Inter-layer synchronization in non-identical multi-layer networks

TL;DR

This paper develops an approximate analytical framework to understand inter-layer synchronization in non-identical multi-layer networks, validated through numerical simulations and experiments with electronic circuits, revealing the impact of structural differences.

Contribution

It introduces an inertial term to model structural differences in multi-layer networks and validates the approach with numerical and experimental results.

Findings

Approximate analytical model captures effects of structural differences.

Betweenness centrality of missing links influences synchronization.

Experimental validation with electronic circuits confirms theoretical predictions.

Abstract

Inter-layer synchronization is a dynamical state occurring in multi-layer networks composed of identical nodes. The state corresponds to have all layers synchronized, with nodes in each layer which do not necessarily evolve in unison. So far, the study of such a solution has been restricted to the case in which all layers had an identical connectivity structure. When layers are not identical, the inter-layer synchronous state is no longer a stable solution of the system. Nevertheless, when layers differ in just a few links, an approximate treatment is still feasible, and allows one to gather information on whether and how the system may wander around an inter-layer synchronous configuration. We report the details of an approximate analytical treatment for a two-layer multiplex, which results in the introduction of an extra inertial term accounting for structural differences. Numerical validation of the predictions highlights the usefulness of our approach, especially for small or moderate topological differences in the intra-layer coupling. Moreover, we identify a non-trivial relationship between the betweenness centrality of the missing links and the intra-layer coupling strength. Finally, by the use of two multiplexed identical layers of electronic circuits in a chaotic regime, we study the loss of inter-layer synchronization as a function of the betweenness centrality of the removed links.