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

Journal article

We study a generic family of nonlinear dynamics on undirected networks generalizing 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 of investigating dynamical behavior on families of networks made of trees of motifs.

Authors: Homs-Dones, M; Devriendt, K; Lambiotte, R

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Applied Dynamical Systems
• Volume: 20
• Issue: 3
• Pages: 1544-1570
• Publication date: 2021-09-09
• Acceptance date: 2021-06-02
• DOI: 10.1137/20M1376844
• EISSN: 1536-0040
• Language: English
• Keywords: consensus dynamics; network science; nonlinear dynamics; fixed points; effective resistance; trees