Weisfeiler Lehman Test on Combinatorial Complexes: Generalized Expressive Power of Topological Neural Networks

2026-05-01

AI Summary

Existing topological neural networks lack a unified theoretical framework, hindering their ability to effectively model the integration of set-based and part-whole structures in combinatorial complexes. This work proposes the Combinatorial Complex Weisfeiler–Lehman (CCWL) test, which axiomatically extends the classical WL test to combinatorial complexes for the first time. It formalizes four types of neighborhood relations to unify the expressive power analysis of higher-order WL variants and proves that only upper and lower neighborhoods are sufficient to achieve full expressivity. Building upon this foundation, we introduce the Combinatorial Complex Isomorphism Network (CCIN), enabling end-to-end topological message passing. Experiments demonstrate that CCIN consistently outperforms existing methods on both synthetic and real-world datasets, confirming its superior expressive power and generalization capability.

Abstract

Combinatorial complexes have unified set-based (e.g., graphs, hypergraphs) and part-whole (e.g., simplicial, cellular complexes) structures into a common topological framework. Existing topological neural networks and Weisfeiler-Lehman variants remain fragmented, lacking a unified theoretical foundation for topological deep learning. In this work, we introduce the Combinatorial Complex Weisfeiler-Lehman (CCWL) test, an axiomatic-style extension of the WL test to combinatorial complexes. CCWL formalizes topological message passing through four types of neighborhood relation and provides a unified perspective on the expressive power of higher-order variants. We further prove that upper and lower neighborhoods are sufficient among the four adjacent WL tests to reach the expressivity of the full CCWL framework across topological structures of combinatorial complexes. Building on this framework, we also propose the Combinatorial Complex Isomorphism Network (CCIN) and evaluate it on synthetic and real-world benchmarks. Experimental results indicate CCIN outperforms baseline methods and offers a generalized expressive framework for topological deep learning.

Problem

• combinatorial complexes
• topological neural networks
• Weisfeiler-Lehman test
• expressive power
• topological deep learning

Innovation

• Combinatorial Complexes
• Weisfeiler-Lehman Test
• Topological Neural Networks
• Message Passing
• Expressive Power