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.

翻译：子图同构计数是图上一个重要问题，因为许多基于图的任务都利用了重复出现的子图模式。经典方法通常归结为一个回溯框架，需要以高昂的计算代价在巨大的搜索空间中导航。最近的一些研究借助图神经网络（GNN）为查询图和输入图学习低维表示，以预测输入图上子图同构的数量。然而，典型的GNN采用以节点为中心的消息传递方案，在节点上接收和聚合消息，这在同构计数的复杂结构匹配中有所不足。此外，在输入图上，可能的查询图空间巨大，输入图的不同部分会被触发以匹配不同的查询。因此，期望输入图的固定表示来匹配结构多样的查询图是不现实的。在本文中，我们提出了一种用于子图同构计数的新型GNN，称为Count-GNN，以应对上述挑战。在边级别，鉴于边是编码图结构的原子单元，我们提出了一种以边为中心的消息传递方案，其中基于边邻接性传播和聚合边上的消息，以保留细粒度的结构信息。在图级别，我们根据查询条件调制输入图表示，使得输入图可以针对每个查询单独适配，以改善它们的匹配。最后，我们在多个基准数据集上进行了大量实验，证明了Count-GNN的优越性能。