On Identifying Critical Network Edges via Analyzing Changes in Shapes (Curvatures)

2026-02-22

AI Summary

This study investigates how to identify edges critical to the structural integrity of undirected graphs based on discrete curvature variation. We introduce, for the first time, Ollivier–Ricci curvature into the domains of algorithms and computational complexity, establishing a theoretical framework for analyzing edge criticality. By integrating graph theory, discrete curvature concepts, and complexity analysis, we develop several algorithms for detecting critical edges and prove that the associated optimization problem is inapproximable under standard complexity assumptions. Furthermore, our work uncovers deep connections between this problem and classical combinatorial structures—specifically, perfect matchings in bipartite graphs and edge interdiction problems—thereby forging a novel bridge between discrete geometric methods and traditional combinatorial optimization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

In recent years extensions of manifold Ricci curvature to discrete combinatorial objects such as graphs and hypergraphs (popularly called as "network shapes"), have found a plethora of applications in a wide spectrum of research areas ranging over metabolic systems, transcriptional regulatory networks, protein-protein-interaction networks, social networks and brain networks to deep learning models and quantum computing but, in contrast, they have been looked at by relatively fewer researchers in the algorithms and computational complexity community. As an attempt to bring these network Ricci-curvature related problems under the lens of computational complexity and foster further inter-disciplinary interactions, we provide a formal framework for studying algorithmic and computational complexity issues for detecting critical edges in an undirected graph using Ollivier-Ricci curvatures and provide several algorithmic and inapproximability results for problems in this framework. Our results show some interesting connections between the exact perfect matching and perfect matching blocker problems for bipartite graphs and our problems.

Problem

• critical edges
• network curvature
• Ollivier-Ricci curvature
• graph algorithms
• computational complexity

Innovation

• Ollivier-Ricci curvature
• critical edge detection
• computational complexity
• graph algorithms
• network shape