MultiNet: Adaptive Multi-Viewed Subgraph Convolutional Networks for Graph Classification
Authors: Xinya Qin
Lu Bai
Lixin Cui
Ming Li
Hangyuan Du
Edwin Hancock
Venue: NeurIPS 2025 poster ()
Keywords: Graph Convolutional Networks
Graph Classification
Over-smoothing
Primary area: general_machine_learning

Abstract: The problem of over-smoothing has emerged as a fundamental issue for Graph Convolutional Networks (GCNs). While existing efforts primarily focus on enhancing the discriminability of node representations for node classification, they tend to overlook the over-smoothing at the graph level, significantly influencing the performance of graph classification. In this paper, we provide an explanation of the graph-level over-smoothing phenomenon and propose a novel Adaptive Multi-Viewed Subgraph Convolutional Network (MultiNet) to address this challenge. Specifically, the MultiNet introduces a local subgraph convolution module that adaptively divides each input graph into multiple subgraph views. Then a number of subgraph-based view-specific convolution operations are applied to constrain the extent of node information propagation over the original global graph structure, not only mitigating the over-smoothing issue but also generating more discriminative local node representations. Moreover, we develop an alignment-based readout that establishes correspondences between nodes over different graphs, thereby effectively preserving the local node-level structure information and improving the discriminative ability of the resulting graph-level representations. Theoretical analysis and empirical studies show that the MultiNet mitigates the graph-level over-smoothing and achieves excellent performance for graph classification.

MultiNet: Adaptive Multi-Viewed Subgraph Convolutional Networks for Graph Classification
Xinya Qin, Lu Bai', Cui², Ming Li Hangyuan Du Edwin R. Hancock
School of Artificial Intelligence, Beijing Normal University, Beijing, China. School of Information, Central University of Finance and Economics, Beijing, China. $Zhejiang Institute of Optoelectronics, Jinhua, China.
*Zhejiang Key Laboratory of Intelligent Education Technology and Application, Zhejiang Normal University, Jinhua, China. School of Computer and Information Technology, Shanxi University, Taiyuan, China. 6 Department of Computer Science, University of York, York, United Kingdom. XinyaQin @
bailu @ bnu.edu.cn
Abstract
The problem of over-smoothing has emerged as fundamental issue for Graph Convolutional Networks (GCNs). While existing efforts primarily focus on en- hancing the discriminability of node representations for node classification, they tend to overlook the over-smoothing at the graph level, significantly influencing the performance of graph classification. In this paper, we provide explanation of the graph-level over-smoothing phenomenon and propose a novel Adaptive Multi- Viewed Subgraph Convolutional Network (MultiNet) to address this challenge. Specifically, the MultiNet introduces local subgraph convolution module that adaptively divides each input graph into multiple subgraph views. Then number of subgraph-based view-specific convolution operations are applied to constrain the extent of node information propagation over the original global graph structure, not only mitigating the over-smoothing issue but also generating more discriminative local node representations. Moreover, we develop that establishes correspondences between nodes over different graphs, thereby effectively preserving the local node-level structure information and improving the discriminative ability of the resulting graph-level representations. Theoretical analysis and empirical studies show that the MultiNet mitigates the graph-level over-smoothing and achieves excellent performance for graph classification.
alignment-based readout
1 Introduction
Graph data analysis has become an important research area for deep learning, and has been widely employed in various fields, including bioinformatics [36] 17], social networks [21] [22,, and recom- However, traditional neural networks are originally designed for grid-like data, such as images sequences. They struggle to handle graphs due to the irregular structures and Euclidean properties of graph data. To address these limitations, Graph Neural Networks (GNNs) have emerged powerful tools for learning graph-structured data.
mendation systems [3 9
In recent years, Graph Convolutional Networks (GCNs) have become the most popular architecture in GNN research. The carly spectral-based GCNs, such as the Spectral GNN [7]. ChebNet [13], and Vanilla GCN [18), interpret graphs as signals and define convolution operations in the spectral domain. However, these methods are computationally expensive and rely on the eigenvalue spectrum of the graph Laplacian matrix, making them less scalable. To improve the efficiency, an alternative
Thhe Corresponding Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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line of spatial based GCNs has been introduced recently, such as the GraphSAGE [66] and GIN 34 that perform convolutions by aggregating the information from neighboring nodes without spectral decomposition. While these spatial GCN approaches are usually computationally efficient, they still suffer from several challenges.
One notable challenge for most existing GCNs is the over-smoothing phenomenon 20 24 g where
an increasing number of layers leads excessive propagation of node features across the global graph structure. Since the information diffuses through multiple hops, the initially discriminative local node-level structural features tend lose their discriminative power, resulting increasingly similar node representations. This homogenization weakens the ability of GCNs to capture local node differences and ultimately degrades the performance on node classification. To overcome the shortcoming, variety of mitigation strategies have been developed. including the normalization (2
regularization [40] dynamic connections 26 27
and residual connections
Nevertheless, most existing studies mainly investigate the convergence of node representations within single graph structure. The impact of over-smoothing at the graph level has not been sufficiently explored by now. In this paper, we argue that deep GCNs also tend to generate extremely similar representations for different graphs, leading the graph-level over-smoothing and ultimately influencing the performance of graph classification. This issue is due to the following two key factors. First, when the node information propagates over multiple layers, the resulting node representations tend to be increasingly similar, leading to significant information loss of local distinct nodes. For graphs with similar node information distributions, this will cause the node representations over all graphs to converge. Second, the unordered nature of graph nodes forces GNNs to employ permutation-invanee invariant global readout functions, such as summation, mean, or max pooling, for generating graph-level representations. However, the simplicity of these operations treats all nodes equally, thereby further ignoring the distinct structural information of local nodes associated with similar node representations. As result, the graphs with structurally distinct local-level node features may still produce similar graph-level representations, thereby decreasing the discriminative power of graph classification models.
To indicate how the graph-level wee-shnt phenomenon influences the performance of GCNs, Figure reports the graph classification accuracy and the average cosine distance between graph representations (defined Section 5.3) with varying network depths, associated with three popular GCN models. The results reveal that both the classification accuracies and the representation diversities simultaneously degrade with the deeper network layers (i.e., more than eight layers), and indicate that the models struggle to distinguish between different graph structures. This observation provides empirical evidence that the graph-level over-smoothing arising in GCNs has a negative influence on graph classification.
Accuracy (%)
90
GIN GraphSAGE
GCN
Average Distance
0.8
GIN GraphSAGE
GCN
85
80
0.6
75
0.4
70
0.2
65
60
0.0
2
16
32
2
8
32
Layers
Layers
(a) The graph classification accuracy.
(b) The average distance.
Figure 1: The impact of increasing layers on the MUTAG dataset.
The aim of this paper to develop novel Adaptive Multi-Viewed Subgraph Convolutional Network (MultiNet), that can mitigate the influence of the graph-level over-smoothing problem for graph classification tasks. The key innovation lies in local subgraph convolution module as well as an alignment-basse readout mechanism. The former can restrict the excessive message propagation, preserving local distinct node features and alleviating the feature homogenization. The latter maintains the local information and structural information for the resulting graph-level representations, enhancing the ability distinguish different graphs. The main contributions of this work are summarized as follows:

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We propose local subgraph convolution module that can adaptively divide the original input graph into multiple structurally distinct subgraph views. Within each subgraph view, we define an attention-basee based convolution operation to capture the discriminative local node features. Then we introduce a feature fusion strategy that integrates the complementary information from different subgraph perspectives. This design can effectively preserve the local distinct node information and alleviate the node-level over-smoothing within each individual graph.
