and significantly lower than JKNet and DenseGCN. Regarding training time, PSNR roughly at the same level as JKNet, and time shorter than that of DenseNet.
selected the GCN backbone for experiment. We
Table 5: Comparison of different methods across various datasets and memory consumption (MB) and training time (ms / epoch) on Ogbn-arxiv. The best performance for each dataset is in bold, while the second best is underlined.
Method
Phy
Flickr
Ogbn-arxiv
Memory
Time
GCN
95.32 + 0.11
51.40 + 0.33
64.37 + 0.41
2421
30. 10
ResGCN
95.61 + 0.18
51.90 0.16
66.32 + 0.59
2463
36.06
GCNII
95.90 0.14
46.18
0.21
61.43 + 1.63
2525
33.33
JKNet
95.88 1 0.15
51.65 0.31
60.46 1.21
2921
40.05
DenseGCN
95.50 1 0.12
52.18
0.25
1..4611.5
3131
52.24
PSNR-GCN
95.92 0.17
52.47 0.16
67.81 0.57
2539
42.93
6
Conclusion and Future Work
In this paper, we addressed the oversmoothing in Graph Neural Networks (GNNs) with focus on residual methods. We revisit the oversmoothing from the perspective of overlapping neighborhood subgraphs, explaining why residual methods can alleviate it. Our analysis revealed that current residual methods often lack node adaptivity and struggle with information loss in high-order neigh- borhoods subgraphs. To overcome these limitations, we introduce the Node-Ade Ande-Adaptive Residual Module (PSNR). This innovative module uses graph encoder to learn the posterior distribution of residual coefficients for each node at different layers, enabling fine-grained.

---

node-adaptive neighborhood subgraph aggregation with minimal overhead. Extensive experiments confirmed that the PSNR module can effectively mitigate oversmoothing and improve performance. particularly in scenarios requiring deep networks. Despite the significant progress made by PSNR, training a deeper GNN remains challenging, prompting the need for further research in the future.
Acknowledgements
This work was supported by National Science and Technology Major Project (No. 10222D0115101) National Natural Science Foundation of China Project (No. 623B2086, No. U21A20427), Project (No. WU2022A009) from the Center of Synthetic Biology and Integrated Bioengineering of Westlake University and Integrated Bioengineering of Westlake University and Project (No. WU2023C019) from the Westlake University Industries of the Future Research Funding. Carl Yang was not supported by any funds from China.
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12

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A Fully Observed Node Classification
Table 6: Node classification accuracy (%) are highlighted.
GCN backbone. The best results across different layers
Model
Layer
Layer
Layer
Layer
Layer
Layer
GCN
1..5641.57
78.52:42.00
77.27=2.29
72.1 72.11116.30
46.09=12.10
30.76::1.30
ResGCN
79.39+1.112
79.80++1.55
80.10 1.46
80. 109.91
11.27433.68
36.48+7.55
GCNII
77.40:11449
76.47:+1.96
76.39=1.88
1..1
75.76=1.83
76.27:::.46
Cora
DenseGCN
74.94-22.46
67.06:42.02
67.58=2.66
05.08:3.71
59.98=2.30
34.37-15.69
JKNNtt
78.44++2.10
79:42441448
79.36=2.23
77.1842.17
77.33+2.88
51.58:t7.29
DropMeesago-GCN
80,00:12.22
19.88:1181
18.14=1.65
76.40:11.22
1.018=1.06
30.34:10.63
DeProp-GCN
79.52.42.63
78.44422.39
33.02-5.64
31 31.28443.224
30.20=0.01
30.20:-0.01
Half-Hop-GGN
78.92:42.08
80,74::1.80
80.24=1.61
46.66-+13.37
43.68-7.14
31.03:47.39
PSNR-GCN
80,59:1.57
81.01:1.63
80.55=1.57
18.26:11.36
76.34:22.8
71.75
GCN
66.00012.37
64.13-422.11
64.13-2.10
58.52+33.11
9:3.99
27.52.+-.004
Res-GCN
66.1111.66
66.40:12.229
65.97=1.93
65.46:22002
38.7073.77
33.74::..79
GCNI
6..69
16:18:11..7
66.50=1.77
66.31:12.08
66.27 1.92
6..50
Citeseer
Dense-GCN
62.54122.56
58.99 38.99:13.74
54,35=5.42
50.10:15.24
49.09
31.43
JKNet
65.49:12.32
64.40:122228
63.77:11877
6..8
29.66:17.51
DropMessageGGCN
67.66--1..8 8.44
67.04422.20
63.84
59,82+22.75
22.68+3.77
21. 1,722
Half-Hop-GGN
67.76:1:2.19
67.46:11.99
67.02=2.553
34.48.13.93
8..3
24.14:13.48
DePro-GGNN
67.24442.02
64.86
29,08 10.03
21.72++0.91
21.94-1 1.18
20.91++1.13
PSNR-GCN
68.06:t2.
66.03:+1.99
65. 1,559
65.81:22..33
65.52 1.76
05.85
GCN
90.19-F0.83
88.81+00.22
86.93-
84.47-00.5
71.40++4.93
36.74--4.57
Res-GCN
90.86++0.99
90.63::0.96
89.97=0.90
40:0000000
1.501
39.43..
GCNII
90.34:1.778
90.02:10.81
90.07=0.77
90.09-10.94
90.17
89.93-10.70
Dense-GCN
89.01=11099
89.99-11.27
90.28
19.33+1.733
88.92-110 1.08
88.40+1.33
JKNet
90.81
90,94:10.899
90.53=1.41
89.52=1.11
88.81 1.14
18,44
DropMessage-GGr GCN
90.6412..34
89.18:11.76
89.38 1.52
85.62417.01
87.94
14.4411.13
Half-Hop-GCN
90.30+1.83
89.30:1:2.33
88.98= 1.81
82.98111.8
54 54.60-4.55
15.36+10.227
DeProp-GCN
90.98
69.10:1:2233
71.24=4.18
72.34:12.26
02.60=0.03
22.60:10.01
PSNR-GON
91.23::1.00
90.70:t:149
00.26=1.17
90.26:10.98
90,52+1.02
90.30+:0.88
GCN
91.22:10.73
90.49:10.76
88.10=1.02
80.05:14.25
8.080
46.57119.64
Res-GCM GCN
91.31 +:0.55
90.97:10.78
90.71 0.78
85, $3.98-122.38
65.42:46.75
16.15-10.43
91.02-40.93
90.98-00.22
91.02=0.70
91,20=+081
Photo
Dense GCN
90.38:11002
90.07:1:1.76
89.34=1.48
87.77+12.00
8.
68.22
JKNet
87.26+-1.77
87.96++1.91
84,39 84.39-2.76
90.8541..0 19
90. 1.20
87. 47.93-23 23-96--.
Drop
91,40110900
90,22:1:1119
87.57=2.74
87.82:11.21
8..12:11.05
$0.40:11.07
Half-Hop-GiCN
52.00+17.288
67.48+-24.10
89.32-1.11
83.66+-3.00
64.40+4.30
39.00+ 9.42
DeProp-GCN
91.38:11.3
89.50-15.31
78.12=3.66
1.886-99922
87.74-4.119
84.43-22.31
PSNR GCN
91.44110.22
91,20++11.3
91.39-0.76 0.76
91.11++0.68
91.51 0.66
91.49
GCN
67.70:0,17
56 30+:2.28
48.37=1.56
46.81+22, 06
39 39.62=2.115
33,27 1122211..80 86
Res-GCN
70.28:11.14
70.70:11.54
69.18
51.64+22.11
42.61=3.60
31.42-22.73
GCNII
69.45+1163
1..
69.62-1.61
69.23+11.21
68.87=2.16
Chameloon
Dense-GCN
67.82:1127
64.59 1.299
63.36=1.50
64.39-11.37
30.63
28. 72.14.6666
JKNet
68.26111.15
67.97:11..59
6.761=1.76
67.78:11.60
66.4612.25
$2.98-1.2.79
DropMessge-GCN
65.91+2.43
60.50-12.88
47.20-2.00
12.66=11.60
34.56=1.61
1.
Half-Hop-GGN
60.07:12.34
55.72-13.12
$2.64=1.99
43.19:12.55
36.26:22.2
29.3411:.20
DeProp- CCN GCN
61.51111.00
58.25:12.22
46.46=2.118
31.8812..2
30.81:2.007
33.36:1:0.96
PSNR-GCN
71.5111.90
70.744.2.24
71.40=183
66.00+t1.97
65.82:22.02
69.67.14.59
GCN
4..43
41.68-+1.80
37.43
33.17-+1.22
31.99 1.14
29.83 1.98
Res-GCN
51.7244:.63
52.43-+1.72
50.38=105 1.90
38.35+11.56
26. 26.30-22.06
22. 22.66+11.00
GCNI
1.55
18.28:11.59
45.71=200
45.37-11.72
1..06:11.50
45.32.1 .85
Squirrel
Dense-GCN
49.574-1.58
49.08-+1.36
49,29
JKNet
1.4
49.11+2.27
46.62=1.44
24.79-13.88
45.73 1.80
41.75+ 1.95
DropMessage-CC GCN
45.71+1..76
38.6844-0009
26.67
25.71++1.15
23.77 1.18
20,06 20.06--0.23
Half-Hop-GCN
42.78++1.01
41.87+1.36
37.84-1.68
60-42.11112
26.36=1.66
1.50
DeProp-GCN
44.07+11.07
40,9% +.29 1.29
33.38 1.70
28 0...33--0.966
28.4
29. 2,78
PSNR-GCN
54.9541..73
54.13+11.11
50.68=1.45
50.22-46.32
50.06-0.997
50.24411.58
13

