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

Abstract

We present analytical results for the distribution of first hitting (FH) times of random walks (RWs) on random regular graphs (RRGs) of degree c ⩾ 3 and a finite size N. Starting from a random initial node at time t = 1, at each time step t ⩾ 2 an RW hops randomly into one of the c neighbors 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. The first time at which the RW enters a node that has already been visited before is called the FH time or the first intersection length. The FH event may take place either by backtracking (BT) to the previous node or by retracing (RET), namely stepping into a node which has been visited two or more time steps earlier. We calculate the tail distribution P(T FH > t) of FH times as well as its mean ⟨T FH⟩ and variance Var(T FH). We also calculate the probabilities P BT and P RET that the FH event will occur via the BT scenario or via the RET scenario, respectively. We show that in dilute networks the dominant FH scenario is BT while in dense networks the dominant scenario is RET and calculate the conditional distributions P(T FH = t|BT) and P(T FH = t|RET), for the two scenarios. The analytical results are in excellent agreement with the results obtained from computer simulations. Considering the FH event as a termination mechanism of the RW trajectories, these results provide useful insight into the general problem of survival analysis and the statistics of mortality rates when two or more termination scenarios coexist.