Graph Homomorphism Distortion: A Metric to Distinguish Them All and in the Latent Space Bind Them

TL;DR

This paper introduces a new graph homomorphism distortion metric that incorporates node features, enabling better assessment of graph similarity and enhancing the expressivity of graph neural networks.

Contribution

It proposes a novel pseudo-metric based on graph homomorphisms that considers features, complementing existing measures and improving neural network performance.

Findings

Efficient computation under certain assumptions.

Complementarity with 1-WL measure.

Structural encodings improve GNN predictive power.

Abstract

A large driver of the complexity of graph learning is the interplay between structure and features. When analyzing the expressivity of graph neural networks, however, existing approaches ignore features in favor of structure, making it nigh-impossible to assess to what extent two graphs with close features should be considered similar. We address this by developing a new (pseudo-)metric based on graph homomorphisms. Inspired by concepts from metric geometry, our graph homomorphism distortion measures the minimal worst-case distortion that node features of one graph are subjected to when mapping one graph to another. We demonstrate the utility of our novel measure by showing that (i.) it can be efficiently calculated under some additional assumptions, (ii.) it complements existing expressivity measures like $1$-WL, and (iii.) it permits defining structural encodings, which improve the predictive capabilities of graph neural networks.