We develop an alignment-based readout mechanism to overcome the limitations of traditional global readout functions in preserving local features and structural information. By establishing consistent cluster-level correspondences across different graphs, our method enables the use of a more expressive readout function during graph aggregation, thereby enhancing the discriminative capability of the learned graph representations. This mechanism effectively preserves local features and structural distinctions extracted from the local subgraph convolution modules within the graph-leve -evel representation, thus alleviating the graph-level over-smoothing problem.
We provide both theoretical and empirical analyses, and demonstrate that the proposed MultiNet can effectively alleviate the graph-level over-smoothing. Experiments on benchmark datasets indicate that the MultiNet can outperform state-of-the arr methods on graph classification tasks.
This paper is organized as follows. Section 2 discusses the node-level and the graph-level over- smoothing problem. Section details the MultiNet model. Section 4 anaalyzes the mitigation effect of the MultiNet. Section 5 presents experiments and results. Section concludes this work.
2 The Over-smoothing Problem for GCNs
2.1 The Node-level Over-Smoothing Phenomenon
The over-smoothing phenomenon is a fundamental challenge in GCNs, where repeated message passing causes node representations to become indistinguishable, ultimately impairing performance on node-level tasks and potentially affecting graph-level tasks. Various works provide different perspectives on the underlying mechanisms of the node-level over-smoothing. For example, [20] models graph convolutional layers form of Laplacian smoothing, showing that deep networks drive the node representations toward a space where they become indistinguishable. From a spectral viewpoint, [24] interprets each graph convolution layer as low-pass filter that attenuates the high- frequency components of the graph signal. While effective for noise reduction, excessive filtering blurs discriminative information, thereby hindering effective classification.
To address this issue, various strategies have been introduced. DropEdge (25) applies stochastic edge removal as form of implicit regularization. PairNorm [40] stabilizes feature diversity by preserving pairwise distances between node embeddings, GraphCON (26] reformulates message passing as nonlinear dynamical system to enhance representation stability. G2 (27) introduces a gradient-based gating mechanism that adaptively halts message propagation per node. Residual-based methods like 19) and GCNII [TO] incorporate skip connections to retain initial features throughout the network. However, these approaches primarily target the node-level representations and do not explicitly restrict information diffusion. In this work, we revisit the over-smoothin problem from a graph-level perspective and present method that effectively alleviates this phenomenon.
2.2 The Graph-Level Over-smoothing Phenomenon
While prior studies have primarily focused on the node-level over-smoothing problem, the graph-level over-smoothing phenomenon remains underexplored. Although [3 introduces a strategy to alleviate over-smoothing and applies it to graph classification tasks, still primarily targets the node-level issue and lacks experiments that characterize the graph-level over-smoothing, We argue that excessive convolution across same-domain graphs can lead to indistinguishable graph representations, where the representations of different graphs become excessively similar after the readout stage, making it difficult for the model to differentiate between graphs and thereby impairing graph classification performance. Here we begin with formal definition, followed by theoretical analysis.
Definition 1 (The Graph-Level Over-Smoothing) Let {G1, G2..... GN } denote set of undirected and connected graphs, and let H IR Nxd be the graph representation matrix after readout
3

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operation and H; E Ie be the graph representation of graph G,. Define a similarity measure Xxd ::: NRd + Rzo satisfying the following properties:
a) 3e Rd s.t. H c for all graphs E {1, + (HH = 0, for H E ER
b) μ( + 2) (H) + z)) for all H, 2 E NRxx
If the representations of different graphs after multiple graph convolution layers tend to become indistinguishable, i.e., p)H) - O, we say that the graph-level ove-ssmog has occurred.
Assumption 1 (The Node Feature Distribution Consistency) Given any two graphs G., Gj g with initial node feature matrices X; and X,, we assume that their feature distributions are statistically consistent, especially in datasets drawn from a shared domain or labeling space. Specifically, the average node feature vectors I;, 2, of X; and X, satisfy 2, R Ty, indicating that the node features across graphs follow similar distributions.
In graph classification datasets, samples typically originate from similar domains, and node features represented as sparse one-hot encodings. Hence, it is reasonable to assume that the distributions of node features across different graphs are largely consistent. In subsection [3.3 and Table 2 we provide empirical evidence showing that node features in graph classification datasets from specific domain exhibit distributional consistency, thereby supporting the validity of Assumption 1.
Theorem 1 Under Assumption 1, excessive and unconstrained message passing, together with overly simple readout functions, tends to lead to the graph-level wer-smoothing.
Proof: According to the analysis in [20]. the spectral radius of the normalized adjacency matrix A, is 1, and its eigenvalues {Ao [ satisfy A E (-1, 1] [8], where n, is the number of nodes in graph G,, Therefore, as the number of layers increases, the terms corresponding to eigenvalues with |A&l < decay exponentially. As L - cc, only the spectral component corresponding to remains. leading to the over-smoothing problem. In this case,
the dominant eigenvalue 0 considering the simplest form of graph convolution layer, where the activation functions are linear and the weight matrix is assumed be the identity matrix, the node representations x,4) after L layer convolutions will eventually converge to the direction related to the average feature vectors, i.c.,
lim x(() = 1,, L-000
where I,, is the all-ones vector. This convergence indicates that the feature differences between nodes are smoothed out, leading to the loss of local-level information. Moreover, common readout functions (such summation, averaging. maximizing) treat all nodes in the graph equally, losing the ability to capture the structural information of the graph. For instance, consider the averaging readout used to obtain the graph-level representations:
H,
This (L)
where i the node index. Since node features tend to converge, using such global readout function is equivalent to directly averaging the original features, rendering the convolution operation ineffective for distinguishing different graphs. According to Assumption 1, x, R Iyy we have H, ~ H,, which leads to (H) + 0. Therefore, the graph representations tend to become indistinguishable, and the graph-level over-smoothing has occurred.
3 MultiNet: The Adaptive Multi-Viewed Subgraph Convolutional Network
3.1 The Overall Framework
In this paper, we propose a novel MultiNet to alleviate the graph-level over-smoothing problem. As shown in Figure 2 the overall framework of the MultiNet consists of three main components: the Global Graph Convolution, the Local Graph Convolution, and the Alignment-Bann -aseed Readout. First, the Global Graph Convolution layer performs initial feature abstraction. Then, the Local Subgraph Convolution Module divides the graph into multiple subgraph views and applies adaptive subgraph convolutions to extract discriminative local features, followed by integrating information Readout module aligns nodes across different graphs to
across views. Finally, the Alignment- generate a more discriminative graph-level representation.

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Local Subgraph Convelution Module
Alignment- -Based
Readout Module
Malti-View Division
Adaptive Subgraph
Cross- View
Convolation
Feature Fusion
Views
Alignment
Weight Sharing Convolatin
Global Graph Convolution Layer
Readut
Attention Matrix
Attention Weights
Figure 2: Overall framework of the proposed MultiNet.
3.2 The Global Graph Convolution Operation
Let G(A, X) denote the input sample graph with n nodes, where IR"x" is the adjacency matrix and X E Rn is the node feature matrix. For attributed graphs, the node features initialized using one-hot encodings of node labels. For unattributed graphs, we use one-hot encodings of node degrees instead. In our proposed MultiNet, we begin by applying a global graph convolution layer to the input graph G(A,X) X) to perform initial feature extraction, producing the initial node features. For notational convenience, we denote the output of the global graph convolution X RRaxd Here we employ the basic convolution operation of the Vanilla GCN defined by [88. i.e.,
x² (AXWe)
Here Wa Rdxd(0) is the learnable weight matrix of the graph convolution layer, and o(-) is non-linear activation function (e.g., ReLU). The normalized adjacency matrix Á computed as Á = 1/2 AD-1/2 1/2, where À A includes self-loops and D is the degree matrix of A.
