On Graph-Informed Distance Metrics for Comparing Graph Partitions

2026-07-30

AI Summary

Existing methods for comparing graph partitions often disregard the underlying graph topology, making it difficult to accurately capture the cohesion within and separation between communities. This work proposes a graph-aware distance framework that constructs a topology-respecting metric by inducing edge partitions, enabling meaningful comparison of continuous graph partitions. The framework adheres to a local graph-aware refinement criterion and is theoretically shown—under the stochastic block model—to be sensitive to topological perturbations. Specifically, the proposed distance almost surely increases with stronger perturbations in both intra- and inter-community splitting scenarios, significantly outperforming conventional metrics such as variation of information, van Dongen distance, and binary cut distance. This provides a structurally consistent and topology-sensitive criterion for evaluating graph partitions.

Abstract

Comparing graph partitions is fundamental to the analysis of network-structured data, yet existing measures for comparing graph partitions typically rely on graph-agnostic indices that treat vertices as exchangeable, ignoring the underlying graph topology that encodes essential information about community cohesion and separation. We propose a general construction of graph-informed distances that compares vertex partitions through induced edge partitions and yields valid metrics on the space of contiguous graph partitions. As special cases, we develop graph-informed versions of variation of information and the van Dongen distance together with a binary cut-based companion distance, and show that these distances satisfy a natural local graph-aware refinement criterion. Under stochastic block models, we prove that stronger topological disruptions incur asymptotically larger distances almost surely in both inter-community and intra-community split settings. These results provide a simple and principled framework to compare graph partitions while respecting the underlying graph structure.

Problem

• graph partitions
• graph topology
• distance metrics
• community structure
• graph-aware comparison

Innovation

• graph-informed distance
• graph partition comparison
• induced edge partition
• variation of information
• stochastic block model