Topological Measure Locating the Effective Crossover between Segregation and Integration in a Modular Network

TL;DR

This paper introduces a new topological measure to identify the crossover point between segregation and integration in modular networks, linking it to dynamical synchronization phenomena and network functionality.

Contribution

It presents a novel, easily computable topological measure that correlates network structure with dynamical synchronization transitions and complexity.

Findings

The measure accurately locates the structural-dynamical crossover.

It correlates with a peak in a proposed dynamical complexity index.

The measure provides insights into network functionality and task performance.

Abstract

We introduce an easily computable topological measure which locates the effective crossover between segregation and integration in a modular network. Segregation corresponds to the degree of network modularity, while integration is expressed in terms of the algebraic connectivity of an associated hyper-graph. The rigorous treatment of the simplified case of cliques of equal size that are gradually rewired until they become completely merged, allows us to show that this topological crossover can be made to coincide with a dynamical crossover from cluster to global synchronization of a system of coupled phase oscillators. The dynamical crossover is signaled by a peak in the product of the measures of intra-cluster and global synchronization, which we propose as a dynamical measure of complexity. This quantity is much easier to compute than the entropy (of the average frequencies of the oscillators), and displays a behavior which closely mimics that of the dynamical complexity index based on the latter. The proposed toplogical measure simultaneously provides information on the dynamical behavior, sheds light on the interplay between modularity vs total integration and shows how this affects the capability of the network to perform both local and distributed dynamical tasks.