Learning to Count Isomorphisms with Graph Neural Networks
Xingtong Yu, Zemin Liu, Yuan Fang, Xinming Zhang
TL;DR
This work tackles subgraph isomorphism counting, a #P-complete task, by introducing Count-GNN, an edge-centric graph neural network that explicitly models fine-grained structure and adapts to diverse queries via query-conditioned graph modulation. The model uses edge-level message passing with edge adjacency, and modulates input-edge representations with FiLM factors generated from the query, yielding a query-specific input graph representation that improves matching. A counter module combines a flexible Match function with a ReLU-based output to estimate the count, and the training objective incorporates FiLM regularization and standard weight decay. Theoretical results show Count-GNN is at least as expressive as node-centric GNNs and strictly more powerful in capturing graph structure; empirically, Count-GNN achieves superior accuracy and substantial speedups over exact and other learning-based counters across multiple datasets, demonstrating practical impact for scalable subgraph counting.
Abstract
Subgraph isomorphism counting is an important problem on graphs, as many graph-based tasks exploit recurring subgraph patterns. Classical methods usually boil down to a backtracking framework that needs to navigate a huge search space with prohibitive computational costs. Some recent studies resort to graph neural networks (GNNs) to learn a low-dimensional representation for both the query and input graphs, in order to predict the number of subgraph isomorphisms on the input graph. However, typical GNNs employ a node-centric message passing scheme that receives and aggregates messages on nodes, which is inadequate in complex structure matching for isomorphism counting. Moreover, on an input graph, the space of possible query graphs is enormous, and different parts of the input graph will be triggered to match different queries. Thus, expecting a fixed representation of the input graph to match diversely structured query graphs is unrealistic. In this paper, we propose a novel GNN called Count-GNN for subgraph isomorphism counting, to deal with the above challenges. At the edge level, given that an edge is an atomic unit of encoding graph structures, we propose an edge-centric message passing scheme, where messages on edges are propagated and aggregated based on the edge adjacency to preserve fine-grained structural information. At the graph level, we modulate the input graph representation conditioned on the query, so that the input graph can be adapted to each query individually to improve their matching. Finally, we conduct extensive experiments on a number of benchmark datasets to demonstrate the superior performance of Count-GNN.