Network Synchronization with Convexity

In this paper, we establish some 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 some 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 regarding exact or approximate synchronization over the considered complex networks. These results show when and how nonlinear node self-dynamics may cooperate with the linear consensus coupling, which eventually leads to network synchronization conditions under much relaxed connectivity requirements.