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

Abstract

We study a generic family of non-linear dynamics on 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 sub-networks 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 sub-networks instead. Applied recursively, this property opens the possibility to investigate the dynamical behaviour on families of networks made of trees of motifs, including hypertrees.

Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian

“…Many of the linear synchronization/consensus models on networks are related to the graph Laplacian or its generalizations [15,44,58]. As such, it is natural to consider nonlinear dynamics whose linearizations correspond to an appropriate Laplacian.…”