  We study diffusion and consensus dynamics in a Network of Networks model. In
this model, there is a collection of sub-networks, connected to one another
using a small number of links. We consider a setting where the links between
networks have small weights, or are used less frequently than links within each
sub-network. Using spectral perturbation theory, we analyze the diffusion rate
and convergence rate of the investigated systems. Our analysis shows that the
first order approximation of the diffusion and convergence rates is independent
of the topologies of the individual graphs; the rates depend only on the number
of nodes in each graph and the topology of the connecting edges. The second
order analysis shows a relationship between the diffusion and convergence rates
and the information centrality of the connecting nodes within each sub-network.
We further highlight these theoretical results through numerical examples.
