A modified Ricci flow on arbitrary weighted graph

TL;DR

This paper introduces a modified Ricci flow for weighted graphs, demonstrating its effectiveness in community detection tasks without the need for iterative surgery, outperforming existing methods.

Contribution

It proposes a new modified Ricci flow on arbitrary weighted graphs with proven global existence and uniqueness, applied successfully to community detection without iterative surgery.

Findings

Outperforms existing community detection algorithms in ARI, NMI, and Q metrics.

Requires only one surgery at the end of the process, simplifying the algorithm.

Effective on real-world and artificial networks.

Abstract

In this paper, we propose a modified Ricci flow, as well as a quasi-normalized Ricci flow, on arbitrary weighted graph. Each of these two flows has a unique global solution. In particular, these global existence and uniqueness results do not require an exit condition proposed by Bai et al in a recent work [2]. As applications, these two Ricci flows are applied to community detection for complex networks, including Karate Club, American football games, Facebook, as well as artificial networks. In our algorithms, unlike in [5,15], there is no need to perform surgery at every iteration, only one surgery needs to be performed after the last iteration. From three commonly used criteria for evaluating community detection algorithms, ARI, NMI and Q, we conclude that our algorithms outperform existing algorithms, including Ollivier's Ricci flow [5], normalized Ollivier's Ricci flow and normalized Lin-Lu-Yau's Ricci flow [15]. The codes for our algorithms are available at https://github.com/mjc191812/Modified-Ricci-Flow.