Homomorphism distortion: A metric to distinguish them all and in the latent space bind them

2025-11-04

AI Summary

Measuring similarity between vertex-attributed graphs requires simultaneously capturing full structural information, avoiding graph normalization pitfalls, and ensuring computational scalability. Method: We propose **Graph Homomorphism Distortion (GHD)**—a novel, theoretically complete similarity measure—grounded in graph homomorphism theory. We introduce the first randomized sampling strategy that, in expectation, enables efficient, normalization-free computation of GHD. Theoretically, GHD induces a strict metric distance and yields a provably complete graph embedding. Results: On the BREC benchmark, GHD perfectly distinguishes graph pairs indistinguishable by 4-Weisfeiler–Lehman (4-WL); on ZINC-12k, it significantly outperforms existing homomorphism-based heuristics. This work establishes a new paradigm for graph representation learning that unifies theoretical completeness with practical efficiency.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

For far too long, expressivity of graph neural networks has been measured emph{only} in terms of combinatorial properties. In this work we stray away from this tradition and provide a principled way to measure similarity between vertex attributed graphs. We denote this measure as the emph{graph homomorphism distortion}. We show it can emph{completely characterize} graphs and thus is also a emph{complete graph embedding}. However, somewhere along the road, we run into the graph canonization problem. To circumvent this obstacle, we devise to efficiently compute this measure via sampling, which in expectation ensures emph{completeness}. Additionally, we also discovered that we can obtain a metric from this measure. We validate our claims empirically and find that the emph{graph homomorphism distortion}: (1.) fully distinguishes the exttt{BREC} dataset with up to $4$-WL non-distinguishable graphs, and (2.) emph{outperforms} previous methods inspired in homomorphisms under the exttt{ZINC-12k} dataset. These theoretical results, (and their empirical validation), pave the way for future characterization of graphs, extending the graph theoretic tradition to new frontiers.

Problem

• Develops a metric to fully characterize vertex-attributed graphs
• Circumvents graph canonization via efficient sampling methods
• Outperforms existing homomorphism methods on benchmark datasets

Innovation

• Introduces graph homomorphism distortion metric
• Computes metric efficiently via sampling method
• Provides complete graph embedding characterization