Weisfeiler and Leman Go Infinite: Spectral and Combinatorial Pre-Colorings

TL;DR

This paper demonstrates that by relaxing uniform pre-coloring and using spectral features, the Weisfeiler-Leman test's expressive power can be increased infinitely, improving graph isomorphism testing and GNN capabilities.

Contribution

It introduces a spectral pre-coloring method that enhances the WL test's expressive power beyond traditional limits, with theoretical and experimental validation.

Findings

Spectral pre-coloring increases WL test's expressive power.

The proposed method outperforms standard WL in experiments.

Code and experiments are publicly available.

Abstract

Graph isomorphism testing is usually approached via the comparison of graph invariants. Two popular alternatives that offer a good trade-off between expressive power and computational efficiency are combinatorial (i.e., obtained via the Weisfeiler-Leman (WL) test) and spectral invariants. While the exact power of the latter is still an open question, the former is regularly criticized for its limited power, when a standard configuration of uniform pre-coloring is used. This drawback hinders the applicability of Message Passing Graph Neural Networks (MPGNNs), whose expressive power is upper bounded by the WL test. Relaxing the assumption of uniform pre-coloring, we show that one can increase the expressive power of the WL test ad infinitum. Following that, we propose an efficient pre-coloring based on spectral features that provably increase the expressive power of the vanilla WL test. The above claims are accompanied by extensive synthetic and real data experiments. The code to reproduce our experiments is available at https://github.com/TPFI22/Spectral-and-Combinatorial