Structure of complex networks: Quantifying edge-to-edge relations by failure-induced flow redistribution

TL;DR

This paper introduces graph-theoretical measures based on flow redistribution after edge failures to analyze the dynamic relationships between edges in complex networks, applicable across various real-world systems.

Contribution

It presents a novel edge-centric framework using the pseudo-inverse of the Laplacian to quantify edge-to-edge relations and embeddedness in complex networks.

Findings

Reveals non-local edge interactions through flow redistribution measures.

Applies framework successfully to power grids, traffic, and neuronal networks.

Provides insights into edge importance and network robustness.

Abstract

The analysis of complex networks has so far revolved mainly around the role of nodes and communities of nodes. However, the dynamics of interconnected systems is commonly focalised on edge processes, and a dual edge-centric perspective can often prove more natural. Here we present graph-theoretical measures to quantify edge-to-edge relations inspired by the notion of flow redistribution induced by edge failures. Our measures, which are related to the pseudo-inverse of the Laplacian of the network, are global and reveal the dynamical interplay between the edges of a network, including potentially non-local interactions. Our framework also allows us to define the embeddedness of an edge, a measure of how strongly an edge features in the weighted cuts of the network. We showcase the general applicability of our edge-centric framework through analyses of the Iberian Power grid, traffic flow in road networks, and the C. elegans neuronal network.