---

Table Node classification accuracy (%) on GAT backbone. The best results across different layers
are highlighted.
Datasets
Model
Layer 2
Layer
Layer 8
Layer
Layer 32
Layer 64
GAT
79.21 1.31
78.83: 1.78
41.48-1-.84
30.20:+0.00
30.20=0000
30.20:+0..00
Res-GAT
79.42=1.44
1.97
18.8
18.
77.80=2.25
30.22:10.12
InitRes GAT
79.32- 1.65
79.39 1.45
79.00+1.97
79.16=t1..6
79.67=1.71
79.42-+:1.7
Cora
Dense-GAT
13.39=1552
69.07=2.77
58.85:43.02
60.3442.12
39.4423.03
55.80:1..12
JK-GAT
19.32=1.006
80.04=1.61
77.28:+1..2
1.55
1.25
77.58:22.11
DroMMsssge-GGal GAT
80.36+1.92
79.20++0.22
76.54442.49
12.04=11.02
30.17-+0.06
GAT
69.06-13394
30,22:4-0.06
31.20-+0.02
30.20+0.011
30.20-40.01
Half Hop-GAT GAT
77.20-1.86
76.98=11.42
77.24111.60
75.84:11.68
70.50=:4.69
30.20 10:20:10
PSNR-GAT
80.47=1622
80,22=00.8
79.9611
80.02::1.64
1.74
19.69:11.29
GAT
67.12=1.59
65.14=3.00
21.9042.440
21.10=0.000
21.10=0.00
21.10:10000
Res-GAT
67.19=1991
65.65#2.221
63.26:12.31
63.4711118
62.82.22.22
22. 1..:6111311
InitRes-GAT
66.84=2.52
64.15#2.44
65.80 1,9
64,89+2.65
64.69-2.49
65.11+22.65
Citeseer
Dense-GAN GAT
61.63-2.53
54.84+22.3
50.69+3.79
49.21+3348
4..24+
36.34433.2
JK-GAT
65.83=2.21
64.79:12.46
63.4442.51
65.00:1143
63.57+2.27
62. 2.226
DropMessago-GAt
67.82-2.04
66 26.54
65. 104:3322
38,20:22.55
21.90=1.64
20. 2.:90
DeProp-GAT
61.16=3.50
11.52:11.25
23.20:42.77
1..12:1.776
21.00=1.67
20.50:10.92
Half-Hop-GAT
66.74=1998
66.70=2..58
63.12:43.49
60.00:13.30
43.98
24.32:1::446
PSNR PSTR-GAT
68.01 18.01-2.14
65.61
66.46-42,14
65.48
16.36-22.25
65.50-+-2.35
GAT
89.43=1662
17.23=10.14
72.634555.0 30
38.81
22.60=0.00
22. 60:00.00000
Re-GAT
89,48-1
09.92440.86
23.10+1.455
22.46-4-2.22
InitRes GAT GAT
19.61=1022
77.82=5.90
19.4344..23
18,50=:117
88. 1.22
86. 2.:
CS
Dense-GAT
89.72-0.86
88.81
88.28:10.7
87.01=1.52
1.
87.23--1.78
JK-GAT
89.77=1.02
90.10=0094
90.03++.226
89.6841411
DropMesago-GAT
90.67=1.68
1..20=
1..58:11.66
85,68+1..2 1.29
1..58=1.40
11.43
DeProp-GAT
87.34- 1.42
84.18: 1.99
72.26442.77
72.36-42..33
$2. 02.60=0.02
22. 60-00000
Half Hop-
89.66=14m
1.84
84.1242.27
84.75:+2.96
5.92
29.61
PSNR-GAT
10.38=1.12
86.31
14.4
83.84:11.3
81.85
79.85:15.24
GAT
89.64-0.84
12.4=18..66
51.13++10.00
27.97-47.71
25 00.0000.00
25.40-0.000
Res-GAT
89.03=1.35
89.30:14.68
91.40:10.74
18.60:1..99
24.61
26. 16:1881888
InitRes-GAT
88.92-2.22
31.74
91.23:+1.22
9..72+1140
11.53:1.06
91.11=1.21
Photo
Dense-GAr
88.92 1.66
88.10+1.50
87.03-+2.22
16.79+1.97
86.24
85.63++2.15
JK-GAT
87.55=1.39
90.82:11224
90.35++1.11
90.30+1..77
90.29:1.49
90.07
DropMsssage GAT
91.98 0.90
90.29:00.97
82.154-2.82
1.59
85. 0.99
82.15
DeProp-GAT
89.76=1
87.98:1
82.5411.01
81.25:42.88
82.26
80.00:12.11
Holf-Hop-GAT
89.92-0.77
18:48.1.889
16.1777
52.12411.5
1.084
45.51
PSNR-GAT
90.93=1.44 1.42
91448:0.94
1..33
91.05:10.95
91.6410.6
91.3710996
GAT
68.04=11 1.36
48.37=3.77
26. 8616.117
22.86:10.00
22 0.00
22. 86:10.00
Res-GAT
70.90-133
67.87433.43
H2.66-0.94
6..68++1133
27.54+3.63
27.27-43.44
InitRes GAT
69.87=1.65
61.63:5.02
1.81
1..34
6..62.1..32
68.53:11.222
Chameleon
Dense-GAr
67.36-11 1.95
66.87-2.007
66.21422.13
61.95+2.14
62.32+2.10
61.71+1.60
JK-GAT
67.98-1.71
65.71:1140
66.75:11.88
66.99412.23
1.42
66.31-11.56
DropMessagoGATT
63.23=2223
56.98=2.54
48.53422.95
1..00:11.93
1..1911.82
33.75:22.23
DeProp-GAT
64,23-3.22
19
22.27742.77
41. 11.19-22.7
32.29+2.77
32.26 12.-26--
Half-Hop-GAT
62.86=2.04
60.3412.01
50.5847.11
43.05::1.74
33.75=2.37
32.277+11422
GAT
71.29-1.64 1.64
72.04+1.88
71.58+1..9
71.78+1.51
71.47+2.54
72.24+1.66
GAT
47,93 1.99
32.96 1.85
20.57:+1.02
20.00:10.00
20.00=0.000
20.00+10.00
Res-GAT InitRes GAT
52.87=1.68
49.87
55.98122112
10.31
13.07
22 22.01+1.220
51.29 1.42
43.92:1:.777
19.53
1.7
1.83
50.46-11.775
Squirrll
Dense-GAT
50.25=0.88
49.57
16.71111.69
17.8
47.54=1.58
46.87-11.65
JK-GAT
50.43-145
44.1
49.08-L0.79
49.31++1.99
48.87 1.74
49 06.56-1.32
DropMessage GAT
45.23=1.5
30.69+1.84
19.44++1.34
2.550
25.47
DeProp-GAT
45.76-1.32
46.29:3.24
29.994-0.28
28.9740.56
28.91 +0.23
20.00++0.04
Half Hop-Hopp-a... GAT
43.46=1557
47.84:1..36
42.53-41.53
29.19:11.50
1.54
23.79:10.65
PSNR-GAT
57.81-2.08
60.8511.61
59.58:12.09
60.43:12.20
60.00:2.20
60.20:11.53
14

