Edge-based analysis of networks: Curvatures of graphs and hypergraphs

TL;DR

This paper introduces a network analysis approach based on edge and hyperedge curvatures to quantify local structural properties, enabling insights into biological networks like protein interactions and metabolic pathways.

Contribution

It develops a systematic edge-based framework using network curvatures for analyzing complex networks, focusing on relations rather than elements.

Findings

Effective characterization of local network structures.

Application to biological networks demonstrates practical utility.

Provides new tools for analyzing directed and hypergraph networks.

Abstract

The relations, rather than the elements, constitute the structure of networks. We therefore develop a systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is based on structural properties of (hyper)edges, instead of vertices. For that purpose, we utilize so-called network curvatures. These curvatures quantify the local structural properties of (hyper)edges, that is, how, and how well, they are connected to others. In the case of directed networks, they assess the input they receive and the output they produce, and relations between them. With those tools, we can investigate biological networks. As examples, we apply our methods here to protein-protein interaction, transcriptional regulatory and metabolic networks.