  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.