---

B Residual Connection Defination
Common residual connection for GNNs and their corresponding GNNs are described below.
Res. Res is composed of multiple residual blocks containing few stacked layers. Taking the initial input of the n-th residual block X., and the stacked nonlinear layers within the residual block as F(X):
X F(X.. + X..
where residual mapping and identity mapping refer to F(X) and X on the right side the above equation, respectively. Inspired by Res, Guohao Li & Matthias Müller(2019) proposed residual connection learning framework for GCN and called this model ResGCN which can be simply described as follows:
H2 a DD à H H W + H
InitialRes. InitialRes is proposed for the first time in APPNP, unlike Res that carries information from the previous layers, it constructs connection to the initial representation Xy at each layer:
Xa+1 = (1 -1- (1-a) H(X,) + aXo.
where H(X) denotes the aggregation operation within one layer. InitialRes ensures that each node's representation retains at least an a-sized portion of the initial feature information. Correspondingly: APPNP can be formulated as:
H2 (1- a1 AD H HH.
Based on APPNP, GCNII introduces identity mapping from Res make up for the deficiency in APPNP.
Dense. Dense proposes more efficient way features between layers. The input is the outputs of all previous layers of the network and at each layer Dense concats them together:
X H(X,, XI,... X..);
where [-] denotes the concatenation of the feature map for the output of layers to Inspired by Dense, DenseGCN applies similar idea to GCN, i.e., let the output of the k-th layer contains transformations from all previous GCN layers exploit the information from different GCN layers:
H AGG dense H, H......... H
JK. At the last layer, JK sifts from all previous representations X1 Xx] and combines them:
X Xoutput AGG(X, XN).
The AGG operation includes concatenation, Maxpooling and LSTM-attention. When it is introduced to GNN, i.e., JKNet, can be formulated
H Houtput AGG HA-1).
15

---

C SMV for Node Groups of Different Degrees
GCN-2
GCN-4
GCN-8
GCN-16
GCN-32
GCN-64
Cora
Citeseer
Pubmed
0.3
SMV
0.2
2
25
2
2
2
24
2
2
2
2
2
Degree
Degree
Degree
Result of GCN.
GAT-2
GAT-4
GAT-8
GAT-16
GAT-32
GAT-64
Cora
Citeseer
Pubmed
0.35
0.3
0.30
0.3
SMV
0.2
0.2
0.20
0.1
0.15
0.0
0.0
22
2
2
20
2
2
2
Degree
Degree
Degree
Result of GAT.
D Derivation of the closed-form formulas in the table
ResGCN: We can write the recursive formula for ResGCN in the following form:
H& (I + R-
(6)
turn, the following form can obtained by recursion:
H4 (I N
(7)
Using the binomial theorem, we can obtain the closed-form formula for ResGCN as follows:
H CN'H..
(8)
j=0
APPNP: According the recurrence formula of APPNP:
H = oH + (1 a) NH
(9)
To obtain the closed-form formula, we can add term T to both sides of the equation:
H, + T (1 a) NH oH + T.
(10)
We aim to transform the equation into the following form:
H T (1 a) N (H+-1 T).
(11)
Then we need to make sure that there exists a very that satisfies the following equation:
(1 (1-ah a) NT = oH + T,
(12)
which can be transformed into the following form:
((1 - a) N - I)T aH.
(13)
We proof the following lemma:
16

---

Lemma Given that a E (0, (1 a) N is invertible
Proof Proving that (1 a) is invertible is equivalent to demonstrating that it does not possess eigenvalue of Consider the Rayleigh quotient of -x) - I:
X (D- x
X (1- = (1
1.
(14)
X
X
From spectral graph theory, we can know the following equation holds:
X;
X
X
D
LD 1
x
Σ
> 0.
(15)
Vd,
+1
We can decompose into D
A, then have:
x D GD
X
DD
x
> 0,
(16)
X
X
which is equivalent to:
x AD-AAD-
x
X
=1.
(17)
X
X
Combining Eq. and Ineq. 17 can obtain:
x ((1 [ -IX < <1
- 1=
(18)
X
Hence, O can't 't be the eigenvalue of (1 a) N - I. Therefore, (1 N is invertible.
Since Lemma holds, we can derive the concrete form of T:
a ((L - a) N - - H.
(19)
Thus we can keep recurring from Eq. I and obtain the following equation:
H + T = ((1- - N)* (H+T)
(20)
which also can be written as:
H = ((1 a) N)* H (1 T -
(21)
For the second and third terms in Eq. 21
we write (1 a1 as (1 a) N I+ I. We can use the
binomial theorem write ((1 ((1-a) ax) N)* k Σ
a) 1y² then Eq. 21 can be written as (11
==
H ((1 a1 H
Σ -
(22)
Bring in the specific form of T and further derive the closed-form formula of APPNP:
H ((1 a) N) H t Σ
((1o)) IJ'H
(23)
1=0
= -11 H ΣΣ
(1
(24)
i=
17

