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

Ziang Chen, Qiao Zhang, Runzhong Wang

TL;DR

This work analyzes the expressive power of subgraph-enhanced graph neural networks (GNNs) and establishes universal approximation results for $k$-hop subgraph GNNs on graphs whose cycle lengths are bounded by $2k+1$. It shows that such networks can approximate any permutation-invariant/equivariant continuous function under mild assumptions, and extends the results to standard $k$-hop GNNs without subgraph structure under a $k$-separability condition, with a bound of $2k-1$ cycles. The theory is complemented by numerical experiments using Graphormer-based architectures on the ZINC dataset, demonstrating that increasing the aggregation distance $k$ up to about 3 captures most relevant cycle information. The findings clarify how the neighborhood aggregation distance interacts with cycle size, guiding practical design choices for expressive GNNs in graphs with bounded cycles.

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.

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

TL;DR

This work analyzes the expressive power of subgraph-enhanced graph neural networks (GNNs) and establishes universal approximation results for -hop subgraph GNNs on graphs whose cycle lengths are bounded by . It shows that such networks can approximate any permutation-invariant/equivariant continuous function under mild assumptions, and extends the results to standard -hop GNNs without subgraph structure under a -separability condition, with a bound of cycles. The theory is complemented by numerical experiments using Graphormer-based architectures on the ZINC dataset, demonstrating that increasing the aggregation distance up to about 3 captures most relevant cycle information. The findings clarify how the neighborhood aggregation distance interacts with cycle size, guiding practical design choices for expressive GNNs in graphs with bounded cycles.

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 -hop subgraph GNNs that aggregate information from neighbors with distances up to and incorporate the subgraph structure. We prove that under appropriate assumptions, the -hop subgraph GNNs can approximate any permutation-invariant/equivariant continuous function over graphs without cycles of length greater than within any error tolerance. We also provide an extension to -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.

Paper Structure

This paper contains 19 sections, 11 theorems, 12 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Organization
  4. Subgraph graph neural networks and the Weisfeiler-Lehman test
  5. Motivation
  6. $k$-hop subgraph GNNs
  7. $k$-hop subgraph WL test and the equivalent separation power
  8. Expressive power of $k$-hop subgraph GNNs
  9. $1$-hop subgraph GNNs
  10. $k$-hop subgraph GNNs with $k\geq 2$
  11. An extension to $k$-hop GNNs
  12. Numerical Experiments
  13. Experimental Setting and Dataset
  14. $k$-hop Graphormer Implementation Details
  15. Results and Discussions
  16. ...and 4 more sections

Key Result

Theorem 2.5

For any $(G,H),(\hat{G},\hat{H})\in\mathcal{G}_{n,m}$ and any $k>0$, the following are equivalent: Moreover, $(G,H)\stackrel{k,v}{\sim}(\hat{G},\hat{H})$ if and only if $F_v(G,H)=F_v(\hat{G},\hat{H})$ for any $F_v\in\mathcal{F}_{k,v}$.

Figures (6)

  • Figure 1: Two non-isomorphic graphs that cannot be distinguished by MP-GNNs or the WL test.
  • Figure 2: $2$-hop subgraphs rooted at $v_1$ for graphs in Figure \ref{['fig:not_iso_WL']}
  • Figure 3: Two non-isomorphic $3$-separable graphs indistinguishable by the classic WL test, but distinguishable by the $3$-hop subgraph WL test.
  • Figure 4: The $k$-strong separability assumption is necessary in \ref{['thm2:GNN-khop']}
  • Figure 5: Statistics on the longest cycle length across all testing molecules in the ZINC dataset indicate that most molecules have a longest cycle of 6, aligning with the chemical intuition that 6-membered rings are particularly stable. The peaks observed at 9 and 10 also support the prevalence of common fused ring systems.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Definition 2.1: Space of graphs with vertex features
  • Definition 2.2: Spaces of $k$-hop subgraph GNNs
  • Definition 2.3: Permutation-invariant and permutation-equivariant functions
  • Definition 2.4
  • Theorem 2.5
  • Corollary 2.6
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Definition 3.3
  • ...and 18 more