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We present analytical results for the distribution of cover times of randomwalks (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 ofthe time steps the RW may visit a new, yet-unvisited node, while in other timesteps it may revisit a node that has already been visited before. The covertime $T_{\rm C}$ is the number of time steps required for the RW to visit everysingle node in the network at least once. We derive a master equation for thedistribution $P_t(S=s)$ of the number of distinct nodes $s$ visited by an RW upto time $t$ and solve it analytically. Inserting $s=N$ we obtain the cumulativedistribution 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 thenetwork. Taking the large network limit, we show that $P(T_{\rm C} \le t)$converges to a Gumbel distribution. We calculate the distribution of partialcover (PC) times $P( T_{{\rm PC},k} = t )$, which is the probability that attime $t$ an RW will complete visiting $k$ distinct nodes. We also calculate thedistribution of random cover (RC) times $P( T_{{\rm RC},k} = t )$, which is theprobability that at time $t$ an RW will complete visiting all the nodes in asubgraph of $k$ randomly pre-selected nodes at least once. The analyticalresults for the distributions of cover times are found to be in very goodagreement with the results obtained from computer simulations.

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