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Title: Statistical properties of non linear random walks on networks

Abstract: Random walks simulate various interacting entities (the nodes) which exchange ’particles’ according to the topological structure defined by the links and they are considered to introduce a dynamics on networks. These stochastic dynamical systems can be applied to model complex systems like transportation networks, ecological systems, neural networks, economic systems etc.. In the linear case the statistical properties are defined by single particles dynamics and the equilibrium distribution depends on the network structure. We introduce non linear effects by assuming a finite transportation capacity of the links or a finite capacity in the nodes. As a consequence the transition probabilities depend on the dynamical state of the network and one cannot derive statistical properties of the system from single particle dynamics. We show that non-linear effects can be described by introducing an entropic force among the node states which allows to derive a master equation for the evolution of the probability distribution of the node population. This entropic force has a relevant effect on stationary distribution and the relaxation time scale depends on the numerosity of the population so that the thermodynamics limit is non trivial and the lifetime of the transient state is very long. Using numerical simulation we derive an analytical form of the entropic force and we study the dependence of the stationary distribution on the network topology.