A Centrality Measure Using Magnitude Homology

2026-07-17

AI Summary

This study addresses the problem of effectively measuring node centrality in graphs from geometric and topological perspectives. To this end, it introduces magnitude homology—a novel application in graph centrality analysis—and proposes a local centrality measure grounded in relative homology: the importance of a node is quantified by the change in magnitude homology resulting from its removal. The proposed measure satisfies several natural axioms, exhibits favorable theoretical properties, and demonstrates unique effectiveness in experiments, offering complementary insights to classical centrality metrics. This work thus provides a new topological lens for evaluating node importance in complex networks.

Abstract

The magnitude of a metric space constitutes an expressive invariant that subsumes numerous different geometrical-topological invariants. Building on recent advances in magnitude homology, i.e., a bigraded homology theory that recovers the magnitude, we develop a novel local measure of the centrality or importance of nodes in a graph. Our measure is inspired by the concept of relative homology as it considers the change in magnitude homology when removing a vertex. We show that our proposed measure satisfies several properties a centrality measure is reasonably expected to respect and demonstrate that we introduce a new perspective on centrality by comparing to several established centrality measures.

Problem

• centrality
• magnitude homology
• graph
• node importance
• metric space

Innovation

• magnitude homology
• centrality measure
• relative homology
• metric space
• graph topology