Nonlinear consensus on networks: equilibria, effective resistance and trees of motifs

TL;DR

This paper analyzes nonlinear consensus dynamics on networks, deriving equilibrium conditions and stability criteria using effective resistance, and introduces a recursive approach for networks composed of interconnected motifs like trees.

Contribution

It provides a unified framework for understanding equilibria and stability in nonlinear consensus models based on network topology, including a recursive method for complex networks.

Findings

Equilibrium points expressed via network topology and effective resistance.

Stability classified using weighted effective resistance.

Recursive analysis for networks built from smaller subnetworks.

Abstract

We study a generic family of nonlinear dynamics on undirected networks generalising linear consensus. We find a compact expression for its equilibrium points in terms of the topology of the network and classify their stability using the effective resistance of the underlying graph equipped with appropriate weights. Our general results are applied to some specific networks, namely trees, cycles and complete graphs. When a network is formed by the union of two subnetworks joined in a single node, we show that the equilibrium points and stability in the whole network can be found by simply studying the smaller subnetworks instead. Applied recursively, this property opens the possibility to investigate the dynamical behaviour on families of networks made of trees of motifs.