---

E Derivation of the closed-forg formula of PSNR-GCN
For each diagonal element As, of As, is trivial obtain:
0 < AR,ū <.
To derive the closed-form formula of PSNR-GCN, we need proof the following lemma first.
Lemma 2 Set all the diagonal elements of A satisfy Aa < 1, then (AN + 1) invertible.
Proof 2 Proving that AN + 1 is invertible is equivalent to demonstrating that its determinant is not equal to O. Because all the diagonal elements of A satisfy 0 < Au < 1, then A is invertible And due
|AN +
| + A-
(25)
Therefore, proving that determinant not equal to 0 equivalent to demonstrating that IN + A-1 is not equal to O, and further equivalent to demonstrating that N + A does not have eigenvalue 0f
Consider the Rayleigh quotient of N + +-
X N + -
R1
(26)
X
Split Eq. 26 and we can derive:
X'NX X
X'A-XX
R
(27)
X
The second term of Eq. 27 can be easily written as follows:
X-A-IX
N EE1
Ag
X
2i=1
Since O < Ag 1, therefore Aui -1 > 1, then
A
(28)
X
For the first item, we write specific form as follows:
X'NX
X
D
AD-
x
(29)
X
X
From spectral graph theory, we know that the following formula holds:
xT (A+
Σ
X:
X
> 0.
(30)
Vd++1
vd,+1/
+1
(NENS)EE
Further mathematically transforming this formula, we can get the following form:
XT D (A +D)
X
X
D À A 1A+D-21 21 D + X
(31)
XTX
T (D 1 AD X
X D $ DD 1 :)x x
2X IX
(32)
X"X
X
X
X (D ADD X
D
Xx
+1
X > 0.
(33)
X
18

---

Further, we get the following result:
X
D
AAADD
X
2 Ix
-.
(34)
X
X
It is trivial to obtain:
2X' -
2 Li=1
(d: 1) - x²
(35)
X
Di=1 xx
N
Combining Eq. 27
Ineq. 28
Inq.. 34 and Ineq. 33
we can get the following inequality:
x
D
AD 1 +
X
> O.
(36)
X
can be obtained that the eigenvalue of D 1 AD + A- is greater than 0, so not an
eigenvalue of it. Further, AN 1 is invertible.
Now, we derive the closed form of the formula. Given the following recursive formula:
H1 + Ak-1 Hi - 1-12 110D-112HA H/2H H
(37)
where H = 1- H/2H A₁ d (2)
(m)) (m..((n) xk
S(g (1)
,,0 After
mathematical transformation, Eq. 37 can be written as:
H = (I + A4-1) H.
A 1 A/ H4
(38)
Set N = 1-1/2AD 1/2 then Eq. 38 can be abbreviated as:
H₂ (I + 1 H AH-
(39)
We try to modify Eq. 39 to the form that is more suitable for obtaining the closed form of the formula:
Hg + M N (H + Mą-1).
(40)
To verify whether there exists such M that satisfies the equation, need to solve the following
equation:
- NM (I A&-1) H1 M
(41)
which is equivalent to the following form:
- (A + M = (I+ ( H1.
(42)
Based on the definition, all the diagonal elements of A satisfy 0 < X < 1, according to Lemma (A+-1N N I) is invertible. Then M (AsINN I) (I Ax-1) H,, which means such
M that we require exists.
First, we perform the following mathematical transformation on Eq. 40
H& M = - (Ha-1 + M + M MR-2),
(43)
which can be split into the following form:
H4 + M = - - (Ha-1 + M + (-A+-IN) (M+ - -
(44)
Let N denotes -Ap-1 N, so the formula can be simply written as:
H M N (H2-1 + MR-2) + N (Ma-1 MR-2).
(45)
We first use Eq. 40 to recurse once, then derive the following formula:
H M = Ñ - (Hg-2 Mą-a) + Ñ (M₃- MM--)
(46)
By analogy, continuing to split and iterate, can get the closed form of the output of the k-th layer:
11 H N (M, M. M
II
N. (H, + M,) - MM-1-
(47)
j=
1
19

---

F
Comparing with other subgraph-based methods
is worth noting that a lot of previous work mentions "subgraph". However, our approach is fundamentally different from these approaches.
Relation with other subgraph- aassraph-based methods. While there are existing works subgraphs, they primarily focus graph classification tasks, aiming to learn representations of entire graphs. Given the limited capacity of GNNs effectively represent the entire graphs, these subgraph-basee approaches employ various strategies to leverage information from multiple subgraph structures within the overall graph to improve the representation.
related to
In contrast, our focus is on node-level tasks, specifically, enhancing the representations individual nodes. In context, naturally refer to neighborhood subgraphs different orders of ego- networks centered on a node (which can be regarded as subgraphs of the entire graph). Our work uncovers the relationship between these subgraphs and over-smoothing, well as how to utilize them to enhance node representations. This distinction fundamentally sets our approach apart from other subgraph methods.
G Experiment Setup
G.1 Details of Datasets
The dataset statistics are shown in Table 8 and details on dataset splits are summarized as follows:
Experiment 5.3 & 5.4 For Cora, Citeseer, Coauthor-CS,
randomly select 20 nodes per class for training set, 500 nodes for validation and 1000 nodes for testing. For Chameleon and Squirel, randomly divide each class's nodes into 60%, 20%, and 20% as the train, validation, and test sets, respectively.
Amazon-Photee
Experiment 5.5
We follow the widely used semi-supervised setting in [1.
Experiment 5.6 For larger datasets, We randomly divide each class's nodes into 20%, 20%, and 60% as the train, validation, and test sets, respectively.
Table 8: Dataset statistics of real-world datasets.
Cora
Citeseer
Pubmed
Amazon-Photo
Ogbu-arsiv
Chameleon
Squirrel
Coauthor-CS
Coauthor-Pay
#Nodes
2708
3527
19717
7630
169343
2277
5201
18353
34493
89250
#Fdges
5429
4732
119081
126842
1166243
36101
217073
81894
247962
899756
#eatuues
AClasses
7
G.2 Parameter Settings
We summarized the hyperparameters used in different experiments in Table 9
Table 9: Hyperparameters for experiments
Experiment
Learning
Drupout
Weight Decay
Hidden
Altention
Mox Epoch
Early Stopping Eguch
Experiment
GAT
[0.01, 0.001)
10.01,0.00 0.001)
0.0065
500
500
100
Experiment 5.
GUN
GAT
[0.0,, 0.001)
10.01, 0,001)
0.5
0.5
0.0055
000055
128
32
1000
1000
1000
1000
Expesiment 3.6
GCN
[0.01, 0.001)
0.5
0.0005
128
500
100
H. The Analysis of PSNR Residual Coefficients
We conducted an empirical study using an 8-layer PSNR-GCN trained on the Cora dataset to obtain the best-performing model. We saved the mean and standard deviation of the learned residual coefficient distribution for each layer. Nodes were evenly divided into four groups based on their degree, with each group containing similar number of nodes, and the average mean and standard
20

