Synchronization of interconnected heterogeneous networks: The role of network sizes

Huixin Zhang,Weidong Zhang,Jianxi Gao

doi:10.1038/s41598-019-42636-6

Huixin Zhang, Weidong Zhang + Show 1 more

Open Access

https://doi.org/10.1038/s41598-019-42636-6

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Abstract

Increasing evidence shows that real networks interact with each other, forming a network of networks (NONs). Synchronization, a ubiquitous process in natural and engineering systems, has fascinatingly gained rising attentions in the context of NONs. Despite efforts to study the synchronization of NONs, it is still a challenge to understand how do the network sizes affect the synchronization and its phase diagram of NONs coupled with nonlinear dynamics. Here, we model such NONs as star-like motifs to analytically derive the critical values of both the internal and the external coupling strengths, at which a phase transition from synchronization to incoherence occurs. Our results show that the critical values strongly depend on the network sizes. Reducing the difference between network sizes will enhance the synchronization of the whole system, which indicates the irrationality of previous studies that assume the network sizes to be the same. The optimal connection strategy also changes as the network sizes change, a discovery contradicting to the previous conclusion that connecting the high-degree nodes of each network is always the most effective strategy to achieve synchronization unchangeably. This finding emphasizes the crucial role of network sizes which has been neglected in the previous studies and could contribute to the design of a global synchronized system.

Highlights

The present high attention to network science roots in its capability of explaining the behavior of complex real systems, including Internet[1,2], sociology[3], biology[4], transportation[5], and power grids[6]
We propose a mathematical framework to study the synchronization of network of networks (NONs), with nonlinear dynamics and fast-changing network sizes
We mainly focus on the case of a NON composed of two networks, which can be extended to the case of multiple networks by introducing new network layers and the connections between the networks

Summary

Introduction

The present high attention to network science roots in its capability of explaining the behavior of complex real systems, including Internet[1,2], sociology[3], biology[4], transportation[5], and power grids[6]. Three key factors dominate the synchronization of NONs: the connection strategy, i.e. the linkage between different networks; the dynamics, i.e. the universally nonlinear behavior in the real world; and the structure, i.e. the connections between a certain amount of nodes. While research on these three factors have led to some important findings, some scientific gaps remain. The number of the generating units of the grid-connected microgrids is significantly different from that of the traditional power grid, and the former number might vary by time due to unstable resources Another example is the plant-animal mutualistic network[35]. Our results light up the important role of network sizes and may help to design a global synchronized system

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