Random spreading phenomena in annealed small world networks

Here we study simple random walk dynamics in an annealed version of a small world network (SWN) consisting of N nodes. This is done by calculating the mean number of distinct sites visited, S(t), and the return probability, P-00(t), as functions of time t. The former is a key quantity both from the statistical physics point of view and especially for characterizing the efficiency of the network connectedness. Our results for S(t) shows features similar to the SWN with quenched disorder, but with a crossover time that is inversely proportional to the probability p of making a long-range jump instead of being proportional to p(-2) as in quenched case. We have also carried out simulations on a modified annealed model where the crossover time behaves as p(-2) due to specific time dependent transition probabilities and we present an approximate self-consistent solution to it. (C) 2002 Elsevier Science B.V. All rights reserved.