A close relation between hitting times of the simple random walk on a graph, the Kirchhoff index, resistance-centrality, and related invariants of unicyclic graphs is displayed. Combining with the graph transformations and some other techniques, sharp upper and lower bounds on the cover cost (resp. reverse cover cost) of a vertex in an $n$-vertex unicyclic graph are determined. All the corresponding extremal graphs are identified, respectively.
We exhibit a close connection between hitting times of the simple random walk on a graph, the Wiener index, and related graph invariants. In the case of trees we obtain a simple identity relating hitting times to the Wiener index. It is well known that the vertices of any graph can be put in a linear preorder so that vertices appearing earlier in the preorder are "easier to reach" by a random walk, but "more difficult to get out of". We define various other natural preorders and study their relationships. These preorders coincide when the graph is a tree, but not necessarily otherwise. Our tre…
We study a multiple-urn version of the Ehrenfest model. In this setting, we denote the n urns by Urn 1 to Urn n, where n>=2. Initially, M balls are randomly placed in the n urns. At each subsequent step, a ball is selected and put into the other n-1 urns with equal probability. The expected hitting time leading to a change of the M balls' status is computed using the method of stopping times. As a corollary, we obtain the expected hitting time of moving all the M balls from Urn 1 to Urn 2. This proves a conjecture which was recently made in Chen et al.(2017).
The hitting time, h_uv, of a random walk on a finite graph G, is the expected time for the walk to reach vertex v given that it started at vertex u. We present two methods of calculating the hitting time between vertices of finite graphs, along with applications to specific classes of graphs, including grids, trees, and the 'tadpole' graphs.
High-Precision Framework for Expected Hitting Times Analysis in the Dice-Sum Process
Department of Mathematics, Faculty of Science, Mahidol University（玛希隆大学理学院数学系）；Centre of Excellence in Mathematics, MHESI（泰国高等教育与研究创新部数学卓越中心）；Science Division, Mahidol University International College（玛希隆大学国际学院科学学部）
We prove expectation and concentration results for the following random variables on an Erdős-Rényi random graph $\mathcal{G}\left(n,p\right)$ in the sparsely connected regime $\log n + \log\log \log n \leq np < n^{1/10}$: effective resistances, random walk hitting and commute times, the Kirchoff index, cover cost, random target times, the mean hitting time and Kemeny's constant. For the effective resistance between two vertices our concentration result extends further to $np\geq c\log n, \; c>0$. To achieve these results, we show that a strong connectedness property holds with high probabilit…
The discrete Green's function (without boundary) $\mathbb{G}$ is a pseudo-inverse of the combinatorial Laplace operator of a graph $G=(V,E)$. We reveal the intimate connection between Green's function and the theory of exact stopping rules for random walks on graphs. We give an elementary formula for Green's function in terms of state-to-state hitting times of the underlying graph. Namely, $\mathbb{G}(i,j) = π_j \left( \sum_{k \in V} π_k H(k,j) - H(i,j) \right)$ where $π_i$ is the stationary distribution at vertex $i$ and $H(i,j)$ is the expected hitting time for a random walk starting from ve…
In this paper, we are concerned with mean hitting time $\langle\mathcal{H}\rangle$ for random walks on recursive growth tree networks that are built based on an arbitrary tree as the seed via implementing various primitive graphic operations, and propose a series of combinatorial techniques that are called Mapping Transformation to exactly determine the associated $\langle\mathcal{H}\rangle-$polynomial. Our formulas can be able to completely cover the previously published results in some well-studied and specific cases where a single edge or a star is often chose to serve as seed for creating …
Given an irreducible discrete-time Markov chain on a finite state space, we consider the largest expected hitting time $T(α)$ of a set of stationary measure at least $α$ for $α\in(0,1)$. We obtain tight inequalities among the values of $T(α)$ for different choices of $α$. One consequence is that $T(α) \le T(1/2)/α$ for all $α< 1/2$. As a corollary we have that, if the chain is lazy in a certain sense as well as reversible, then $T(1/2)$ is equivalent to the chain's mixing time, answering a question of Peres. We furthermore demonstrate that the inequalities we establish give an almost everywher…
The time it takes a random walker in a lattice to reach the origin from another vertex $x$, has infinite mean. If the walker can restart the walk at $x$ at will, then the minimum expected hitting time $T(x,0)$ (minimized over restarting strategies) is finite; it was called the ``grade'' of $x$ by Dumitriu, Tetali and Winkler. They showed that, in a more general setting, the grade (a variant of the ``Gittins index'') plays a crucial role in control problems involving several Markov chains. Here we establish several conjectures of Dumitriu et al on the asymptotics of the grade in Euclidean latti…
A survey is presented of known results concerning simple random walk on the class of distance-regular graphs. One of the highlights is that electric resistance and hitting times between points can be explicitly calculated and given strong bounds for, which leads in turn to bounds on cover times, mixing times, etc. Also discussed are harmonic functions, moments of hitting and cover times, the Green's function, and the cutoff phenomenon. The main goal of the paper is to present these graphs as a natural setting in which to study simple random walk, and to stimulate further research in the field.
