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

We present analytical results for the distribution of cover times of random walks (RWs) on random regular graphs consisting of $N$ nodes of degree $c$ ($c \ge 3$). Starting from a random initial node at time $t=1$, at each time step $t \ge 2$ an RW hops into a random neighbor of its previous node. In some of the time steps the RW may visit a new, yet-unvisited node, while in other time steps it may revisit a node that has already been visited before. The cover time $T_{\rm C}$ is the number of time steps required for the RW to visit every single node in the network at least once. We derive a master equation for the distribution $P_t(S=s)$ of the number of distinct nodes $s$ visited by an RW up to time $t$ and solve it analytically. Inserting $s=N$ we obtain the cumulative distribution of cover times, namely the probability $P(T_{\rm C} \le t) = P_t(S=N)$ that up to time $t$ an RW will visit all the $N$ nodes in the network. Taking the large network limit, we show that $P(T_{\rm C} \le t)$ converges to a Gumbel distribution. We calculate the distribution of partial cover (PC) times $P( T_{{\rm PC},k} = t )$, which is the probability that at time $t$ an RW will complete visiting $k$ distinct nodes. We also calculate the distribution of random cover (RC) times $P( T_{{\rm RC},k} = t )$, which is the probability that at time $t$ an RW will complete visiting all the nodes in a subgraph of $k$ randomly pre-selected nodes at least once. The analytical results for the distributions of cover times are found to be in very good agreement with the results obtained from computer simulations.