Weisfeiler and Lehman Go Cellular: CW Networks

TL;DR

This paper introduces CW Networks, a new class of graph neural networks based on regular cell complexes, which extend simplicial complexes to improve expressivity and modeling of higher-order structures, achieving state-of-the-art results.

Contribution

The paper extends theoretical results from simplicial complexes to regular cell complexes, creating a flexible, hierarchical message passing framework called CW Networks that surpasses traditional GNNs in expressivity.

Findings

CW Networks are more powerful than the WL test.

CW Networks achieve state-of-the-art results on molecular datasets.

The architecture models higher-order signals and compresses node distances.

Abstract

Graph Neural Networks (GNNs) are limited in their expressive power, struggle with long-range interactions and lack a principled way to model higher-order structures. These problems can be attributed to the strong coupling between the computational graph and the input graph structure. The recently proposed Message Passing Simplicial Networks naturally decouple these elements by performing message passing on the clique complex of the graph. Nevertheless, these models can be severely constrained by the rigid combinatorial structure of Simplicial Complexes (SCs). In this work, we extend recent theoretical results on SCs to regular Cell Complexes, topological objects that flexibly subsume SCs and graphs. We show that this generalisation provides a powerful set of graph "lifting" transformations, each leading to a unique hierarchical message passing procedure. The resulting methods, which we collectively call CW Networks (CWNs), are strictly more powerful than the WL test and not less powerful than the 3-WL test. In particular, we demonstrate the effectiveness of one such scheme, based on rings, when applied to molecular graph problems. The proposed architecture benefits from provably larger expressivity than commonly used GNNs, principled modelling of higher-order signals and from compressing the distances between nodes. We demonstrate that our model achieves state-of-the-art results on a variety of molecular datasets.