Localization phase transition in stochastic dynamics on networks with hub topology

Abstract

Dynamics among central sources (hubs) providing a resource and large number of components enjoying and contributing to this resource describes many real life situations. Modeling, controlling, and balancing this dynamics is a challenging problem that arises in many scientific disciplines. We analyze a stochastic dynamical system exhibiting this dynamics with a multiplicative noise. We show that this model can be solved exactly by passing to variables that describe the mass ratio between the components and the hub. We derive a deterministic equation for the average mass ratio in the absence of noise on the hub. This equation describes logistic growth. We derive the phase diagram of the model with and without noise on the hub. We show that when there in no noise on the hub there is no localization phase. In the presence of noise on the hub, we identify two regimes by deriving the equilibrium distribution of the process. The first regime describes balance between the non-hub components and the hub, in the second regime the resource is concentrated mainly on the hub. We generalize the results to a system with multiple hubs. We show that there is less concentration on the hubs as the number of hubs increases, and in the limit of infinite hubs the average mass ratio grows or decays exponentially. Surprisingly, in the limit of large number of components the transition values do not depend on the amount of resource given by the non-hub nodes. We propose an interesting application of this model in the context of porous media using Magnetic Resonance (MR) techniques.