Return probability for random walks on scale-free complex trees

The time course of random processes usually differs depending on the topology of complex networks which are a substrate for the process. However, as this Letter demonstrates, the first-return as well as the survival probabilities for random walks on the scale-free (SF) trees decay in time according to the same invariant power-law behavior. This means that both quantities are independent of the node power-law degree distributions which are distinguished by different scaling exponents. It is also shown here that the crucial property of the networks, affecting the dynamics of random walks, is their tree-like topology and not SF architecture. All analytical results quantifying these predictions have been verified through extensive computer simulations.

► We show that the first return of random walks on scale-free trees is recurrent.
► As a consequence the survival probability of random walks is smaller than one.
► This behavior is independent of the degree distribution of the scale-free trees.
► It strongly depends on the network tree-like topology with a mean degree two.