Synchronization in interdependent networks

TL;DR

This paper investigates how interdependence between a 1D network and a Watts-Strogatz small-world network affects synchronization, revealing that weak coupling suppresses and strong coupling enhances synchronization, with complex reentrant behaviors.

Contribution

It provides an analytical and numerical study of synchronization in interdependent networks, highlighting the nonmonotonic effects of inter-network coupling strength on synchronization behavior.

Findings

Weak inter-network coupling suppresses synchronization in the 1D network.

Strong coupling enhances synchronization as partial synchronization increases.

Reentrant behavior observed in the intermediate coupling range.

Abstract

We explore the synchronization behavior in interdependent systems, where the one-dimensional (1D) network (the intranetwork coupling strength $J_{\rm I}$) is ferromagnetically intercoupled (the strength $J$) to the Watts-Strogatz (WS) small-world network (the intranetwork coupling strength $J_{\rm II}$). In the absence of the internetwork coupling ($J = 0$), the former network is well known not to exhibit the synchronized phase at any finite coupling strength, whereas the latter displays the mean-field transition. Through an analytic approach based on the mean-field approximation, it is found that for the weakly coupled 1D network ($J_{\rm I} \ll 1$) the increase of $J$ suppresses synchrony, because the nonsynchronized 1D network becomes a heavier burden for the synchronization process of the WS network. As the coupling in the 1D network becomes stronger, it is revealed by the renormalization group (RG) argument that the synchronization is enhanced as $J_{\rm I}$ is increased, implying that the more enhanced partial synchronization in the 1D network makes the burden lighter. Extensive numerical simulations confirm these expected behaviors, while exhibiting a reentrant behavior in the intermediate range of $J_{\rm I}$. The nonmonotonic change of the critical value of $J_{\rm II}$ is also compared with the result from the numerical RG calculation.