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Ricci Matrix Comparison for Graph Alignment: A DMC Variation

Ashley Wang, Peter Chin

TL;DR

This work studies curvature-based graph alignment by introducing Ricci Matrix Comparison (RMC), a curvature-based variant of the existing Degree Matrix Comparison (DMC). RMC substitutes neighbor-degrees with Forman-Ricci curvatures $Ric(e)$ of adjacent edges and uses the Hungarian algorithm to compute node correspondences, supported by a theoretical link to the graph Laplacian $L = D - A$ via a Curvature-Laplacian relation. Empirical results on discretized tori and the line graph of a PPI network demonstrate that RMC can recover geometric identities and achieve high accuracy (80-90% range), illustrating the potential of a differential-geometric view for graph alignment. The work lays groundwork for broader validation and theoretical guarantees for curvature-based alignment and suggests future work on diverse networks and deeper connections to topological invariants such as the Euler characteristic.

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.

Ricci Matrix Comparison for Graph Alignment: A DMC Variation

TL;DR

This work studies curvature-based graph alignment by introducing Ricci Matrix Comparison (RMC), a curvature-based variant of the existing Degree Matrix Comparison (DMC). RMC substitutes neighbor-degrees with Forman-Ricci curvatures of adjacent edges and uses the Hungarian algorithm to compute node correspondences, supported by a theoretical link to the graph Laplacian via a Curvature-Laplacian relation. Empirical results on discretized tori and the line graph of a PPI network demonstrate that RMC can recover geometric identities and achieve high accuracy (80-90% range), illustrating the potential of a differential-geometric view for graph alignment. The work lays groundwork for broader validation and theoretical guarantees for curvature-based alignment and suggests future work on diverse networks and deeper connections to topological invariants such as the Euler characteristic.

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.

Paper Structure

This paper contains 25 sections, 23 equations, 8 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Forman-Ricci Curvature
  3. Tiling Methods to Construct a Torus
  4. Two-Dimensional Tiling
  5. Three-Dimensional Tiling
  6. Ricci Matrix Comparison
  7. Degree Matrix Comparison
  8. Definition 4.1
  9. Ricci Matrix Comparison
  10. Example 4.1
  11. Theoretical Motivation
  12. Graph Laplacian
  13. Variation of Signature Row Vector from a Degree Matrix
  14. The "Curvature-Laplacian" Equation
  15. Lemma 5.1
  16. ...and 10 more sections

Figures (8)

  • Figure 1: Equilateral triangular tiling. Blue hexagons indicate triangles that form a hexagonal ring.
  • Figure 2: Square tiling for an approximate ring.
  • Figure 3: Triangulating a prism within three-dimensional torus.
  • Figure 4: Example graph for demonstrating Ricci row vector construction.
  • Figure 5: Illustration of one-dimensional graph.
  • ...and 3 more figures

Theorems & Definitions (1)

  • proof