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

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.

arXiv comment27 pages, 4 figures. Preprint - To be published in SIAM Journal on Applied Dynamical Systems

References in corpus (9)

• Statistical physics of social dynamics
• Synchronization in complex networks
• Link communities reveal multiscale complexity in networks
• Bipartite Graphs as Models of Complex Networks
• The unreasonable effectiveness of tree-based theory for networks with clustering
• The Key Player Problem in Complex Oscillator Networks and Electric Power Grids: Resistance Centralities Identify Local Vulnerabilities
• Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis
• Multistability of Phase-Locking in Equal-Frequency Kuramoto Models on Planar Graphs
• Non-linear network dynamics with consensus-dissensus bifurcation

Cited by in corpus (2)

• Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian
• Non-linear network dynamics with consensus-dissensus bifurcation