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 times of random walks (RWs) on random regular graphs (RRGs) of degree $c \ge 3$ and a finite size $N$. Starting from a random initial node at time $t=1$, at each time step $t \ge 2$ 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 first hitting time or the first intersection length. The first hitting 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 $P( T_{\rm FH} > t )$ of first hitting (FH) times as well as its mean $\langle T_{\rm FH} \rangle$ and variance ${\rm Var}(T_{\rm FH})$. We also calculate the probabilities $P_{\rm BT}$ and $P_{\rm RET}$ that the first hitting event will occur via the backtracking scenario or via the retracing scenario, respectively. We show that in dilute networks the dominant first hitting scenario is backtracking while in dense networks the dominant scenario is retracing and calculate the conditional distributions $P(T_{\rm FH}=t| {\rm BT})$ and $P(T_{\rm FH}=t| {\rm RET})$, for the two scenarios. The analytical results are in excellent agreement with the results obtained from computer simulations. Considering the first hitting 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.