On Graph-Informed Distance Metrics for Comparing Graph Partitions

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.