Spin model of two random walkers in complex networks

Abstract

In this paper, we design a spin model to optimize the search in complex networks using two random walkers by reducing the wasted time in revisiting nodes. A good measure of the global performance of the searching process is the Mean Cover Time (MCT), which is the time needed from the start of the searching process to the end that all sites in the network are reached by at least one walker. We use a three-state spin model to minimize the MCT for two random walkers. In the model, each site in the network is described by a three-state spin, with unvisited sites having a spin state defined by white color, and the visited sites in spin states with non-white colors (red and blue in the case of two walkers). The visit of a site by a walker changes the state of the spin. We introduce a repulsive interaction between spins to model the interactions between walkers. Numerical results using the Erdős-Rényi (ER) network, the Watts-Strogatz (WS) network and three small-world real network datasets show satisfactory results in reducing the MCT. For small-world networks, both the artificial WS network and real-world network datasets show the existence of the critical repulsion strength which minimizes the MCT. We also provide a heuristic explanation for the presence of the critical repulsion that minimizes the MCT for the WS network and its absence in the ER network. Our model provides guidance to future research on multiple random walkers over complex networks, with potential applications for efficient information spreading in social networks.