Network Synchronization with Convexity

TL;DR

This paper derives new synchronization conditions for complex networks with nonlinear, nonidentical node dynamics and switching directed graphs, highlighting how convexity and linear couplings facilitate synchronization.

Contribution

It introduces integral convexity conditions on nonlinear node dynamics and establishes new synchronization criteria under relaxed connectivity assumptions.

Findings

Synchronization achieved under integral convexity of node dynamics

Conditions for exact and approximate synchronization

Relaxed connectivity requirements for network synchronization

Abstract

In this paper, we establish a few new synchronization conditions for complex networks with nonlinear and nonidentical self-dynamics with switching directed communication graphs. In light of the recent works on distributed sub-gradient methods, we impose integral convexity for the nonlinear node self-dynamics in the sense that the self-dynamics of a given node is the gradient of some concave function corresponding to that node. The node couplings are assumed to be linear but with switching directed communication graphs. Several sufficient and/or necessary conditions are established for exact or approximate synchronization over the considered complex networks. These results show when and how nonlinear node self-dynamics may cooperate with the linear diffusive coupling, which eventually leads to network synchronization conditions under relaxed connectivity requirements.