Small-world networks: Links with long-tailed distributions

Small-world networks (SWN), obtained by randomly adding to a regular struct ure additional links (AL), are of current interest. In this paper we explor e (based on physical models) a new variant of SWN, in which the probability of realizing an AL depends on the chemical distance between the connected sites. We assume a power-law probability distribution and study random walk ers on the network, focusing especially on their probability of being at th e ori,origin. We connect the results to Levy flights, which follow from a m ean-field variant of our model.