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

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.