We study diffusion and consensus dynamics in a network of networks model. In this model, there is a collection of subnetworks, 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 subnetwork. 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 subnetwork. We further highlight these theoretical results through numerical examples.