3.3 The Local Subgraph Convolution Module
We define novel local subgraph convolution module to adaptively capture local structural patterss from multiple subgraph views. Specifically, the proposed local subgraph convolution module consists of three key components, i.e., the Multi- View Division, the Adaptive Subgraph Convolution, and the Cross- View Feature Fusion.
3.3.1 The Multi- View Subgraph Division
We propose a multi-view division strategy that adaptively assigns the graph into multiple local subgraphs. Specifically,
graph convolution layer learn attention weight matrix P E maxmm where each element P.j represents the attention weight that softly assigns node of the original graph the j-th subgraph view. larger value of P.ş implies that node is more important in the j-th subgraph view. This process enables diverse local substructure modeling and emphasizes the relative importance of nodes across different perspectives. Formally,
P = softmax (Ax (0) Wp),
(4)
where Wp E od xm is a learnable weight matrix for view assignment. The row-wise softmax function ensures that the attention weights across all views for each node sum 1.
In the 1-th layer of subgraph convolution, we maintain a firee-dimensional node representation tensor H()) E R ()) We denote the slice corresponding to view by H!!) Hg B so
that the tensor can be written compactly as ( [H( ..... H (0) To initialize the view-specific
5

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feature matrices, we weight the original node features using the learned attention matrix:
g (diag(P;)) X
j= 1...... m,
(5)
where P3 IR" denotes the j-th column of the assignment matrix P. and diag(P;) is the diagonal matrix of Pj.
3.3.2 The Adaptive Subgraph Convolution Operation
Unlike the standard aggregation scheme of GCNs, we adopt a constrained message-passing strategy, where the information propagation of each node is regulated by its attention score. While this operation shares similarities with the Graph Attention Network (GAT, [32], the attention scores here are learned globally, reflecting the overall importance of each node within this view, rather than computed from local neighborhoods. The updated representation is computed as
H [++1) o (A (diag H W())
(6)
where W Rdd0 xd(++3) denotes the convolution weight the 1-th layer. The convolution weight shared across subgraph views. Here, we the normalized adjacency of the original graph, A, and the view-specific effect is realized via the node-wise gating matrix diag(P))
3.3.3 The Cross- View Feature Fusion
After L convolutions, fuse features from all subgraph views to obtain comprehensive node representations. Specifically, we concatenate all view-specific node features along the feature dimension, resulting in a matrix size n x md("), which is then passed through a Multi-Layer Perceptron (MLP) to produce a unified node feature matrix E Rxd i.e.
Z = MLP(( ( ... (1
3.4 The Alignment- based Readout
we propose structure-aware graph readout mechanism based alignment. This mechanism helps avoid the limitations of simple global readout functions, which tend to neglect the distinet structural information and local differences between nodes, thus alleviating the graph-level over-smoothing phenomenon. By establishing reliable ordered correspondence for node clusters, this mechanism leverages more expressive MLP for graph-level readout.
Inspired by some works that study node alignment mechanisms in graphs 2 12
Specifically, we first assign nodes to clusters. ensure that clusters are consistently aligned across different graph samples, we learn shared cluster assignment function parameterized by Ws E T x* which generates an assignment probability matrix S E IR=x* for each graph through graph convolution layer, i.e.,
S softmar( (anear((A ws).
(8)
Then, this matrix to perform alignment for node features, i.e.,
=T 7.
Since the aligned node features 2 E Raxd now order-consistent across samples, we first flatten 7 into vector of dimension rl' and then apply Multi-Layer MLP to assign different importance weights to the aligned clusters, producing the graph-level representation y, i.c.,
MLP(ffatten( 2)).
(10)
4 Theoretical Analysis: How the MultiNet Works?
In this section, we theoretically analyze why the proposed MultiNet effectively mitigates the graph- level over-smoothing phenomenon. The proposed MultiNet mitigates the graph-level mvor-smo phenomenon from two aspects: First, the local subgraph convolution module limits the extent of node information propagation, reducing the loss of local node-level information and alleviating the

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Original Graphs
I-layer Conv
L-layer Conv
Readou:
Original Graphs
Subgraphs
L-layer Conv Align-Readout
Different atoms
Node features and graph representations
The sabgraph regions emphasized by different views
Figure 3: Toy examples to compare the conventional GCNs (left) and our MultiNet (right).
node-level over-smoothing problem. Second, the structure-aware readout mechanism aligns nodes across different graphs and employs a more expressive MLP for graph readout, thereby reducing the loss of structural information and mitigating the graph-level over-smoothing phenomenon.
Specifically, in the local subgraph convolution module, the propagation kernel in the j-th view defined in Equation can be equivalently expressed as:
A₁ Å (diag(P,)).
(11)
Since P3 is not a vector with all ones, the spectral radius of Aú is smaller than that Å, i.e.
P(A,) < -
(12)
where the spectral radius is denoted by p(.). As the number of layers increases, node features do not collapse the dominant eigenvector direction, and instead retain contributions from multiple eigenvector components. From the perspective of information propagation, this design effectively limits the extent of feature diffusion, allowing the model focus specific local regions with higher attention weights, thus mitigating uncontrolled propagation and preserving distinct local node information. Information propagation through low-attention nodes is significantly suppressed by the attention weights, limiting its influence. While deeper layers may reduce feature magnitudes, this scaling issue can be addressed by normalization methods such as LayerNorm. Moreover, distinct attention vectors Pj in each view guide the model to focus on different local regions. Fusing these diverse perspectives further enhances the expressiveness and robustness of node representations.
Moreover, unlike global readout functions and traditional matrix decomposition or low tow-rank approxi- mation methods, the alignment-baase readout mechanism does not simply compress node features. Instead, it adaptively establishes correspondences between node clusters, enabling the use more expressive readout function such as MIP. This design preserves permutation invariance while maintaining distinct structural information, thereby enhancing the discriminative power of graph representations and effectively mitigating the graph-level over-smoothing problem.
Here we select two representative toy examples of molecular graphs, ethanol and dimethyl ether, to compare the conventional GCNs with our proposed MultiNet model (see Figure 3), As shown on the left, in the traditional GCNs, multi-layer message passing causes node features to become homogenized, resulting in the loss of crucial local-level information. Subsequent readout operations (e.g., sum or mean pooling) further overlook structural differences between molecules, ultimately rendering the two distinct molecular graphs indistinguishable. In contrast, shown the right, our MultiNet leverages a subgraph convolution module that adaptively identifies and focuses on discriminative functional groups, guiding the convolutional process toward structurally distinet local regions and effectively reducing the similarity among node features. Furthermore, by incorporating a structure-awee aware node alignment strategy and employing more expressive MLP, our method enhances the discriminative power of the final graph representations.
Experiments
5.1
The Experimental Setup
evaluate the proposed MultiNet model on several benchmark graph classification datasets, in- cluding MUTAG, PTC. MR, ENZYMES, PROTEINS. DD, IMDB-B, and IMDB-M. These datasets
PROTEINNS
7

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Table 1: Classification accuracy (In + standard error) on benchmark datasets.
