Core detection via Ricci curvature flows on weighted graphs

TL;DR

This paper introduces a Ricci curvature flow-based method for core detection in graphs, providing bounds on weights and outperforming traditional centrality measures, with practical applications in network analysis.

Contribution

It derives bounds for weights in Ricci curvature flows and demonstrates a novel, more effective core detection method compared to existing approaches.

Findings

Weights remain bounded during Ricci curvature flows, preventing overflow or zeroing.

The Ricci flow method outperforms PageRank and centrality measures in core detection.

Codes are available at https://github.com/12tangze12/core-detection-via-Ricci-flow.

Abstract

Graph Ricci curvature is crucial as it geometrically quantifies network structure. It pinpoints bottlenecks via negative curvature, identifies cohesive communities with positive curvature, and highlights robust hubs. This guides network analysis, resilience assessment, flow optimization, and effective algorithm design. In this paper, we derived upper and lower bounds for the weights along several kinds of discrete Ricci curvature flows. As an application, we utilized discrete Ricci curvature flows to detect the core subgraph of a finite undirected graph. The novelty of this work has two aspects. Firstly, along the Ricci curvature flow, the bounds for weights determine the minimum number of iterations required to ensure weights remain between two prescribed positive constants. In particular, for any fixed graph, we conclude weights can not overflow and can not be treated as zero, as long as the iteration does not exceed a certain number of times; Secondly, it demonstrates that our Ricci curvature flow method for identifying core subgraphs outperforms prior approaches, such as page rank, degree centrality, betweenness centrality and closeness centrality. The codes for our algorithms are available at https://github.com/12tangze12/core-detection-via-Ricci-flow.