  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.
