Convergence Analysis of Signed Nonlinear Networks

TL;DR

This paper investigates the convergence behavior of signed nonlinear networks with passive node dynamics and nonlinear edge functions, extending classical signed network theory and establishing conditions for convergence and clustering.

Contribution

It generalizes signed networks to nonlinear edge functions using passivity theory and characterizes convergence conditions with equivalent edge functions.

Findings

Networks with positive edges reach consensus or form clusters.

Adding non-positive edges still allows convergence under certain conditions.

Equivalent edge functions generalize effective resistance in nonlinear networks.

Abstract

This work 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 work is to generalize the classical notion of signed networks for graphs with scalar weights to graphs with nonlinear edge functions using notions from 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 non-positive 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 non-positive 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 effective resistance. We finally characterize the relationship between the convergence property and the equivalent edge function, when a non-positive 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.