Method
MUTAG
PTC_MR
ENZYMES
PROTHINS
IMDH-B
Rank
DGCNN
84.0=6.7
58.3+7.0
38.9+5.7
72.9+3.5
76.6+44.3
69.24:3.0
45.6+3.4
12.40
DifiPool
79.8+7.1
6088+7.0
59.5+5.6
73.7+3.5
75.0+3.5
68.443.3
45.6+3.4
10.80
ECC
75.4=6.2
$5.743.3
29.548.2
7233334
72.6=4.1
67.742.8
43.5+3.1
15.60
GIN
84.7-6.7
58.845.5
$9.614.5
73.344.0
75.3+229
71.243.9
48,5+3333
7.40
GraphSAGE
83.6499.6
60.1+4.7
58.2+6.0
73.044.5
72.9+200
68.8+4.55
47.6+:355
10.80
DGK
82.66=1445
1..32=1.13
$3.4=0.9
7168==0.50
78.50=0.22
66.96:0.56
44,55:00.52
10.57
I-RWNN
89.244.33
56.745.2
74.7+3.3
77.624.7
70.844.8
47,8+3.88
6.33
2-RWNN
88.1=:4.8
57,4499
74.1=22.8
76.94446
70.6::4.4
48.8:1299
7.16
3-RWNN
88.6+4.11
$7.6+6.3
74.3+33.
77.4449
70.7+:3.9
47,8++355
6.50
GKNN-WI.
85.73+42.70
59.29-42.54
74.94++1.10
69.70+22.0
47.87++1.78
7,00
GKNN-GL
85.24=22.28
1..4
1..36=1112
69.90=2.20
4..67=1.22
7,40
RWGK
80.77=0.72
55.91=0.37
22.37=0.35
74.20=0.40
71.70=0.47
67.94=0.77
46.72-10.30
13,14
SPGK
03.38:0.31
56.55-10.53
29.00=00448
75.10=
06.25-1.26
71.26::1004
51.33-+0.57
6,71
GK
81.66=00.11
24.87+0.22
71.67+0.55
0.26
65.87=0.98
45,42+0.87
14.17
WLSK
02.8
56.05=0.51
52.75=0.44
73.52=0.43
0.378=0.36
7188
49.50=00.49
7.14
JTQK
05,5
$7.39210:66
56.41=0042
72.86-1041
79.4
72.45=0.81
00
6.00
ASK
87.50=0.65
70,38=0.22
50.12.0.51
9.33
EDBMK
86.35
56.75
36.85
78.19
8.25
QBMK 08.55=00.4
09.38=0.36
77.60=
5.00
MultiNet 89.81=1446
62.65-10.88
1.58
76.40:10.87
78.90=0.51
76.4920.60
51.9::0..25
2.28
cover two primary domains: bioinformatics (Bio) and social networks (SN). All experiments are implemented in PyTorch and PyTorch Geometric, and executed GPU (24GB VRAM). For all datasets, we use a local subgraph convolution module with to 4 layers, set the number of views 8, the node embedding dimension to 32, and the number of aligned nodes during readout to 8, with ReLU as the activation function. The model is trained using the Adam optimizer, with hyperparameters such learning rate and number of epochs tuned via validation. To ensure statistical robustness, we perform ten runs of 10-fold cross-validation and report the mean classification accuracy along with the standard deviation
NVIDIA GeForce RTX 3090
5.2 Experiments on Graph Classification
We compare the MultiNet with several advanced GNNs and graph kernels. Specifically, The advanced GNNs include: five baseline models (the DGCNN 33) DifiPool (38,, ECC 30,, GIN
34]
and GGE (16)) as well as six additional advanced models (the DGK (37] P-RWNN
23
(with p 1, 3), GKNN-WL, and GKNN-GL (1). The graph kernels include: the RWGK
15
SPGK 6). GK (29). WLSK (28) JTQK [4 (where 2), ASK [5], EDBMK [35), and QBMK 3
The classification accuracy and standard error reported in Table [ and the last column represents the average rank. For the baseline deep learning methods, we report the results from the fair comparison 14] or reproduce them following its evaluation protocol. For other models, adapt their best results from their original papers.
The MultiNet achieves the best near-best classification accuracy most benchmark datasets, demonstrating the effectiveness of the proposed multi-view adaptive propagation mechanism in enhancing the performance of graph classification. The advantages of the MultiNet are summarized as follows: 1) Compared with the graph kernel methods, the MultiNet provides an end-to-end framework that adaptively extracts features from different views, eliminating the need for handcrafted kernel design and feature engineering. 2) Compared to the traditional GNN-based methods, our MultiNet restricts the extent of message propagation through node-level attention weights, enabling the model to focus on view-specific critical local features. This effectively suppresses redundant neighbor information and reduces noise. Moreover, MultiNet maintains multiple parallel subgraph convolutions. allowing information to propagate adaptively in different directions. This design captures diverse structural patterns and improves generalization across various types of graph data. 3) Compared to the models that use simple readout functions (e-g., the GIN), the MultiNet introduces an alignment- based readout strategy that considers local node features and the overall graph structure when generating graph representations, thereby further enhancing the expressiveness.
"Code is available at http:://gihhub.oom/iiiiggg/////d
8

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Table AD of initial node features for different datasets.
Dataset
MUTAG
PTC_MR
ENZYMES
PROTEINS
DD>
IMDR-B
IMDB-M
AD
0.0331
0.1608
0.1315
0.1857
0.1437
0.6320
0.7328
Accuracy 0%
Ascurwyy
8
0/5
Layers
Layers
Layers
Layers
(a) MUTAG(Acc).
(b) PTCCMR(Acc).
(c) PROTEINS(Acc).
(d)
6.3
Sesrage Diplaace
2.4
94
TXX
DepASaCE
OepiZAzE
16
22
04
0.2
10
32
04
0.0
28
52
10
22
04
Layers
Layers
(e) MUTAG(AD).
(1) PTC. (R(A).
(g) PROTEINS(AD).
(h) IMDB-B(AD).
Figure 4: Classification accuracies (%) and the AD values on the four datasets.
5.3 Evaluations on Mitigating the Over-Smoothing
We conduct experiments to validate the effectiveness of MultiNet in mitigating the graph-level over- smoothing. To quantify the differences between graph representations, we adopt the Average Cosine Distance (AD). Given the graph-level representation matrix H, AD is defined as:
N
N
H Hy ||
m(H)=
(13)
N2
Although this metric does not satisfy conditions a) and b) in Definition 1, primary concern whether graph representations converge to similar directions. Therefore, we choose a metric that emphasizes directionality rather than vector magnitudes, which is similar those used in the node-level over-smoothing analysis. A higher AD value indicates greater diversity among graph representations, suggesting that over-smoothing is less severe. To analyze the impact of initial features, we compute the average node features for each dataset and calculate their corresponding AD values (see Table 2) The results show that the AD values for the Bio datasets are below 0.2, supporting Assumption that the initial node features in these datasets are highly similar.
Subsequently, we compare the proposed MultiNet with three standard GCNs (including the Vanilla GCN, GIN, and GraphSAGE) on three Bio datasets (including the MUTAG, PTC_MR, and PRO- TEINS datasets) and an SN dataset: the IMDB B dataset. We set the number of convolutional layers to 2, 4, 8, 16, 32, and 64, and report both classification accuracy (%) and the AD values in Figure 4 In the MultiNet, the number of layers refers the depth of the local subgraph convolution module. The results show that our model already outperforms other baselines under shallow configurations (e.g., L 2 ~ 4), demonstrating that the proposed subgraph convolution and the alignment- readout are inherently effective without requiring deeper networks. As the network depth increases, the MultiNet experiences a smaller drop accuracy and consistently maintains a higher AD value. In contrast, the AD values of the baseline models decrease significantly, indicating a more severe over-smoothing problem. These findings demonstrate that the MultiNet effectively mitigates the over-smoothing problem on the graph level. Even though Assumption does not hold clearly on the IMDB-B dataset, the graph-level smootsigg remains evident. This suggests that local node-level features and structural information play crucial role in graph classification, which are effectively captured by the MultiNet. Notably, under deeper configurations (e.g., L=16 or L=32), the MultiNet achieves competitive or even superior classification accuracy compared to its shallower variants on certain datasets, which further demonstrates its ability to mitigate over-smoothing while preserving the ad advantages of deeper propagation.

