On the Expressive Power of Subgraph Graph Neural Networks for Graphs with Bounded Cycles

TL;DR

This paper demonstrates that $k$-hop subgraph GNNs can approximate any continuous permutation-invariant function on graphs with bounded cycles, enhancing their expressive power beyond traditional GNNs.

Contribution

It proves the expressive capabilities of $k$-hop subgraph GNNs on graphs with bounded cycles and extends the analysis to GNNs without subgraph structure.

Findings

$k$-hop subgraph GNNs can approximate any permutation-invariant function on certain graphs.

Theoretical validation of the relationship between aggregation distance and cycle size.

Experimental results support the theoretical claims on benchmark datasets.

Abstract

Graph neural networks (GNNs) have been widely used in graph-related contexts. It is known that the separation power of GNNs is equivalent to that of the Weisfeiler-Lehman (WL) test; hence, GNNs are imperfect at identifying all non-isomorphic graphs, which severely limits their expressive power. This work investigates $k$-hop subgraph GNNs that aggregate information from neighbors with distances up to $k$ and incorporate the subgraph structure. We prove that under appropriate assumptions, the $k$-hop subgraph GNNs can approximate any permutation-invariant/equivariant continuous function over graphs without cycles of length greater than $2k+1$ within any error tolerance. We also provide an extension to $k$-hop GNNs without incorporating the subgraph structure. Our numerical experiments on established benchmarks and novel architectures validate our theory on the relationship between the information aggregation distance and the cycle size.