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

TL;DR

This paper explores the computational complexity of identifying critical edges in networks using Ricci curvature, providing algorithms and inapproximability results to advance interdisciplinary research.

Contribution

It introduces a formal framework for analyzing algorithmic complexity of critical edge detection via Ricci curvature in graphs, linking to classical combinatorial problems.

Findings

Established connections between Ricci curvature-based problems and perfect matching problems.

Provided algorithms and inapproximability results for detecting critical edges.

Linked network curvature analysis to well-known combinatorial packing and covering problems.

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 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 our problems, the exact perfect matching and perfect matching blocker problems for bipartite graphs and two well-known combinatorial packing/covering problems.