Communicability Angles Reveal Critical Edges for Network Consensus Dynamics

TL;DR

This paper investigates how the removal of certain edges, identified by communicability angles, significantly delays consensus in networks, revealing key structural influences on dynamical processes.

Contribution

It introduces the use of communicability angles to identify critical edges affecting consensus time and provides theoretical justification linking network topology to dynamics.

Findings

Removing edges with large communicability angles increases consensus time by 5.68 times.

Edges with high betweenness centrality also significantly delay consensus, by 3.70 times.

Network density and average distance-sum explain over 80% of the variance in consensus time.

Abstract

We consider the question of determining how the topological structure influences a consensus dynamical process taking place on a network. By considering a large dataset of real-world networks we first determine that the removal of edges according to their communicability angle -an angle between position vectors of the nodes in an Euclidean communicability space- increases the average time of consensus by a factor of 5.68 in real-world networks. The edge betweenness centrality also identifies -in a smaller proportion- those critical edges for the consensus dynamics, i.e., its removal increases the time of consensus by a factor of 3.70. We justify theoretically these findings on the basis of the role played by the algebraic connectivity and the isoperimetric number of networks on the dynamical process studied, and their connections with the properties mentioned before. Finally, we study the role played by global topological parameters of networks on the consensus dynamics. We determine that the network density and the average distance-sum -an analogous of the node degree for shortest-path distances, account for more than 80% of the variance of the average time of consensus in the real-world networks studied.