Convergence Analysis of Signed Nonlinear Networks

This paper analyzes the convergence properties of signed networks with nonlinear edge functions. We consider diffusively coupled networks comprised of maximal equilibrium-independent passive (MEIP) dynamics on the nodes, and a general class of nonlinear coupling functions on the edges. The first contribution of this paper is to generalize the classical notion of signed networks for graphs with scalar weights to graphs with nonlinear edge functions using notions from the passivity theory. We show that the output of the network can finally form one or several steady-state clusters if all edges are positive and, in particular, all nodes can reach an output agreement if there is a connected subnetwork spanning all nodes and strictly positive edges. When there are nonpositive edges added to the network, we show that the tension of the network still converges to the equilibria of the edge functions if the relative outputs of the nodes connected by nonpositive edges converge to their equilibria. Furthermore, we establish the equivalent circuit models for signed nonlinear networks, and define the concept of equivalent edge functions, which is a generalization of the notion of the effective resistance. We finally characterize the relationship between the convergence property and the equivalent edge function, when a nonpositive edge is added to a strictly positive network comprised of nonlinear integrators. We show that the convergence of the network is always guaranteed, if the sum of the equivalent edge function of the previous network and the new edge function is passive.