Weisfeiler and Leman Go Infinite: Spectral and Combinatorial Pre-Colorings
This work addresses a bottleneck in graph isomorphism testing and MPGNNs, offering a novel approach to improve expressive power, though it appears incremental as it builds on existing WL test frameworks.
The paper tackles the limited expressive power of the Weisfeiler-Leman (WL) test and Message Passing Graph Neural Networks (MPGNNs) by relaxing the assumption of uniform pre-coloring, showing that expressive power can be increased ad infinitum, and proposes an efficient spectral-based pre-coloring method that provably enhances the vanilla WL test, supported by extensive synthetic and real data experiments.
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