Diffusive Dynamics on a Network

The effect of the topology of a finite sized interacting system, modelled into a network, on its dynamics is an interesting question. Various methods have been proposed in the literature for building different equivalence classes or families of networks based on the structural aspects of the system. It is observed that similarities in the local statistics of two networks are not sufficient to predict the dynamics on them. We suggest a global statistic, the Moore and Shannon's network reliability polynomial that depends on both the structure and the dynamics to explore the behaviour of a diffusive process on a network. It gives the probability that a system composed of many different interacting components has a desired property. The computation of the reliability polynomial exactly is often NP-hard, but estimating it using Monte-Carlo simulation is feasible even for graphs with hundreds of millions of edges. The probabilistic nature of the polynomial allows the mapping of parameters of one network on to another using a simple transformation inspired by renormalization group approaches that keeps the dynamical phenomenon invariant.