Ricci Matrix Comparison for Graph Alignment: A DMC Variation

TL;DR

This paper introduces Ricci Matrix Comparison, a geometric approach to graph alignment using Ricci curvature, demonstrating high accuracy in complex network and torus alignment tasks.

Contribution

It proposes a novel Ricci-curvature-based graph alignment method, extending the Degree Matrix Comparison approach with theoretical and experimental validation.

Findings

Achieved 80-90+% accuracy in complex network alignment

Demonstrated effectiveness in identifying holes in tori

Validated the potential of differential geometry in graph alignment

Abstract

The graph alignment problem explores the concept of node correspondence and its optimality. In this paper, we focus on purely geometric graph alignment methods, namely our newly proposed Ricci Matrix Comparison (RMC) and its original form, Degree Matrix Comparison (DMC). To formulate a Ricci-curvature-based graph alignment situation, we start with discussing different ideas of constructing one of the most typical and important topological objects, the torus, and then move on to introducing the RMC based on DMC with theoretical motivations. Lastly, we will present to the reader experimental results on a torus and a complex protein-protein interaction network that indicate the potential of applying a differential-geometric view to graph alignment. Results show that a direct variation of DMC using Ricci curvature can help with identifying holes in tori and aligning line graphs of a complex network at 80-90+% accuracy. This paper contributes a new perspective to the field of graph alignment and partially shows the validity of the previous DMC method.