Can Graph Neural Networks Count Substructures?

TL;DR

This paper investigates the expressive power of graph neural networks in counting substructures, revealing their limitations and proposing new models that improve substructure detection in graph data.

Contribution

The paper provides theoretical analysis of GNNs' ability to count substructures, establishes new equivalences, and introduces the Local Relational Pooling model for improved substructure counting.

Findings

MPNNs, 2-WL, and 2-IGNs cannot count induced substructures of 3 or more nodes.

2-WL and 2-IGNs are equivalent in distinguishing non-isomorphic graphs.

Local Relational Pooling effectively counts substructures and performs well on molecular tasks.

Abstract

The ability to detect and count certain substructures in graphs is important for solving many tasks on graph-structured data, especially in the contexts of computational chemistry and biology as well as social network analysis. Inspired by this, we propose to study the expressive power of graph neural networks (GNNs) via their ability to count attributed graph substructures, extending recent works that examine their power in graph isomorphism testing and function approximation. We distinguish between two types of substructure counting: induced-subgraph-count and subgraph-count, and establish both positive and negative answers for popular GNN architectures. Specifically, we prove that Message Passing Neural Networks (MPNNs), 2-Weisfeiler-Lehman (2-WL) and 2-Invariant Graph Networks (2-IGNs) cannot perform induced-subgraph-count of substructures consisting of 3 or more nodes, while they can perform subgraph-count of star-shaped substructures. As an intermediary step, we prove that 2-WL and 2-IGNs are equivalent in distinguishing non-isomorphic graphs, partly answering an open problem raised in Maron et al. (2019). We also prove positive results for k-WL and k-IGNs as well as negative results for k-WL with a finite number of iterations. We then conduct experiments that support the theoretical results for MPNNs and 2-IGNs. Moreover, motivated by substructure counting and inspired by Murphy et al. (2019), we propose the Local Relational Pooling model and demonstrate that it is not only effective for substructure counting but also able to achieve competitive performance on molecular prediction tasks.