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5.4 Ablation Studies
We conduct ablation studies by removing individual modules to evaluate the contribution of each component in the proposed MultiNet. Specifically, compare the performance of the following three variants: (1) w/o L: removing the local subgraph convolution module; (2) w/o F: replacing the feature fusion module with simple feature addition; (3) Avg. Readout: replacing the alignment module with an average readout function; and (4) Sum Readout: replacing the alignment module with a sum readout function. The results in Table Findicate that the MultiNet consistently outperforms all ablated variants across benchmark datasets. These findings demonstrate the effectiveness of each component in our framework. The local subgraph convolution module enables adaptive multi-view feature extraction. The feature fusion module effectively integrates information from diverse perspectives. Meanwhile, the alignment module enhances the expressiveness of the graph-leve representations. Together, these modules collectively contribute to the superior classification performance.
Table 3: Ablation experimental accuracy (In % + standard error).
Method
MUTAG
PTC_MR
ENZYMES
PROTEINS
DD
IMDB-B
IMDB-M
MultiNet
89.81::1.46
62.65:10.88
5.583:1.55
76.40:10.87
78.9010.51
76.49-0.60 0.60
0..93::..25
w/o
83.39+1.04
58.83+00.44
27.53+-105
75.01+0.40
78.60+0.45
74.73-0.6
50.61+0.43
w/o
88.131:2.13
59.5112.19
50.1711.20
74.83:1.90
78.5810.36
72.20-1.06
50.80:1145
Avg. Readout
16.117
61.79+1.51
52.334-2.09
74.3340.22
78.63+0.25
73.52-1.05
50.13+0.32
Sum Readout
85,55+0..6
62.53+00.88
29.17+1.77
73.51#1142
7844++0.22
73,50=1.92
47.13+1143
5.5 Impact of the Number of Subgraphs
We evaluate the impact of the number of subgraphs model performance by conducting extensive experiments on the MUTAG, PROTEINS, and IMDB-B datasets, varying the number of subgraphs among 2, 4, 8, 16, and 32. The results in Figure show that setting the number 8 yields the highest accuracy across all three datasets, suggesting that a moderate division effectively captures rich structural information and consistently enhances performance. However, further increasing the number of subgraphs (to 16 and 32) leads to performance degradation, possibly due to the introduction of noise or overfitting from excessive division.
90
76
70
314
74
16
32
Subgrapas
Subgmabh
Subgrapās
(a) MUTAG.
(b) PROTEINS.
(c) IMDB-B.
Figure 5: The experimental results of different numbers of subgraphs.
Conclusions
In this paper, we propose a novel MultiNet to mitigate the graph-level over-smoothing issue en- countered in graph classification tasks. By incorporating local subgraph convolution module, the MultiNet adaptively divides the input graph into multiple subgraph views. The proposed method extracts more discriminative local features by focusing on view-specific regions with higher attention weights, followed by cross-view node feature fusion to obtain comprehensive node representations. Additionally, we introduce an alignment-bun based readout mechanism that enhances the quality of the graph representations. Through theoretical analysis and experimental results, we demonstrate that the MultiNet effectively mitigates the graph-level over-smoothing and consistently outperforms existing state-of-the-an methods on graph classification tasks. In future work, we aim to extend our model to explore the syoo-smog problem in node classification tasks and investigate sparsification-ban based approaches to further reduce its spatial complexity.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (Nos. 62576371, T2122020, 62172370 and 62576198). Ming Li acknowledged the supports from the Jinhua Science and Technology Plan (No. 2023-3-003a). This work is also supported by Open Project Foundation of Key Laboratory of Computation Intelligence and Chinese Information Processing of Ministry of Education and Key Laboratory of Data Intelligence and Cognitive Computing of Shanxi Province.
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A Technical Appendices and Supplementary Material
A.1 Statistics of the Datasets
In Table 4 we summarize the statistics of the benchmark graph classification datasets used in this work, including the maximum (Max #Nodes) and average number of nodes (Mean #Nodes), total number of graphs (#Graphs), number of classes (#Classes), and the domain of each dataset.
Table 4: Statisties of the datasets used in experiments
Dataset
MUTAG
PTC_MR
ENZYMES
PROTEINS
DD
IMDB-B
IMDB-M
Max #Nodes
28
64
126
620
5748
136
89
Mcan #Nodes
17.93
14.29
32.63
39.06
284.3
19.77
13
#Graphs
188
344
600
1113
1178
1000
1500
#Classes
2
Domain
Bio
Bio
Bio
Bio
Bio
SN
SN
A.2 Visualization
To intuitively demonstrate the effectiveness of multi-view subgraph division, we showease examples from real-world datasets in Figure 6 including MUTAG, PROTEINS, and IMDB-B. Each row corresponds to the same graph instance, while each column shows a specific subgraph view. The intensity of the node color represents the attention weights. The variation in color across different subgraphs suggests that the model adaptively regulates information propagation between subgraphs by capturing features from distinct local regions. This helps preserve local information and mitigates the over-smoothing of node features.
Example foo MITAG
Exmples Grom PROTEINS
The Allection of Nodess
Exumples from IMDRE
Figure 6: Examples of multi-view subgraph division from different datasets.