---

deviation for each group across different layers are reported in Table 10 The following observations can be obtained from the table: the mean residual coefficient increases with the number of layers. suggesting that PSNR effectively retains high-order subgraph information. In certain layers, the variance of the residual coefficients rises, indicating that added randomness helps mitigate information loss in higher-order subgraphs. In the shallow layers, the mean values show no significant differences across node degrees; however, deeper layers, nodes with higher degrees tend to exhibit lower mean values, indicating that these nodes retain more initial information due to significant subgraph overlap. All of these observations align with our expectations, illustrating how residual coefficients adapt based node degree and layer depth.
Table 10: Summary of means and standard deviations for different layers
Layer
Croup
Group
Group
Group
Group
Group 2 Group
Group
Layer 0
0,003
0,0799
0.0066
0:009
0002
00000
0,00022
0.0001
Layer
DO009
0.0017
0,0015
0.0016
0.3102
0.312
0.3150
0.2410
Layer 2
000000
0,0000
0.0003
000088
DOFBO
00131
0.0446
0.0170
Layer 3
0.5582
0.8077
0.5972
0.4979
00084
DOT18
0.0105
0.008
Layer
L4898
5.4511--5
Layer 5
2,4080
2,3974
2,3981
2,1414
00062
0O081
0.0078
0.0059
6
5,2347
5,3148
3,2070
3.8177
00006
00089
0.0075
0.0089
Layer
11.0787
11,1253
10.9247
8,7238
0.1612
0,2099
0,2128
0.0677
Comparison of Experimental Results of Different GraphEncoders.
In practice, in addition to SAGE, encoders such as GAT and GCN can also be utilized. To provide further insight, we have included results for different encoders on the classical semi-supervised node classification task. The results are summarized the table blow.
Table 11: Different GraphEncoder performance for SSNC task (layer 2)
Graph Encoder
Cora
Citeseer
CS
Photo
Chamelcon
Squirrel
GCN
1.6
68.46
90.52+t0.82
91.56:+0.74
72.02+:1600
16.14+1.51
GAT
1.63 1.55
68.77:+1.9
90.61+0.89
91.18+0.92
71.97+1.28
56.24+1.11
SAGE
68.06=22.12
91.23+1.00
91.44+
71.51+1.90
54.95=11.73
Table 12: Different GraphEncoder performance for SSNC task (layer 4)
Graph Encoder
Cora
Citeseer
CS
Photo
Chameleon
Squirrel
GCN
81.65.1 1.70
68.11 1.24
90.6610.770
91.1411.990
71.58.12.07
56.34 1.48
GAT
82.21 1.411
67.96.1..00
90.5710.88
91.1710.81
71.2911..75
56.50
SAGE
11.0
66.03
90.70=1.49
91.20
70.74
54.13+1.41
The results indicate that each encoder has its strengths and performs differently across various datasets, demonstrating superior performance compared to the baseline and emphasizing the potential of PSNR.
21

