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

Abstract

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 $T_{\mathrm{C}}$ 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 $P(T_{\mathrm{C}}\le t)=P_t(S=N)$ that up to time $t$ an RW will visit all the $N$ nodes in the network. Taking the large network limit, we show that $P(T_{\mathrm{C}}\le t)$ converges to a Gumbel distribution. We calculate the distribution of partial cover (PC) times $P(T_{\mathrm{PC},k}=t)$, 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 $P(T_{\mathrm{RC},k}=t)$, 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.

arXiv comment47 pages, 11 figures. arXiv admin note: text overlap with arXiv:2102.12195, arXiv:2106.10449

References in corpus (36)

• Epidemic processes in complex networks
• Random graphs with arbitrary degree distributions and their applications
• Diffusion with Stochastic Resetting
• Random walks and diffusion on networks
• Random walks on graphs: ideas, techniques and results
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Non-backtracking random walks mix faster
• First Passage Properties of the Erdos-Renyi Random Graph
• Exploring Complex Networks through Random Walks
• Cover time for random walks on arbitrary complex networks
• Random walk and trapping processes on scale-free networks
• Return times of random walk on generalized random graphs
• Depletion-Controlled Starvation of a Diffusing Forager
• Arrival Time Statistics in Global Disease Spread
• Distribution of diameters for Erdös-Rényi random graphs
• Analytical results for the distribution of first return times of random walks on random regular graphs
• Distance distribution in configuration model networks
• Self-avoiding walks and connective constants in small-world networks
• Role of Depletion on the Dynamics of a Diffusing Forager
• Distribution of shortest cycle lengths in random networks
• Universality classes of foraging with resource renewal
• Takeover times for a simple model of network infection
• The distribution of path lengths of self avoiding walks on Erdős-Rényi networks
• The average number of distinct sites visited by a random walker on random graphs
• Return probabilities and hitting times of random walks on sparse Erdos-Renyi graphs
• Kinetic growth walks on complex networks
• How fast does a random walk cover a torus?
• Random walks on networks: cumulative distribution of cover time
• The distribution of first hitting times of random walks on Erdős-Rényi networks
• The distribution of first hitting times of random walks on directed Erdős-Rényi networks
• Analytical results for the distribution of first hitting times of random walks on random regular graphs
• The Cover Time of Random Walks on Graphs
• Dynamically accelerated cover times
• The distribution of first hitting times of non-backtracking random walks on Erdős-Rényi networks
• The distribution of the maximum number of common neighbors in the random graph
• Random Walks on Complex Networks

Cited by in corpus (6)

• Analytical results for the distribution of first-passage times of random walks on random regular graphs
• Efficient network exploration by means of resetting self-avoiding random walkers
• A Gaussian integral that counts regular graphs
• The joint distribution of first return times and of the number of distinct sites visited by a 1D random walk before returning to the origin
• First return times on sparse random graphs
• Analytical results for the distribution of first return times of non-backtracking random walks on configuration model networks