Analytical results for the distribution of first-passage times of random walks on random regular graphs

Abstract

We present analytical results for the distribution of first-passage (FP) times of random walks (RWs) on random regular graphs that consist of N nodes of degree c ≥ 3. Starting from a random initial node at time t = 0, at each time step t ≥ 1 an RW hops into a random neighbor of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. We calculate the distribution P (T FP = t) of first-passage times from a random initial node i to a random target node j, where j = i. We distinguish between FP trajectories that follow the shortest path (SPATH) from the initial node i to the target node j and FP trajectories, which do not follow the shortest path (¬SPATH). More precisely, the SPATH trajectories are defined as trajectories in which the subnetwork that consists of the nodes and edges along the trajectory is a tree network and the distance ℓ ij between i and j on this subnetwork is the same as in the whole network. The SPATH scenario is probable mainly in case that the distance ℓ ij between the initial node i and the target node j is small. The analytical results are found to be in very good agreement with the results obtained from computer simulations.