---

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Answer: [Yes]
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Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes]
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Justification:
Limitations of our work are discussed in Section 6
Guidelines:
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The authors are encouraged to create separate "I "Limitations" section in their paper.
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Reviews:
[{"review_id": "e3fMyuyPVa", "reviewer": "Reviewer_ZBJB", "summary": "This paper proposes a PSNR module to alleviate the over-smoothing problem faced by graph neural networks when the number of layers increases. The effectiveness of this method is demonstrated through both theoretical analysis and experimental results.", "questions": "1.\tWhy is there no report of variance in the SSNC-MV experiment?\n2.\tThis method can be applied to traditional gnns. Is it orthogonal to other methods that alleviate oversmoothing, such as DropEdge?", "limitations": "The authors have addressed the limitations and potential societal impact.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "1.\tThe motivation is reasonable and the method is well-motivated.\n2.\tThe experiments conducted are comprehensive, utilizing widely adopted datasets.\n3.\tThe paper show the insight of how the residual method can alleviate the over-smoothing issue, which can further promote research and application of the residual method in the field of graph neural networks.", "weaknesses": "1.About the conclusion that subgraph overlapping causes oversmoothing， it does not seem to holds since transformer can access all the node without oversmoothing. \n2.L104: \"Considering nodes with high degrees tend to have larger neighborhood subgraph overlap,\" this conclusion lacks of detailed explanation.\n3.Some minor typo: L86 oversmoothing", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "_We thank the reviewer for the valuable feedback. We are glad that the reviewer appreciates the idea and technical contributions of our work. Below, we address the reviewer’s concerns one by one._\n\n> **Q1: The paper assumes that the posterior distribution of residual coefficients is Gaussian, but it is not clear if this assumption holds in all experimental settings. Further discussion on the choice of distribution would be beneficial.**\n\nA1: We chose the Gaussian distribution due to its commonality and widespread use in statistical modeling. The primary focus of our paper is on employing a graph posterior encoder to model the posterior distribution of node-level residual coefficients. The specific choice of distribution is not central to our main contribution. Our method is versatile and can accommodate any distribution that can be expressed in a reparameterized form, such as the beta distribution. We will include a more detailed discussion on the flexibility of our approach regarding the choice of distribution in the revised manuscript.\n\n> **Q2: L104: \"Considering nodes with high degrees tend to have larger neighborhood subgraph overlap,\" this conclusion lacks of detailed explanation.**\n\nA2: Thank you for pointing out the need for further clarification. Larger neighborhood subgraphs tend to have a higher degree of overlap than smaller neighborhood subgraphs. Take an extreme scenario as an example. As the layer number increases, several larger neighborhood subgraphs expand and cover the entire graph. In this case, these larger neighborhood subgraphs will have complete overlap, while smaller neighborhood subgraphs may be dispersed throughout different parts of the entire graph, resulting in a relatively lower degree of overlap and making it less likely for them to have complete overlap. In conclusion, Nodes with higher degree tend to have larger neighborhood subgraphs overlap.\n\n> **Q3: Some minor typo: L86 oversmoothing.**\n\nA3: Thank you for pointing this out. We will correct the typo in the revised version of the manuscript.\n\n> **Q4: Why is there no report of variance in the SSNC-MV experiment?**\n\nA4: In the SSNC-MV experiment, we used the results of GCN and other baselines directly from the original paper [1], which did not report variance. To maintain consistency with the original results, we also did not report variance in our experiments. \n\n>**Q5: This method be applied to traditional gnns, Is it orthogonal to other methods that alleviate oversmoothing, such as dropedge?**\n\nA5: Indeed, our method is a general and plug-and-play module that is not in conflict with other methods designed to alleviate oversmoothing, such as DropEdge. It can be used in conjunction with these methods to enhance their performance. To validate this, we conducted experiments using DropEdge as an example and tested whether PSNR could further improve DropEdge's performance. We performed tests on the Cora dataset and reported the results in the table below. \n\n**Table Re1:**  To verify the orthogonality of PSNR and DropEdge.\n| Method       | Cora      | Citeseer  | CS | Photo | Chameleon | Squirrel |\n|--------------|-----------|-----------|-------------|--------------|-----------|----------|\n| DropEdge     | 74.31±5.98 | 58.48±3.42 | 85.17±0.96  | 80.37±2.27   | 44.04±1.57 | 33.51±0.93 |\n| PSNR+DropEdge | **78.61±1.63** | **65.27±1.42** | **90.22±1.09**  | **91.22±0.77**   | **62.72±3.22** | **47.22±1.57** |\n\nThe results demonstrate that PSNR can indeed further enhance the performance of DropEdge.\n\n[1] Wei Jin, et al. \"Feature overcorrelation in deep graph neural networks: A new perspective\". SIGKDD, 2022", "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": "", "extra_scores": {}}, {"review_id": "H7xE02Rh2O", "reviewer": "Reviewer_61ZY", "summary": "The authors revisit the problem of over-smoothing in graph neural networks from the perspective of overlapping neighbourhood subgraphs, and propose a node adaptive residual module based on a posteriori sampling to demonstrate the effectiveness of this method from both theoretical and experimental perspectives. In this paper, the adaptivity to different nodes is achieved by learning the residual coefficients of each node, which helps to capture the features and dependencies of the nodes more accurately.", "questions": "1. The technical description is too brief, which is not reader-friendly. For example, how is positional embedding implemented in the text? How is γ initialised?\n2. What is the physical meaning of the residual coefficient? That is, can we interpret the residual coefficients as significance?\n3. Why this a posteriori approach to residual coefficients is used?", "limitations": "The authors should consider further clarification of the dataset, e.g., whether the heterophily of the data may have an impact on this method.", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The problem of over-smoothing has important implications for graph neural networks\n2. This paper is well organised in its entirety\n3. The authors provide some theoretical proofs and code, which supports this paper well.", "weaknesses": "1. The paper needs further polishing, e.g. some formulas are numbered and some are not.\n2. Although the authors provide a complexity analysis, I still have concerns about the efficiency of the algorithm. The authors should consider providing the computation time and number of parameters in Table 4.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "_We thank the reviewer for the valuable feedback. We are glad that the reviewer appreciates the idea and technical contributions of our work. Below, we address the reviewer’s concerns one by one._\n\n>**Q1: The paper needs further polishing, e.g. some formulas are numbered and some are not.**\n\nA1: Thank you for pointing this out. Some formulas were not numbered as they were not referenced later in the paper. However, we understand the importance of consistency and will add numbering to all formulas, including those below L165, L171, L199, and L201.\n\n>**Q2: Although the authors provide a complexity analysis, I still have concerns about the efficiency of the algorithm. The authors should consider providing the computation time and number of parameters in Table 4.**\n\nA2: We would like to thank the reivewer for this valuable suggestion. We have updated Table 4 to include the computation time and number of parameters for our algorithm. The following results were obtained using an A100 80G server, training each model for 500 epochs on the OGBn-Arxiv dataset. The table provides the training time per epoch in milliseconds, and the number of parameters for each method:\n\n**Table Re1:** Training time and parameter count of different residual methods. \n| Method | Training Time (ms/epoch) | Parameter Count |\n|--------|---------------------------|-----------------|\n| GCN    | 30.1087                   | 27,496          |\n| Dense  | **52.2488**                   | **85,096**          |\n| GCNII  | 33.3330                   | 27,496          |\n| JK     | 40.0566                   | 48,040          |\n| Res    | 36.0624                   | 27,496          |\n| PSNR   | 42.9386                   | 31,723          |\n\nAs shown in the table, PSNR introduces a moderate increase in training time compared to other methods, primarily due to the additional graph convolution layer effectively doubling the model’s depth. However, this results in a relatively modest increase in parameter count and memory consumption. Despite the increased runtime, PSNR demonstrates significant improvements in model performance, highlighting its efficiency and effectiveness in improving model performance.\n\n> **Q3: The technical description is too brief, which is not reader-friendly. For example, how is positional embedding implemented in the text? How is $\\gamma$ initialised?**\n\nA3: Thank you for highlighting the need for a more detailed technical description. Here is a more comprehensive explanation:\n\n**Positional Embedding Implementation:** The positional embedding in our method is inspired by the approach used in Transformer models. Specifically, the positional encoding is computed using the following formula:\n$$PE_{(layer,2i)}=sin(layer/10000^{2i/d_{\\mathrm{model}}}) $$\n$$ PE_{(layer,2i+1)}=cos(layer/10000^{2i/d_{\\mathrm{model}}}),$$\nwhere $layer$ represents the layer index, $i$ is the dimension index, and $d_{\\mathrm{model}}$ is the dimension of the embedding vectors. This encoding helps incorporate layer-specific information into the model, allowing a single network layer to capture multi-layer residual coefficient distributions effectively.\n\n**Initialization of $\\gamma$:** The parameter $\\gamma$ is initialized to 0.1 and is treated as a learnable parameter throughout the training process. This initialization allows $\\gamma$ to adjust dynamically as the model trains, optimizing its contribution to the model’s performance.\nWe will include these details in the revised manuscript to ensure a clearer understanding of the technical aspects of our method.\n\n> **Q4:What is the physical meaning of the residual coefficient? That is, can we interpret the residual coefficients as significance?**\n\nA4: The residual coefficient essentially represents the retention factor of features from previous layers, allowing for a trade-off between information from different orders of aggregation. In our work, from the perspective of subgraph aggregation, the residual coefficient can be interpreted as a summation coefficient associated with the results of aggregating features from different-order subgraphs. This means the residual coefficient modulates the influence of various subgraph aggregations, thus controlling how different levels of information are combined and preserved in the final representation.\n\n>**Q5: Why this a posteriori approach to residual coefficients is used?**\n\nA5: As mentioned in L154, not all nodes can learn effective node-level coefficients during the training process. Therefore, we use a posteriori method to model the residual coefficients by learning the posterior distribution of these coefficients. This approach allows us to indirectly capture the effective node-level coefficients, enhancing the model's ability to handle varying levels of information and improve overall performance. \n\n>**Q6: The authors should consider further clarification of the dataset, e.g., whether the heterophily of the data may have an impact on this method.**\n\nA6: Our paper primarily focuses on the issue of over-smoothing, and we did not specifically address the impact of heterophily on our method. However, we agree that it is important to consider how different task settings, including heterophily graphs, might affect our approach. Regarding heterophilic graphs, our method enhances the model's expressive power by accumulating multi-order subgraph information. Moreover, some literature[1] suggests a connection between heterophily and over-smoothing. Therefore, we believe that our method may be effective to some extent in heterophilic scenarios as well. \n\n[1] Cristian Bodnar, et al. 'Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs', NeurIPS 2022", "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": "", "extra_scores": {}}, {"review_id": "ibgouKhRqO", "reviewer": "Reviewer_Uv7R", "summary": "This manuscript focuses on the issue of over-smoothing in Graph Neural Networks (GNNs), which occurs when increasing the number of layers causes node representations to become indistinguishable. The authors explore this problem from the perspective of overlapping neighborhood subgraphs and propose a novel Posterior-Sampling-based, Node-Adaptive Residual module (PSNR) to mitigate it. The paper demonstrates how PSNR integrates multiple orders of neighborhood subgraphs and achieves distinguishability and adaptability, overcoming the limitations of previous residual methods. Theoretical analyses and extensive experiments confirm the effectiveness of the PSNR module across various settings.", "questions": "see weakness", "limitations": "see weakness", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written, making complex concepts and methodologies easy to understand.\n\n2. The motivations and theoretical foundations of the proposed PSNR module are compellingly presented, enhancing the credibility of the research.