Motif-based filtrations for persistent homology: A framework for graph isomorphism and property prediction

Abstract

Determining whether two graphs are isomorphic is a fundamental problem with practical applications in areas such as molecular chemistry or social network analysis, yet it remains a challenging task, with exact solutions often being computationally expensive. We address this task using persistent homology built on motif-based filtrations of graphs, a method from topological data analysis that summarizes the shape of data by tracking the persistence of structural features along filtrations. Specifically, we use edge-weighting schemes based on the densities of triangles, chordless squares, and chordless pentagons, which have been shown to be effective for detecting network dimensionality. Our cycle-density filtrations distinguish non-isomorphic graphs perfectly or nearly perfectly across four demanding graph families, many of which exhibit symmetries. We outperform curvature-based, degree-based, and Vietoris--Rips filtrations, and match or exceed the accuracy of egonet-distance methods while incurring a lower computational cost. The expressive power of our filtrations goes beyond isomorphism testing: because they capture rich structural information from graphs, they consistently achieve top performance on property prediction tasks using real-world data, and exhibit high sensitivity to edge rewiring and removal. Together, these findings establish cycle-density filtrations as an effective and computationally tractable framework for graph comparison and characterization, bridging topological data analysis and network science.