Disentangled Graph Representation Based on Substructure-Aware Graph Optimal Matching Kernel Convolutional Networks

TL;DR

Inspired by CNNs, GOMKCN introduces the Graph Optimal Matching Kernel (GOMK) as a convolutional operator, computing similarities between subgraphs and learnable graph filters, and incorporates local correspondences in similarity measurement, resolving the trade-off between differentiability and accuracy in graph kernels.

Abstract

Graphs effectively characterize relational data, driving graph representation learning methods that uncover underlying predictive information. As state-of-the-art approaches, Graph Neural Networks (GNNs) enable end-to-end learning for diverse tasks. Recent disentangled graph representation learning enhances interpretability by decoupling independent factors in graph data. However, existing methods often implicitly and coarsely characterize graph structures, limiting structural pattern analysis within the graph. This paper proposes the Graph Optimal Matching Kernel Convolutional Network (GOMKCN) to address this limitation. We view graphs as node-centric subgraphs, where each subgraph acts as a structural factor encoding position-specific information. This transforms graph prediction into structural pattern recognition. Inspired by CNNs, GOMKCN introduces the Graph Optimal Matching Kernel (GOMK) as a convolutional operator, computing similarities between subgraphs and learnable graph filters. Mathematically, GOMK maps subgraphs and filters into a Hilbert space, representing graphs as point sets. Disentangled representations emerge from projecting subgraphs onto task-optimized filters, which adaptively capture relevant structural patterns via gradient descent. Crucially, GOMK incorporates local correspondences in similarity measurement, resolving the trade-off between differentiability and accuracy in graph kernels. Experiments validate that GOMKCN achieves superior accuracy and interpretability in graph pattern mining and prediction. The framework advances the theoretical foundation for disentangled graph representation learning.

Figures (12)

• Figure 1: The idea of decoupled representation learning. A graph can be represented as a set of node-centric subgraphs. By mapping these subgraphs into a Hilbert space, representations of individual structural factors and the entire graph can be obtained.
• Figure 2: Key mechanisms in graph kernel. (a) The $R$-Convolution framework calculates the graph similarity by the sum of similarities between all-substructure pairs. However, what matters may be the similarities between matched substructures, denoted by solid lines, with the dashed lines representing the spurious similarities. (b) The Label Refinement Mechanism seeks to encode a tree by mapping all node labels to a new label in an injective manner. Nevertheless, when confronted with various unknown trees, a definitive and differentiable hash function remains to be found.
• Figure 3: The framework of the Graph Optimal Matching Kernel Convolutional Network. For an input graph, the subgraphs centered around each node within $k$ hops are fed into the model for pattern analysis. Each subgraph is compared with the graph filters to generate the disentangled node representation. The model iteratively updates node features in the graphs using the neighbor aggregation mechanism. Concatenating the historical features of a node forms a multi-resolution representation of the subtree rooted at this node. Each subgraph or filter embedding is denoted by the set of subtree embeddings. Structural alignment between subgraphs and graph filters can be accomplished based on the subtree embeddings with the optimal match kernel. The similarity between graphs is measured as the sum of paired local structural similarities.
• Figure 4: The method of $t$-level Subtree Encoding. Firstly, the subgraph is decomposed into subtrees rooted at individual nodes, noting that the same node often appears multiple times. Secondly, the features of nodes at each level are aggregated towards the root with the weight of edges, respectively, forming layer embeddings. Thirdly, the subtree embedding is defined as the sequence of level embedding, and the set of subtree embeddings forms the subgraph embedding. Importantly, The embeddings can be computed by iteratively multiplying the feature matrix with the adjacency matrix without explicitly extracting the subtrees.
• Figure 5: Subtree Matching based on greedy algorithm. Firstly, the similarity between any two subtrees across the subgraph and the filter is calculated by the solid function, as shown in the first row. The darker the color of the square, the higher the similarity between the two subtrees. Secondly, the subtrees within the subgraph sequentially select the most similar counterpart in the filter, as shown in the second and third rows. Once a subtree has been selected, it will no longer participate in subsequent matches. Thirdly, the similarity between the subgraph and the filter is measured by the sum of paired subtrees similarities, indicated by the checkmarks.