21


Reviews:
[{"review_id": "4RNtaDDyJI", "reviewer": "Reviewer_XtUo", "summary": "This work aims to reduce the effects of the graph-level over-smoothing problem for graph classification, and proposes a novel MultiNet model (a new graph neural network, GNN). This paper first analyze how the over-smoothing influences the graph representation learning through the GNN. Then, this paper proposes to perform the attention-based subgraph convolution on adaptive extracted subgraph based views, that focuses on the information propagation within the subgraph view and naturally reduce the information passing over the global structure. The experiments on benchmark datasets show the superior performance for the proposed MultiNet model.", "questions": "1. I feel a little confused for the symbol P^(j) between the equations (11) and (12), should it be P_j ? It is not consistent to other symbol P_j. Maybe, it is a typos?\n2. For the experiments, why some GNN models are not evaluated on the MUTAG and PTC datasets?", "limitations": "I think some theoretical analysis about how the proposed method reduce the graph-level over-smoothing need to be more clear. As I say in the weakness, the information seems still pass between low attention nodes, this may lead to some minor over-smoothing problem.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. An interesting strategy to divide the global graph structure into subgraph-based views, and perform the subgraph convolution to reduce the over-smoothing problem of the convolution operation on the global structure.\n2. Provide mathematical analysis to show how the MultiNet works and limits the over-smoothing over the global structure.\n Weakness:\nIt seems that each subgraph view is not completely isolated from the global graph structure, since the substructure is identified by node attentions. Although the information may be mainly propagated between high attention nodes, the information may still pass between the low attention nodes. The authors did not clearly explain this problem.", "was_revised": true, "final_justification": "accept", "rebuttal": "Dear Reviewer XtUo,\n\nWe sincerely thank you for your careful reading of our paper and for the valuable questions and suggestions. Below, we provide detailed responses to each of your comments. We have made every effort to clarify our contributions and address your concerns with additional explanations, experiments, or theoretical analysis where appropriate.\n\nWe hope that our responses can resolve your concerns and better convey the significance of our work. We would greatly appreciate any additional feedback you may have.\n\nResponse to weakness and limitations: Thanks for the suggestion. In our design, we adopt a soft attention mechanism instead of the hard assignment to ensure differentiability and stable gradient backpropagation. The adaptive node attention allows the model to emphasize important nodes in each subgraph view, guiding it to focus on more discriminative local regions in terms of structure. While some information may still flow between the low-attention nodes, such propagation is effectively suppressed by the attention weights, which significantly limits its influence. This constrained message propagation mechanism mitigates the global feature homogenization and thereby alleviates the graph-level over-smoothing issue. We will further refine our theoretical analysis in section 4 to clearly explain how our method suppresses over-smoothing while preserving the flexible information propagation, as well as discussing the potential information flow among the low-attention nodes.\n\nResponse to Q1: Thanks for the suggestion. The notation $P^(j)$ should be written as $P_j$, which denotes the attention vector for the j-th subgraph view. We will correct this notation in the final version.\n\nResponse to Q2: Thanks for the suggestion. We have clearly indicated the source of the experimental results in Section 5.2. For the baseline deep learning methods, we follow the fair comparison protocol provided in [1], which does not report results on the MUTAG and PTC_MR datasets. Accordingly, we did not include these baselines on the two datasets in our initial submission. To address your concern, we have preliminarily reproduced the results for these models using the official code from [1]. As shown in the table below, our proposed MultiNet still achieves superior performance on both datasets, further demonstrating its effectiveness and robustness. We will update the manuscript to incorporate additional results and relevant discussions. \n\n| Datasets | DGCNN | DiffPool| ECC | GIN| GraphSAGE| MultiNet|\n|----------|------------|------------|-----------|-----------|-----------|--------------|\n| MUTAG| 84.0±6.7| 79.8±7.1| 75.4±6.2| 84.7±6.7| 83.6±9.6| **89.81±1.46** |\n| PTC_MR| 58.3±7.0| 60.8±7.0| 55.7±3.3| 58.8±5.5| 60.1±4.7| **62.65±0.88** |\n\n[1] Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. A Fair Comparison of Graph Neural Networks for Graph Classification. In Proceedings of ICLR, 2020."}, {"review_id": "KjG8Z9utAZ", "reviewer": "Reviewer_xLjw", "summary": "This paper develops a new Adaptive Multi-Viewed Subgraph Convolutional Network (MultiNet) to solve the over-smoothing problem and improve the graph classification performance. The MultiNet is defined based on the subgraph convolution on divided subgraphs from the original global graph. The experiments on standard datasets confirm the performance of the MultiNet. Overall, this paper is well-written, easy to follow and  and technically sound.", "questions": "Some statements need to be more clear. Some experiments are missed. Some technology details need to be well explained. Please refer my rebuttal weakness for details.", "limitations": "Yes.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "**Strengths**\n1. This paper defines an end-to-end framework to adaptively divide the input graph into subgraph views, and perform the local convolution on the subgraphs to limit the information passing among different subgraph views. So, the new MultiNet can avoid the over-smoothing problem. \n2. This paper defines a new local convolution filter on the subgraph view with the attentions of adjacent nodes. The new filter can propagate the node information based on the attention and limit the information of nodes with lower attention.\n3. The authors provide necessary theoretical analysis to confirm the advantages of  the new MultiNet.\n\n**Weaknesses**\n1. The over-smoothing is always a serious problem for node classification, but how the over-smoothing influence the graph classification? More theoretical analysis or descriptions are needed. \n2. Why use the alignment readout, why do not directly use the sum or avg pooing? These operations are widely used for graph neural networks to extract graph representation.\n3. For table 1, the accuracies of some method are missed, why not compare them? \n4. The authors also need to cite some necessary references about graph over-smoothing problem, not only list the references about node over-smoothing.", "was_revised": true, "final_justification": "All my concerns have been well addressed. This work is theoretically solid and proposes a novel method. Therefore, I recommend accepting this paper.", "rebuttal": "Dear Reviewer xLjw,\n\nWe sincerely thank you for your careful reading of our paper and for the valuable questions and suggestions. Below, we provide detailed responses to each of your comments. We have made every effort to clarify our contributions and address your concerns with additional explanations, experiments, or theoretical analysis where appropriate.\n\nWe hope that our responses can resolve your concerns and better convey the significance of our work. We would greatly appreciate any additional feedback you may have.\n\nResponse to W1: Thanks for the suggestion. Although over-smoothing was originally identified in the context of node classification, we argue that it also has a significant impact on graph classification. When stacking multiple graph convolution layers, node features across structurally different regions may become overly similar, thereby diminishing the ability to preserve and distinguish informative local substructures. This issue is further exacerbated by the simple permutation-invariant global readout functions (e.g., sum or mean), which can obscure the structural differences among graphs and reduce the discriminative power of graph-level representations. To address this, we provide a formal definition of graph-level over-smoothing in Section 2.2, along with a theoretical analysis of its causes and implications. In addition, Figure 3 offers an illustrative example showing how over-smoothing degrades the distinctiveness of graph representations in classification tasks. We will strengthen this theoretical discussion further in the revised version.\n\nResponse to W2: Thanks for the suggestion. Traditional pooling operations such as sum, mean, or max are indeed widely used due to their simplicity, but they tend to overlook the local structural and the feature differences among nodes. In contrast, our alignment-based readout learns a cluster assignment matrix that aligns the order of node clusters and preserves the local topology across graphs, enabling the use of a more expressive MLP mapping. This enhances the ability to capture discriminative subgraph structures. We have highlighted these advantages in Section 3.4 and Section 4, and further validated the effectiveness of the alignment readout by replacing it with average and sum readouts in the ablation studies presented in Section 5.4. We will improve the explanation of this design choice in the revised version.