Iterative Ricci-Foster Curvature Flow with GMM-Based Edge Pruning: A Novel Approach to Community Detection

TL;DR

This work introduces a novel community detection method, based on Ricci flow on graphs, that robustly recovers the planted community structure of SBM-s, establishing Ricci-Foster Flow with GMM-clustering as a principled and computationally effective new tool for network analysis.

Abstract

Community detection in complex networks is a fundamental problem, open to new approaches in various scientific settings. We introduce a novel community detection method, based on Ricci flow on graphs. Our technique iteratively updates edge weights (their metric lengths) according to their (combinatorial) Foster version of Ricci curvature computed from effective resistance distance between the nodes. The latter computation is known to be done by pseudo-inverting the graph Laplacian matrix. At that, our approach is alternative to one based on Ollivier-Ricci geometric flow for community detection on graphs, significantly outperforming it in terms of computation time. In our proposed method, iterations of Foster-Ricci flow that highlight network regions of different curvature -- are followed by a Gaussian Mixture Model (GMM) separation heuristic. That allows to classify edges into ''strong'' (intra-community) and ''weak'' (inter-community) groups, followed by a systematic pruning of the former to isolate communities. We benchmark our algorithm on synthetic networks generated from the Stochastic Block Model (SBM), evaluating performance with the Adjusted Rand Index (ARI). Our results demonstrate that proposed framework robustly recovers the planted community structure of SBM-s, establishing Ricci-Foster Flow with GMM-clustering as a principled and computationally effective new tool for network analysis, tested against alternative Ricci-Ollivier flow coupled with spectral clustering.

Figures (1)

• Figure 1: A. Top: Graphs shown before and after pruning edges with the highest weights. Bottom: Distributions of edge weights following iterative Ricci–Foster flow. A Gaussian Mixture Model (GMM) separates the weights into two distinct distributions; all edges corresponding to the rightmost peak are then removed to produce the upper right graph (t-test, p-value$<4.62e-62$) B. Algorithmic performance comparison: Evaluation of the computational efficiency of three clustering approaches—Ricci–Ollivier flow–based clustering, Ricci–Foster flow–based clustering, and spectral clustering.