\emph{Strictly Chordality-$k$ graphs ($SC_k$)} are graphs which are either cycle-free or every induced cycle is of length exactly $k, k \geq 3$. Strictly chordality-3 and strictly chordality-4 graphs are well known chordal and chordal bipartite graphs, respectively. For $k\geq 5$, the study has been recently initiated in \cite{sadagopan} and various structural and algorithmic results are reported. In this paper, we show that maximum independent set (MIS), minimum vertex cover, minimum dominating set, feedback vertex set (FVS), odd cycle transversal (OCT), even cycle transversal (ECT) and Stein…
Given a permutation sigma of the integers {-n,-n+1,...,n} we consider the Markov chain X_{sigma}, which jumps from k to sigma (k\pm 1) equally likely if k\neq -n,n. We prove that the expected hitting time of {-n,n} starting from any point is Theta(n) with high probability when sigma is a uniformly chosen permutation. We prove this by showing that with high probability, the digraph of allowed transitions is an Eulerian expander; we then utilize general estimates of hitting times in directed Eulerian expanders.
Researchers have designed many algorithms to measure the distances between graph nodes, such as average hitting times of random walks, cosine distances from DeepWalk, personalized PageRank, etc. Successful although these algorithms are, still they are either underperforming or too time-consuming to be applicable to huge graphs that we encounter daily in this big data era. To address these issues, here we propose a faster algorithm based on an improved version of random walks that can beat DeepWalk results with more than ten times acceleration. The reason for this significant acceleration is th…
Let T(x,r) denote the first hitting time of the disc of radius r centered at x for Brownian motion on the two dimensional torus. We prove that sup_{x} T(x,r)/|log r|^2 --> 2/pi as r --> 0. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus Z_n^2 is asymptotic to (2n log n)^2/pi. Determining these asymptotics is an essential step toward analyzing the f…
Given a connected graph $G$ with some subset of its vertices excited and a fixed target vertex, in the geodesic-biased random walk on $G$, a random walker moves as follows: from an unexcited vertex, she moves to a uniformly random neighbour, whereas from an excited vertex, she takes one step along some fixed shortest path towards the target vertex. We show, perhaps counterintuitively, that the geodesic-bias can slow the random walker down exponentially: there exist connected, bounded-degree $n$-vertex graphs with excitations where the expected hitting time of a fixed target is at least $\exp (…
We study the biased random walk where at each step of a random walk a "controller" can, with a certain small probability, move the walk to an arbitrary neighbour. This model was introduced by Azar et al. [STOC'1992]; we extend their work to the time dependent setting and consider cover times of this walk. We obtain new bounds on the cover and hitting times. Azar et al. conjectured that the controller can increase the stationary probability of a vertex from $p$ to $p^{1-ε}$; while this conjecture is not true in full generality, we propose a best-possible amended version of this conjecture and c…
