Structural and functional networks in complex systems with delay

TL;DR

This paper derives exact relationships between the topology and functional dynamics of delay-coupled complex networks, revealing how structural properties influence activity patterns and synchronization.

Contribution

It provides exact solutions for the equations linking network structure and function in motifs and directed networks, including a mean-field approximation for uncorrelated networks.

Findings

Clusterization depends on node in-degree

Locking frequency decreases with average degree

Power-law exponents of structural and functional networks are related

Abstract

Functional networks of complex systems are obtained from the analysis of the temporal activity of their components, and are often used to infer their unknown underlying connectivity. We obtain the equations relating topology and function in a system of diffusively delay-coupled elements in complex networks. We solve exactly the resulting equations in motifs (directed structures of three nodes), and in directed networks. The mean-field solution for directed uncorrelated networks shows that the clusterization of the activity is dominated by the in-degree of the nodes, and that the locking frequency decreases with increasing average degree. We find that the exponent of a power law degree distribution of the structural topology, b, is related to the exponent of the associated functional network as a =1/(2-b), for b < 2.