\n\nResponse to W3: Thanks for the suggestion. We have clearly indicated the source of the experimental results in Section 5.2. To ensure a fair comparison, we report the results from the fair comparison [1] for the baseline deep learning methods. For other models, we adapt their best results from their original papers. Since the referenced literature lacked comparative analyses on these specific datasets, we did not initially include their experimental results. Following the experimental protocol established in [1], we have preliminarily supplemented the performance evaluation of baseline deep learning methods on both MUTAG and PTC_MR datasets. As shown in the table below, our proposed MultiNet still achieves superior performance on both datasets, further demonstrating its effectiveness and robustness. We will update the manuscript to incorporate additional results and relevant discussions. \n\n| Datasets | DGCNN | DiffPool| ECC | GIN| GraphSAGE| MultiNet|\n|----------|------------|------------|-----------|-----------|-----------|--------------|\n| MUTAG| 84.0±6.7| 79.8±7.1| 75.4±6.2| 84.7±6.7| 83.6±9.6| **89.81±1.46** |\n| PTC_MR| 58.3±7.0| 60.8±7.0| 55.7±3.3| 58.8±5.5| 60.1±4.7| **62.65±0.88** |\n\n[1] Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. A Fair Comparison of Graph Neural Networks for Graph Classification. In Proceedings of ICLR, 2020.\n\nResponse to W4: Thank you for the suggestion. Most existing literature has primarily focused on the node-level over-smoothing, and there is a notable scarcity of studies addressing the over-smoothing problem at the graph level. In our paper, we take a step forward by providing a strict definition of the graph-level over-smoothing for the first time in Section 2.2. Furthermore, in Section 5.3, we introduce a scientific metric to measure the graph-level over-smoothing and validate it through experiments.\nAlthough the work in [2] proposes a strategy to mitigate over-smoothing and applies it for graph classification tasks, it still focuses on alleviating over-smoothing at the node level and lacks the experiments addressing over-smoothing at the graph level. Thus, our work fills this gap and offers a more comprehensive analysis. We will provide additional explanations in the revised version to enhance our contribution.\n\n[2] Stevan Stanovica, Benoit Gaüzère, and Luc Brun. Graph Neural Networks with maximal independent set-based pooling: Mitigating over-smoothing and over-squashing. Pattern Recognit. Lett., 187, 14–20, 2025."}, {"review_id": "gGI3bP196N", "reviewer": "Reviewer_YyPC", "summary": "This paper introduces MultiNet, a novel Adaptive Multi-Viewed Subgraph Convolutional Network designed to address graph-level over-smoothing in Graph Convolutional Networks (GCNs). By leveraging local subgraph convolution and alignment-based readout mechanisms, MultiNet enhances graph classification performance through preserving local node features and structural information.", "questions": "Q1. How does the paper theoretically or experimentally prove that permutation invariance of traditional readout functions is the primary cause of graph-level over-smoothing?\nQ2. Given that the proposed model still relies on permutation-invariant aggregation, how does it fundamentally address graph-level over-smoothing?\n\nQ3. Does the proposed model leverage deeper propagation to achieve better performance, or does it primarily focus on mitigating performance degradation caused by deeper layers?\n\nQ4. Are there any experiments showing that the model outperforms shallower networks in terms of classification accuracy?\n\nQ5. Why does the paper not include comparisons with state-of-the-art graph-level algorithms? How does the proposed method optimize graph-level representations compared to these recent approaches?\n\nQ6. How does the aggregation process in the proposed model relate to matrix factorization techniques? Can the authors provide theoretical or experimental validation to demonstrate the effectiveness of using matrix factorization-like operations in the model design?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "S1. MultiNet adaptively divides graphs into multiple subgraph views, capturing discriminative local node features and mitigating node-level over-smoothing.\n\nS2. Establishes node correspondences across graphs, enabling expressive readout functions that preserve structural information and reduce graph-level over-smoothing.\n\nS3. Demonstrates effectiveness in mitigating over-smoothing and achieving state-of-the-art graph classification performance on benchmark datasets.\n\nW1. The paper claims that solving node-level over-smoothing cannot fully address graph-level over-smoothing due to the permutation invariance of traditional readout functions. However, the authors do not provide sufficient theoretical or experimental evidence to support this claim. Moreover, the proposed model still relies on permutation-invariant aggregation, which raises questions about whether the issue is truly resolved.\n\nW2. The ultimate goal of addressing over-smoothing is to improve model performance. However, the experimental results show that the proposed model primarily mitigates the performance degradation caused by deeper layers, rather than leveraging deeper propagation to achieve better overall performance. This limits the practical benefits of the model.\n\nW3. The paper lacks comparisons with recent graph-level algorithms. Without such comparisons, it is difficult to evaluate the effectiveness of the proposed method in optimizing graph-level representations.\n\nW4. The aggregation process in the proposed model closely resembles matrix factorization, particularly in the use of the node assignment matrix to map node embeddings to cluster-level embeddings. However, the paper does not explicitly discuss this similarity or provide theoretical validation for the effectiveness of this design. Adding a discussion and validation of matrix factorization approximations would strengthen the paper's contributions.", "was_revised": true, "final_justification": "This rebuttal partially resolved my concerns. Despite my consistent questioning about the permutation-invariant aggregation, I would like to raise my score due to the reviews given by the other reviewers. I will also not be against the acceptance of this paper.", "rebuttal": "Dear Reviewer YyPC,\n\nWe sincerely thank you for your careful reading of our paper and for the valuable questions and suggestions. Below, we provide detailed responses to each of your comments. We have made every effort to clarify our contributions and address your concerns with additional explanations, experiments, or theoretical analysis where appropriate.\n\nWe hope that our responses can resolve your concerns and better convey the significance of our work. We would greatly appreciate any additional feedback you may have.\n\nResponse to Q1: Thanks for the suggestion. We argue that our focus is not on permutation-invariant readout functions causing graph-level over-smoothing, but rather on how overly simplistic global readout functions (e.g., sum or average pooling) can exacerbate the graph-level over-smoothing when the node representations become homogeneous. Specifically, graph classification tasks inherently require the permutation-invariant readouts to ensure the output consistency regardless of the node ordering. However, most traditional GNNs often rely on the overly simplified global pooling schemes that treat all nodes as equally important. This approach ignores both the local node features and the topological structures. When combined with the node-level over-smoothing, it results in indistinguishable graph representations and severely degrades the discriminative power of the model. To support this claim, we have provided a formal mathematical proof in Theorem 1 (Section 2.2), which shows how the simple readout functions can lead to the graph-level representation convergence under the node feature homogenization. In addition, we offer an illustrative example in Figure 3 (left) and empirical evidence in Section 5.4 through the ablation studies comparing alignment readout with the traditional avg/sum pooling. We will revise the relevant discussion in the final version to ensure clarity and rigor. \n\nResponse to Q2: Thanks for the suggestion. We agree that the permutation invariance is a necessary property for the readout functions in graph classification, as the predicted graph labels should not depend on the node ordering. However, we argue that the issue lies not in the permutation invariance itself, but in the simplicity of the traditional permutation-invariant global readout functions, such as sum or average pooling, which treat all nodes equally. The root reason of this issue lies in the disorder of nodes within the graph. To address this, we aim to establish a reliable ordered correspondence for node clusters, which leads us to introduce an alignment-based readout mechanism. Specifically, we learn a cluster assignment matrix S using a shared function across all graphs, which ensures that clusters are aligned in a consistent order across different samples. This aligned structure allows us to apply a non-symmetric MLP at the cluster level, assigning different importance weights to different clusters. This mechanism preserves permutation invariance at the graph level, while avoiding the uniform treatment of nodes, thereby improving the discriminative power of the graph representation and effectively mitigating the graph-level over-smoothing. For this issue, we have clearly explained our motivation in the Introduction and provided a theoretical explanation in Section 4. We will further elaborate on this mechanism in the revised version.\n\nResponse to Q3: Thanks for the suggestion. Our goal is to enhance the discriminative power of graph representations by alleviating the graph-level over-smoothing, rather than relying solely on the deeper propagation for performance gains. As shown in Table 1 and Figure 4, our model already outperforms other baselines under shallow configurations (e.g., L=2~4), demonstrating that the proposed subgraph convolution and the alignment-based readout are inherently effective without requiring deeper networks. Furthermore, the trends in Figure 4 indicate that our model degrades much more slowly as the number of layers increases, effectively alleviating the performance deterioration typically observed in deeper GCNs. Notably, in some datasets, deeper versions (e.g., L=16 or L=32) even outperform shallower ones. This shows that our method not only improves the stability in deeper architectures but also retains the potential to benefit from deeper propagation. Therefore, we believe our model is practically valuable, as it adapts well to both shallow and deep settings, offering robust and competitive graph-level representations. We will follow your suggestion and revise the relevant sections to enhance clarity.