\n\n3. The experimental validation is extensive, covering diverse scenarios such as fully observed node classification, large graph datasets, and missing feature cases, demonstrating the robustness and scalability of the PSNR module.", "weaknesses": "1. What is the difference between the neighborhood subgraphs proposed in the paper and the subgraphs covered in other works?\n\n2. The analysis of the cumulative product term in the formula seems to show a discrepancy of a -1 factor compared to the PSNR-GCN formula. Does this affect the conclusion?\n\n3. There are three recent representative methods mentioned in section 5.4, namely DropMessage, Half-Hop, and DeProp. Why are their results not included in the missing feature setting?\n\n4. What are the specific reasons and insights that guided the selection of GraphEncoder for the GraphEncoder?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "_We thank the reviewer for the valuable feedback. We are glad that the reviewer appreciates the idea and technical contributions of our work. Below, we address the reviewer’s concerns one by one._\n\n>**Q1:  Relation to other subgraphs: What is the difference between the neighborhood subgraphs proposed in the paper and the subgraphs covered in other works?**\n\nA1: The neighborhood subgraphs used in our paper differ significantly from the subgraphs discussed in other works.  Our neighborhood subgraphs represent different-order ego networks for a given node, primarily serving as the information domain for node representations.  In contrast, other methods involving subgraphs, such as those used for graph classification tasks, often rely on sampling subgraphs from the graph to enhance representation capabilities.  These methods address the limited expressive power of GNNs for graph classification by augmenting the representation through subgraph sampling strategies.\nWe have detailed these differences and the context in the Appendix F, and we will include a reference to this discussion in the main text of the revised manuscript.\n\n>**Q2: Details about Proposition 1: The analysis of the cumulative product term in the formula seems to show a discrepancy of a -1 factor compared to the PSNR-GCN formula. Does this affect the conclusion?**\n\nA2: We appreciate your attention to this detail.  The discrepancy where the cumulative product term differs by a factor of -1 from the PSNR-GCN formula does not affect our conclusions.   As discussed in the paper, our analysis focuses on the asymptotic behavior as the number of layers increases.   This asymptotic analysis remains valid regardless of the -1 factor and does not impact the overall conclusions regarding the smoothing behavior.\n\n>**Q3: Details on SSNC-MV: There are three recent representative methods mentioned in section 5.4, namely DropMessage, Half-Hop, and DeProp. Why are their results not included in the missing feature setting?**\n\nA3: Thank you for pointing this out. We have addressed this by including the results of the recent methods DropMessage, Half-Hop, and DeProp in the missing feature setting for the SSNC-MV experiment. The updated results are provided in the table below:\n\n**Table Re1:**  Recent methods performance under SSNC-MV setting.\n|    | GCN Backbone  |              |              | GAT Backbone  |              |              |\n|-------------|---------------|--------------|--------------|---------------|--------------|--------------|\n|       Method         | _Cora_        | _Citeseer_   | _Pubmed_     | _Cora_        | _Citeseer_   | _Pubmed_     |\n| DeProp      | 71.4 / 6   | 59.4 / 2     | 76.1 / 4     | 68.04 / 2     | 48.3 / 2     | 75.88 / 4    |\n| DropMessage | 75.5 / 10     | 61.0 / 6     | 76.4 / 6     | 76.5 / 6      | 61.1 / 8     | 76.6 / 6     |\n| HalfHop     | 73.7 / 8      | 59.48 / 6    | 76.5 / 6     | 76.0 / 20     | 59.6 / 4     | 76.9 / 6     |\n| PSNR        | **77.3 / 20**     | **61.1 / 15**    | **77.0 / 30**    | **77.9 / 8**      | **61.9 / 15**    | **77.3 / 10**    |\n\nAs shown in the table, PSNR continues to achieve superior performance compared to these recent methods, even under the missing feature setting.\n\n>**Q4: Choice of GraphEncoder (SAGE): What are the specific reasons and insights that guided the selection of SAGE for the GraphEncoder?**\n\nA4: The choice of SAGE for the Graph Encoder in our study was not driven by any specific design considerations. In practice, other choices, such as GAT and GCN, could also be used as Graph Encoders. To provide more insight, we have included results for different encoders on standard node classification task.  The results are summarized in the table below:\n\n**Table Re2:**  Different GraphEncoder performance for SSNC task (layer 2).\n\n| Graph Encoder  | Cora         | Citeseer     | CS  | Photo | Chameleon     | Squirrel      |\n|--------|--------------|--------------|--------------|--------------|---------------|---------------|\n| GCN    | **80.98±1.60**   | 68.46±2.28   | 90.52±0.82   | **91.56±0.74**   | **72.02±1.60**    | 56.14±1.51    |\n| GAT    | 80.89±1.63   | **68.77±1.89**   | 90.61±0.89   | 91.18±0.92   | 71.97±1.28    | **56.24±1.11**    |\n| SAGE   | 80.59±1.57   | 68.06±2.12   | **91.23±1.00**   | 91.44±0.82   | 71.51±1.90    | 54.95±1.73    |\n\n\n**Table Re3:**  Different GraphEncoder performance for SSNC task (layer 4).\n\n| Graph Encoder | Cora         | Citeseer     | CS  | Photo | Chameleon     | Squirrel      |\n|--------|--------------|--------------|--------------|--------------|---------------|---------------|\n| GCN    | 81.65±1.70   | **68.11±1.24**   | 90.66±0.70   | 91.14±0.90   | **71.58±2.07**    | 56.34±1.48    |\n| GAT    | **82.21±1.41**   | 67.96±1.20   | 90.57±0.89   | 91.17±0.81   | 71.29±1.75    | **56.50±1.45**    |\n| SAGE   | 81.01±1.63   | 66.03±1.93   | **90.70±1.49**   | **91.20±1.03**   | 70.74±2.24    | 54.13±1.41    |\n\nThe results indicate that each encoder has its own strengths and performs differently across various datasets, demonstrating superior performance compared to the baseline.", "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": "", "extra_scores": {}}, {"review_id": "VNmLjxj01y", "reviewer": "Reviewer_yetc", "summary": "The paper proposes to address over-smoothing through the lens of overlapping neighborhoods and revolves around addressing it in residual GCN line of work. It highlights the lack of adaptability of higher-order neighborhood information as the limitation of previous residual methods. To overcome these drawbacks, it introduces PSNR, which learns node adaptive residual coefficients through posterior inference using a graph encoder.", "questions": "Please refer to the weaknesses.", "limitations": "There is a brief discussion on the limitations.", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The method to use node-level coefficient for residual connection from the neighborhood is novel.\n- The paper is written clear and easy to follow.", "weaknesses": "- The discussion of related works is limited: only the residual methods are discussed. To better position the contribution, it seems necessary to give a brief overview of other types of GNN methods for over-smoothing alleviation. For example, GPR-GNN [1] looks closely related since it also proposes adaptive “GPR-weights” for message aggregation.\n\n- The state-of-the-art baseline methods on the heterophilic datasets, such as GPR-GNN [1] and ACM-GNN [1], are not discussed in the related works or included in the performance comparison.\n\n- Figure 3 shows for deeper architectures, the performance of the vanilla initial residual (init-res) module is almost identical to PSNR, suggesting that init-res already mitigates over-smoothing to a great extent. How can the more complex PSNR module be justified? Meanwhile, the baseline init-res-GCN is missing from Table 3.\n\n- The reported performance of the baselines, such as GCN, GCNII, do not match with the original papers for Citeseer, Cora, Pubmed, ogb-arxiv datasets. The paper reported much worse performance of these methods than what's in the original papers. I wonder if the authors could clarify it.\n\n- Experiment section: It would be interesting to see the empirical study of the residual coefficients. E.g. how does the coefficient change with a) node degree, and b) the layer k?\n\n##References:\n[1] Luan et al. Revisiting Heterophily for Graph Neural Networks. NeurIPS, 2022.\n[2] Chien et al. Adaptive Universal Generalized PageRank Graph Neural Network. ICLR, 2021.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "_We thank the reviewer for the valuable feedback. We are glad that the reviewer appreciates the idea and technical contributions of our work. Below, we address the reviewer’s concerns one by one._\n\n> **Q1:The discussion of related works is limited ...**\n\nA1: Thank you for the suggestion. At line 31, we briefly categorized the methods for alleviating over-smoothing into three types. Considering that residual methods are a significant part of our approach, we focused primarily on them in our discussion. However, we acknowledge the importance of providing a comprehensive overview of related works, such as GPR-GNN. From our perspective, GPR-GNN proposes adaptive “GPR-weights” for different neighborhood subgraph aggregation, which seems related to our method. However, the main difference between GPR-GNN and PSNR-GNN is that PSNR leverages node-adaptive residual coefficients, which enhance the higher-order neighborhood subgraph aggregation that effectively alleviates higher-order information loss. This fundamental distinction differentiates our method from GPR-GNN. **The  results of model performance in General Response Table2 support our claim.**  We will incorporate this broader perspective in the revised manuscript to better position our contribution within the context of existing work.\n\n> **Q2: The state-of-the-art baseline methods on the heterophilic datasets ... are not discussed in the related works or included in the performance comparison.**\n\nA2:  ACM-GNN uses high-pass, low-pass, and identity channels to extract richer localized information for various heterophilic scenarios, while GPR-GNN learns GPR weights that adjust to node label patterns to address feature over-smoothing. Our primary focus is on the over-smoothing problem rather than heterophilic graphs, which is why ACM-GNN was not discussed. GPR-GNN, which also mitigates over-smoothing, has been compared to PSNR-GNN in previous responses.\nTo further substantiate our approach, we conducted comparative experiments with both ACM-GNN and GPR-GNN. **The results are summarized in General Response Table2.** As can be observed from the table, PSNR-GNN outperforms both ACM-GNN and GPR-GNN across most datasets.\n\n> **Q3: Figure 3 shows for deeper architectures, the performance of the vanilla initial residual (init-res) module is almost identical to PSNR...**\n\nA3: In the paper Figure 3 depicts the results for GCNII, as noted in line 278 of the manuscript. GCNII, being a state-of-the-art init-res method, incorporates an identity matrix that allows the initial residuals to be effectively deepened. Therefore, we selected it as the baseline for initial residual methods. Similarly, Table 3 also represents the performance of init-res methods through GCNII.\nTo demonstrate the advantages of our method over initial residuals and GCNII, we conducted additional comparative experiments following the setup of Experiment 1. These experiments include comparisons among the vanilla GCN, the init-res structure, GCNII, and our PSNR method. **The results, presented in General Response Figure 1**, show that the vanilla init-res structure alone cannot effectively deepen GCNs. In contrast, both our method and GCNII significantly enhance the performance of init-res. Moreover, our approach outperforms GCNII in overall performance. This highlights the superiority of the more complex PSNR module.\n\n> **Q4: The reported performance of the baselines, such as GCN, GCNII, do not match with the original papers for Citeseer, Cora, Pubmed, ogb-arxiv datasets. The paper reported much worse performance of these methods than what's in the original papers. I wonder if the authors could clarify it.**\n\nA4: Firstly, our experiments utilized DGL to implement the models, and all methods were trained, validated, and tested within the same code framework. The detailed code and dataset split file are available at the link provided in the paper. Additionally, the inconsistency in results is due to our use of ten random splits of the dataset, rather than using a single split like the GCN and GCNII did on four datasets. Compared to using just one split, the random split approach yields diverse training and testing data, **ensuring that the data distributions for training and testing are more varied to avoid the issue of achieving good performance on just one data split.** This allows for a more fair and comprehensive comparison of the performance of different models.\n\n> **Q5: Experiment section: It would be interesting to see the empirical study of the residual coefficients. E.g. how does the coefficient change with a) node degree, and b) the layer k?**\n\nA5: We conducted an empirical study using an 8-layer PSNR-GCN trained on the Cora dataset to obtain the best-performing model. We saved the mean and standard deviation of the learned residual coefficient distribution for each layer. Nodes were average divided into four groups based on their degree, and the average mean and standard deviation for each group across different layers are **reported in the General Response Table1**.\nFrom the table, we observe the following:\n- The mean residual coefficient increases with the number of layers, indicating that PSNR effectively retains high-order subgraph information.\n- The variance increases in some layers, suggesting that the increased randomness helps mitigate information loss in high-order subgraphs.\n- In the shallow layers, the mean does not show significant differences across node degrees. However, in deeper layers, nodes with higher degrees tend to have a lower mean, indicating that these nodes retain more initial information due to the higher likelihood of subgraph overlap, which increases their distinguishability.\n\nAll these observations align with our expectations, and demonstrate how the residual coefficients adapt with node degree and layer depth.", "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": "", "extra_scores": {}}]

