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

TL;DR

This paper introduces the CCWL test and CCIN model, unifying topological neural networks and Weisfeiler-Lehman variants to enhance the expressive power of topological deep learning on combinatorial complexes.

Contribution

It develops a unified theoretical framework for topological neural networks using the CCWL test and proposes the CCIN model, demonstrating improved performance on benchmarks.

Findings

CCIN outperforms baseline methods on benchmarks.

Upper and lower neighborhoods suffice for full CCWL expressivity.

The framework unifies various topological neural network approaches.

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.