Related papers: On the trace of random walks on random graphs
In the present paper, we give the exact formula for the average hitting time (HT, as an abbreviation) of random walks from one vertex to any other vertex on the some weighted Cayley graphs.
The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighboring vertices in some fixed cyclic order, rather than to a random sequence of…
We prove an explicit formula of hitting times in terms of enumerations of spanning trees for random walks on general connected graphs. We apply the formula to improve Lawler's bound of hitting times for general graphs, prove a sharp bound…
Let $\{G_i\}$ be the random graph process: starting with an empty graph $G_0$ with $n$ vertices, in every step $i \geq 1$ the graph $G_i$ is formed by taking an edge chosen uniformly at random among the non-existing ones and adding it to…
We consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites. The configuration comprising the graph, the starting point of the walk, and the locations of the trap sites is taken to be exactly…
We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in…
Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll k \lesssim \log |G|$. The results of this article supplement those in the three main papers on random Cayley…
We present the analytical and numerical results of a random walk on the family of small-world graphs. The average access time shows a crossover from the regular to random behavior with increasing distance from the starting point of the…
We study random walks on the giant component of Hyperbolic Random Graphs (HRGs), in the regime when the degree distribution obeys a power law with exponent in the range $(2,3)$. In particular, we first focus on the expected time for a…
We derive an exact closed-form analytical expression for the distribution of the cover time for a random walk over an arbitrary graph. In special case, we derive simplified exact expressions for the distributions of cover time for a…
The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on $n$ vertices, is known to be of order $\log n$. In this paper we investigate what happens when the random…
We study nonconcentration of hitting times for simple random walk on finite graphs. We prove that, for every connected graph with $n$ vertices, \[ \operatorname{Var}_x(\tau_y)+\mathbb E_x\tau_y \ge \frac{(\mathbb E_x\tau_y)^2}{1+\log n}, \]…
We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a…
Given an infinite connected regular graph $G=(V,E)$, place at each vertex Pois($\lambda$) walkers performing independent lazy simple random walks on $G$ simultaneously. When two walkers visit the same vertex at the same time they are…
We study random walks on the giant component of the Erd\H{o}s-R\'enyi random graph ${\cal G}(n,p)$ where $p=\lambda/n$ for $\lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently…
We show that the probability that a random graph $G\sim G(n,p)$ contains no Hamilton cycle is $(1+o(1))Pr(\delta (G) < 2)$ for all values of $p = p(n)$. We also prove an analogous result for perfect matchings.
We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph $G\sim {\mathcal G}(n,p)$ in order to find a subgraph which…
We revisit an old minor topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and…
We consider a class of strongly edge-reinforced random walks, where the corresponding reinforcement weight function is nondecreasing. It is known, from Limic and Tarr\`{e}s [Ann. Probab. (2007), to appear], that the attracting edge emerges…
We present analytical results for the distribution of first-passage (FP) times of random walks (RWs) on random regular graphs that consist of $N$ nodes of degree $c \ge 3$. Starting from a random initial node at time $t=0$, at each time…