\n\nResponse to Q4: Thanks for the suggestion. In Figure 4, we conduct a systematic comparison across different network depths. The results show that our method consistently achieves higher classification accuracy than other models under both shallow and deep settings. Notably, under deeper configurations (e.g., L=16 or L=32), our MultiNet achieves competitive or even superior classification accuracy compared to its shallower variants on certain datasets, which further demonstrates its ability to mitigate over-smoothing while preserving the advantages of deeper propagation. We will follow your suggestion and revise the relevant sections to enhance clarity.\n\nResponse to Q5: Thanks for the suggestion. In Table 1 of our paper, we have included over eighteen representative and state-of-the-art graph classification models, such as p-RWNN (2020), GKNN-WL (2024), GKNN-GL (2024), and QBMK (2024), covering both recent graph neural networks and graph kernel approaches. The results demonstrate that our proposed MultiNet consistently achieves superior classification accuracy across various datasets. Furthermore, Section 5.2 provides a comparative analysis of graph-level representation effectiveness, while the ablation studies in Section 5.4 confirm that each component of our model plays a crucial role in enhancing representation quality. We appreciate the suggestion and will include additional comparisons with more recent methods in the revised version to further strengthen the empirical validation. \n\nResponse to Q6: Thanks for the suggestion. Our method is fundamentally different from the traditional matrix factorization in both its motivation and computational process. First, the core objective of our alignment-based readout is to establish the consistent cluster-level correspondences across graphs, thereby overcoming the limitations of the conventional global readout functions in preserving structural information. This design enhances the expressiveness of the graph-level representations by explicitly retaining the local features and structural distinctions, rather than merely compressing node embeddings. Therefore, the process cannot be simply interpreted as the feature compression or the low-rank approximation. Second, in terms of the computational mechanism, the node-to-cluster assignment matrix is adaptively learned in an end-to-end fashion based on both the graph structure and node features. This assignment is not a static matrix nor derived from a predefined decomposition objective. Consequently, we consider our design conceptually distinct from classical matrix factorization techniques. We have already discussed and justified this design choice in the Introduction and further elaborated it in Section 3.4. We will include a more detailed discussion of this distinction in the revised version to improve theoretical clarity."}, {"review_id": "zVKqgghuMx", "reviewer": "Reviewer_7WyT", "summary": "This paper defines a new graph neural network, namely the Adaptive Multi-viewed Subgraph Convolutional Network (MultiNet), for graph classification. Unlike the classical GNNs, the new MultiNet is defined based on the subgraph convolution operation within the substructures. The substructures are divided by assign different attentions on the nodes, and each substructure consists of nodes with higher attention. Therefore, performing the convolution operation on the substructure essentially equals to performing a subgraph convolution, and this strategy can help to reduce the over-smoothing problem. Because, the information passing is limited among different subgraphs. The new MultiNet have better graph classification performance than the classical GNN methods.", "questions": "I think the authors need to answer my following questions, and the paper can be further improved.\n\nFirst, Eq. 4 and Eq. 8 have the same form, what is the difference between them?\n\nSecond, the node-level over-smoothing is always discussed by many works, what is the difference between the node-level and graph-level over-smoothing? \n\nThird, some minor grammar mistakes need to be fixed, such the sentence on row 184, “where the extent to which a node transmits its information to neighbors...”, there are two leading words of subordinate clause (where and which) .\n\nFourth, as I remember the MUTAG and PTC data was well evaluated by many classical methods, including the DGCNN and GIN, why the authors did not include the results？", "limitations": "I have listed some limitations in Questions, such as some grammar and statement problems.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\nAn interesting idea to reduce the over-smoothing for graph classification.\nGood idea to define new subgraph convolution operation with the attention mechanism.   \nOverall, this paper is clearly written and easy to follow.\n\nExcluding the strengths, I still have some major concerns. Please see Questions.", "was_revised": true, "final_justification": "Thank the authors for addressing my concerns. All of my questions have now been resolved. As I mentioned before, this work is valuable, and I will maintain my positive score.", "rebuttal": "Dear Reviewer 7WyT,\n\nWe sincerely thank you for your careful reading of our paper and for the valuable questions and suggestions. Below, we provide detailed responses to each of your comments. We have made every effort to clarify our contributions and address your concerns with additional explanations, experiments, or theoretical analysis where appropriate.\n\nWe hope that our responses can resolve your concerns and better convey the significance of our work. We would greatly appreciate any additional feedback you may have.\n\nResponse to Q1: Thanks for the suggestion. Although Eq. (4) and Eq. (8) appear similar in form, they serve entirely different purposes and carry distinct semantic meanings within our model. The attention matrix P in Eq. (4) is used during the multi-view division stage, where the nodes of the original graph are assigned to m subgraph views based on their attention scores. This process enables diverse local substructure modeling and emphasizes the relative importance of nodes across different perspectives. In contrast, the matrix S in Eq. (8) is introduced at the final representation generation stage as an alignment-based aggregation matrix. It maps the final node embeddings into s globally shared clusters to facilitate structural alignment and produce the graph-level representation. These two matrices correspond to two distinct phases in our architecture. While they may share a similar mathematical form, their usage and semantic roles are fundamentally different. We will clarify this distinction further in the revised version.\n\nResponse to Q2: Thanks for the suggestion. Although the node-level and graph-level over-smoothing are closely related, they differ in their focus. The node-level over-smoothing occurs within a single graph, where repeated graph convolutions lead to overly similar node representations, ultimately degrading the ability to distinguish nodes for node classification tasks. In contrast, the graph-level over-smoothing refers to the phenomenon where the graph-level representations of different graphs become excessively similar after the readout stage, making it difficult for the model to distinguish between graphs and thus harming performance in graph classification. In our work, we have explicitly distinguished these two types of over-smoothing in Section 2, provided a formal definition and theoretical analysis of graph-level over-smoothing in Section 2.2, and offered an illustrative example in Figure 3. We will further enhance the discussion in the revised version.\n\nResponse to Q3: Thanks for the suggestion. We will thoroughly review the entire manuscript to correct any grammatical errors and typographical mistakes.\n\nResponse to Q4: Thanks for the suggestion. We have clearly indicated the source of the experimental results in Section 5.2. For the baseline deep learning methods, we follow the fair comparison protocol provided in [1], which does not report results on the MUTAG and PTC_MR datasets. Accordingly, we did not include these baselines on the two datasets in our initial submission. To address your concern, we have reproduced the results for these models using the official code from [1]. As shown in the table below, our proposed MultiNet still achieves superior performance on both datasets, further demonstrating its effectiveness and robustness. We will update the manuscript to incorporate additional results and relevant discussions. \n\n| Datasets | DGCNN | DiffPool| ECC | GIN| GraphSAGE| MultiNet|\n|----------|------------|------------|-----------|-----------|-----------|--------------|\n| MUTAG| 84.0±6.7| 79.8±7.1| 75.4±6.2| 84.7±6.7| 83.6±9.6| **89.81±1.46** |\n| PTC_MR| 58.3±7.0| 60.8±7.0| 55.7±3.3| 58.8±5.5| 60.1±4.7| **62.65±0.88** |\n\n[1] Federico Errica, Marco Podda, Davide Bacciu, and Alessio Micheli. A Fair Comparison of Graph Neural Networks for Graph Classification. In Proceedings of ICLR, 2020."}]

Author rebuttal:

Meta-review: This paper proposes MultiNet, an adaptive multi-viewed subgraph convolutional network designed to mitigate graph-level over-smoothing and enhance graph classification. Reviewers initially raised questions regarding the distinction between node-level and graph-level over-smoothing, the clarity of theoretical analysis, missing baselines, and the design of the alignment-based readout. The authors’ rebuttal provided thorough clarifications, additional experiments, and theoretical explanations, which effectively addressed the reviewers’ concerns. Overall, I recommend acceptance, as the work is technically solid, though the final version would benefit from clearer exposition of the theoretical contributions and further emphasis on the differences from related matrix factorization approaches.
Decision: Accept (poster)