Cover Time on a Square Lattice by Two Colored Random Walkers

Abstract

We treat the minimization of cover time on a network by M random walkers as a problem of a multi-agent system, where the M agents have local interactions. We introduce a model of local repulsion between the walkers and visited sites in order to minimize the waste of steps in revisiting sites, thereby also the cover time. We particularly perform numerical simulations for the case of two colored random walkers (M = 2), namely Ray and Ben, on a square lattice. The unvisited nodes are colored white, while the nodes first visited by Ray/Ben are colored red/blue. The cover time is the time when there is no more white nodes. The interaction between a walker and a colored site is a repulsive nearest-neighbor interaction, so we can model peer-avoidance (red-blue or blue-red) and self-avoidance (red-red or blue-blue) for the random walk. We investigate the proper combination of the two walkers’ strategies that minimizes the time to cover a square lattice. A strategy is represented with a binary sequence, in which 0 stands for peer avoidance and 1 for self-avoidance. We find that if the absolute difference D between the number of 1’s in Ray’s and Ben’s strategies is zero, or if Ray or Ben is neutral, i.e. half peer-avoiding and half self-avoiding, the sequence combination is likely to be good. In general, a sequence combination’s cover time is positively correlated with D.