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

Martin Carrasco, Olga Zaghen, Erik Bekkers, Bastian Rieck

TL;DR

This work introduces graph homomorphism distortion (HD), a continuous, permutation-invariant distance on vertex-attributed graphs that addresses the binary expressivity limitations of WL tests and standard GNNs. The authors formalize d_{HD} as a (pseudo)-metric based on maximal attribute deviations under homomorphisms between graphs, and show that, under mild assumptions, it yields a complete graph embedding and can be approximated via sampling with an expectation-complete formulation. They prove theoretical properties and demonstrate empirical effectiveness, including full distinction of graphs in the BRE C benchmark and superior performance over prior homomorphism- and subgraph-count-based methods on ZINC-12k in learning setups. By enabling a continuous, characteristic-based representation of graphs, the approach offers a principled path toward richer graph characterizations and practical tools for graph representation learning.

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 \texttt{BREC} dataset with up to $4$-WL non-distinguishable graphs, and (2.) \emph{outperforms} previous methods inspired in homomorphisms under the \texttt{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.

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

TL;DR

This work introduces graph homomorphism distortion (HD), a continuous, permutation-invariant distance on vertex-attributed graphs that addresses the binary expressivity limitations of WL tests and standard GNNs. The authors formalize d_{HD} as a (pseudo)-metric based on maximal attribute deviations under homomorphisms between graphs, and show that, under mild assumptions, it yields a complete graph embedding and can be approximated via sampling with an expectation-complete formulation. They prove theoretical properties and demonstrate empirical effectiveness, including full distinction of graphs in the BRE C benchmark and superior performance over prior homomorphism- and subgraph-count-based methods on ZINC-12k in learning setups. By enabling a continuous, characteristic-based representation of graphs, the approach offers a principled path toward richer graph characterizations and practical tools for graph representation learning.

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 \texttt{BREC} dataset with up to -WL non-distinguishable graphs, and (2.) \emph{outperforms} previous methods inspired in homomorphisms under the \texttt{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.

Paper Structure

This paper contains 71 sections, 25 theorems, 51 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. General statement of interest.
  3. Problem statement and motivation
  4. Our approach and contribution.
  5. Outline.
  6. Background
  7. Graph theory
  8. Graphs.
  9. Morphisms on graphs.
  10. Subgraphs.
  11. Vertex degrees.
  12. Graph classes.
  13. Vertex attributed graphs.
  14. Graph Neural Networks
  15. Related works
  16. ...and 56 more sections

Key Result

proposition thmcounterproposition

The graph homomorphism distortion $d_{\mathrm{HD}}(G, G')$ is a pseudo-metric, i.e. the following properties hold for arbitrary vertex-attributed graphs $G$, $G'$, $G"$ that share the same attribute function co-domain

Figures (6)

  • Figure 1: Resulting distance matrices from $d_{\mathrm{DH}}(\cdot)$ using $\sigma = 10$ and $\gamma = 10$ on class $4$ of BREC containing $4$-vertex distance-regular graphs. The top row shows the evaluation on shortest paths \ref{['fig:sub_cycle_sp']} and \ref{['fig:sub_tree_sp']} and the bottom on random walks \ref{['fig:sub_cycle_rwpe']} and \ref{['fig:sub_tree_rwpe']}. Notice that cycles are not enough to distinguish this class even when using a "better" $f$.
  • Figure 2: Class $\mathrm{B}$ containing basic (up to non $1$-WL distinguishable) graphs. The distance matrices $d_{\mathrm{DH}}(\cdot)$ with $\sigma = 10$ and $\gamma = 10$ are shown. The top row shows the evaluation on SPEs \ref{['fig:basic_sub_cycle_sp']} and \ref{['fig:basic_sub_tree_sp']} and the bottom on RWPEs \ref{['fig:basic_sub_cycle_rwpe']} and \ref{['fig:basic_sub_tree_rwpe']}.
  • Figure 3: Class $\mathrm{C}$ containing CFI graphs. The distance matrices $d_{\mathrm{DH}}(\cdot)$ with parameters $\sigma = 10$ and $\gamma = 10$ are shown. The top row shows the evaluation on SPEs \ref{['fig:cfi_sub_cycle_sp']} and \ref{['fig:sub_tree_sp']} and the bottom on RWPEs \ref{['fig:cfi_sub_cycle_rwpe']} and \ref{['fig:cfi_sub_tree_rwpe']}.
  • Figure 4: Class $\mathrm{E}$ containing extension graphs. The distance matrices $d_{\mathrm{DH}}(\cdot)$ with parameters $\sigma = 10$ and $\gamma = 10$ are shown. The top row shows the evaluation on SPEs \ref{['fig:extension_sub_cycle_sp']} and \ref{['fig:extension_sub_tree_sp']} and the bottom on RWPEs \ref{['fig:extension_sub_cycle_rwpe']} and \ref{['fig:extension_sub_tree_rwpe']}.
  • Figure 5: Class $\mathrm{R}$ containing regular graphs. The distance matrices $d_{\mathrm{DH}}(\cdot)$ with parameters $\sigma = 10$ and $\gamma = 10$ are shown. The top row shows the evaluation on SPEs \ref{['fig:regular_sub_cycle_sp']} and \ref{['fig:regular_sub_tree_sp']} and the bottom on RWPEs \ref{['fig:regular_sub_cycle_rwpe']} and \ref{['fig:regular_sub_tree_rwpe']}.
  • ...and 1 more figures

Theorems & Definitions (53)

  • definition thmcounterdefinition: Graph homomorphism
  • definition thmcounterdefinition: Graph Isomorphism
  • definition thmcounterdefinition: Graph property
  • definition thmcounterdefinition: Graph invariant
  • definition thmcounterdefinition: Tree decomposition
  • definition thmcounterdefinition
  • definition thmcounterdefinition: Homomorphism distortion
  • definition thmcounterdefinition: Attribute distance
  • definition thmcounterdefinition: Pseudo-metric
  • proposition thmcounterproposition
  • ...and 43 more