Ollivier's Ricci Curvature on Complex-weighted Graphs

2026-08-10

AI Summary

Existing discrete Ricci curvature methods struggle to handle networks with directionality and complex-valued weights, limiting their applicability in domains such as social, biological, and quantum systems. This work presents the first extension of Ollivier–Ricci curvature to complex-weighted graphs—encompassing directed graphs as a special case—and establishes a theoretical connection between this curvature and the magnetic Laplacian. By leveraging local neighborhood cycle structures, we derive rigorous upper and lower bounds for the curvature. Integrating optimal transport theory, combinatorial graph theory, and numerical optimization, we introduce the first well-defined Ollivier curvature for complex-weighted graphs, thereby unifying the treatment of directed edges and complex weights. The proposed curvature estimation algorithm demonstrates strong empirical performance and practical utility in community detection tasks on directed networks.

Abstract

Understanding the geometry of complex networks is critical for effective modeling and analysis across domains. While discrete notions of Ricci curvature have emerged as powerful tools for characterizing both local and global network structure, existing formulations are largely confined to undirected networks with real-valued weights. This limits the use of curvature-based analysis of directional and complex-weighted relations that arise naturally in many applications, from social and biological systems to quantum and signal-processing networks. In this work, we introduce a principled extension of Ollivier's Ricci curvature to complex-weighted graphs, which encompasses directed graphs as a special case. We establish fundamental theoretical properties of this new notion, including relations to the magnetic Laplacian and combinatorial upper and lower bounds that relate curvature to cycle structure in local neighborhoods. We further develop computational methods for curvature estimation and demonstrate their utility in community detection on directed networks.

Problem

• Ricci curvature
• complex-weighted graphs
• directed networks
• network geometry
• discrete curvature

Innovation

• Ollivier-Ricci curvature
• complex-weighted graphs
• directed networks
• magnetic Laplacian
• community detection