Author rebuttal:
# General Response
We thank the reviewers for their insightful and constructive reviews of our manuscript. We are encouraged to hear that the reviewers found our **motivation and  theoretical proofs to be reasonable** **(Reviewers 61ZY, ZBJB, Uv7R)** and appreciated the **comprehensiveness and supportiveness of our experiments**, which utilized widely adopted datasets and covered diverse scenarios **(Reviewers ZBJB, Uv7R，61ZY)**. They highlighted **the clarity and organization of the paper**, making complex concepts easy to understand **(Reviewers Uv7R, 61ZY, yetc)**, and **acknowledged the insightful approach** to addressing the over-smoothing problem, which has important implications for graph neural networks **(Reviewers ZBJB, 61ZY)**. Additionally, **the novelty** of using node-level coefficients for residual connections and **the robustness and scalability of our PSNR module** were recognized **(Reviewers Uv7R, yetc)**.

Based on the reviewers’ valuable feedback, we have conducted several additional experiments. Below, we address the issues pointed out by the reviewers and resolve any possible misunderstandings:

**1.Clarifications:**

- **Concept of Neighborhood Subgraphs:** We provide a clearer definition and distinction of neighborhood subgraphs compared to other subgraph-based methods. (Reviewer Uv7R)

- **Impact of the Ignored Factor on Proposition 1:** We discuss how the ignored factor affects the overall conclusions and provide a detailed explanation to ensure clarity. (Reviewer Uv7R)

- **Reason for Assuming the Distribution of the Residual Coefficient to be Gaussian:** We justify our choice of Gaussian distribution for the residual coefficient and its implications. (Reviewer ZBJB)

- **Explanation of 'High Degrees Tend to Have Larger Neighborhood Subgraph Overlap':** We elaborate on why high-degree nodes tend to have larger overlaps in their neighborhood subgraphs. (Reviewer ZBJB)
- **Detailed Implementation of Layer Embedding:** We describe the technical details of how layer embedding is implemented in our model. (Reviewer 61ZY)
- **Possible Impact of Heterophily Graph on PSNR:** We explore the potential effects of heterophilic graphs on the performance of PSNR and provide our insights.  (Reviewer 61ZY)

**2.New Experiment Results:**

- **Additional Results for Three New Methods on the SSNC-MV Task:** We include new comparative results to evaluate the performance of PSNR against recent methods. (Reviewer Uv7R)
- **Performance of PSNR with Different Residual Coefficient Encoders:** We test PSNR with various residual coefficient encoders to demonstrate its flexibility and robustness. (Reviewer Uv7R)
- **Performance of PSNR with Other Methods Like DropEdge:** We evaluate how PSNR performs when combined with other techniques such as DropEdge. (Reviewer ZBJB)
- **Comparison of Computation Time and Number of Parameters:** We provide a detailed analysis of the computation time and parameter count for different models. (Reviewer 61ZY)
- **Analysis of the Learned Coefficient Distribution:** We present an empirical study of the distribution of the learned coefficients across different layers and node degrees.  (Reviewer 61ZY, yetc)

- **Comparison with Other State-of-the-Art Baselines, such as GPR-GNN and ACM-GNN:** We benchmark PSNR against other leading methods to highlight its effectiveness and superiority. (Reviewer yetc)

**3.Minor Issues:** We corrected the typos and other minor flaws mentioned by all the reviewers.


**_Thank you for your time and effort in reviewing our manuscript. We sincerely appreciate your insightful comments, which will certainly enhance the quality of our article. If our response has effectively addressed your concerns and resolved any ambiguities, we respectfully hope that you consider raising the score. Should you have any further questions or need additional clarification, we would be delight to engage in further discussion._**

Meta-review: After discussion, the authors have addressed all the concerns raised by the reviewers, particularly those of the last reviewer who initially gave a negative score. It is suggested that when submitting the final revised submission, the authors can at least provide additional performance comparisons using conventional test methods. We recommend accepting this paper.
Decision: Accept (poster)