Related papers: Kemeny's constant for non-backtracking random walk…

We study the average probability that a discrete-time quantum walk finds a marked vertex on a graph. We first show that, for a regular graph, the spectrum of the transition matrix is determined by the weighted adjacency matrix of an…

For a random walk on a network, the mean first-passage time from a node $i$ to another node $j$ chosen stochastically according to the equilibrium distribution of Markov chain representing the random walk is called Kemeny constant, which is…

We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion…

The connective constant $\mu(G)$ of a graph $G$ is the asymptotic growth rate of the number of self-avoiding walks on $G$ from a given starting vertex. We survey three aspects of the dependence of the connective constant on the underlying…

For a connected graph $G$, the average hitting time $\alpha(G)$ and the Kemeny's constant $\kappa(G)$ are two similar quantities, both measuring the time for the random walk on $G$ to travel between two randomly chosen vertices. We prove…

Let X be a locally finite, connected graph without vertices of degree 1. Non-backtracking random walk moves at each step with equal probability to one of the "forward" neighbours of the actual state, i.e., it does not go back along the…

Consider continuous-time random walks on Cayley graphs where the rate assigned to each edge depends only on the corresponding generator. We show that the limiting speed is monotone increasing in the rates for infinite Cayley graphs that…

Kemeny's constant measures the efficiency of a Markov chain in traversing its states. We investigate whether structure-preserving perturbations to the transition probabilities of a reversible Markov chain can improve its connectivity while…

In their 1960 book on finite Markov chains, Kemeny and Snell established that a certain sum is invariant. The value of this sum has become known as {\it Kemeny's constant}. Various proofs have been given over time, some more technical than…

In this paper, we determine a formula for Kemeny's constant for a graph with multiple bridges, in terms of quantities that are inherent to the subgraphs obtained upon removal of all bridges and that can be computed independently. With the…

Extensions of Kemeny's constant, as derived for irreducible finite Markov chains in discrete time, to Markov renewal processes and Markov chains in continuous time are discussed. Three alternative Kemeny's functions and their variants are…

Random walks are used for modeling various dynamics in, for example, physical, biological, and social contexts. Furthermore, their characteristics provide us with useful information on the phase transition and critical phenomena of even…

We prove expectation and concentration results for the following random variables on an Erd\H{o}s-R\'enyi 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…

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…

We investigate the hitting times of random walks on graphs, where a hitting time is defined as the number of steps required for a random walker to move from one node to another. While much of the existing literature focuses on calculating…

For any given vertices $u$ and $v$ in a graph, the hitting time of a random walk on a finite graph is the number of steps it takes for a random walk to reach vertex $v$ starting at vertex $u$. The expected value of the hitting time is the…

Graph products have been extensively applied to model complex networks with striking properties observed in real-world complex systems. In this paper, we study the hitting times for random walks on a class of graphs generated iteratively by…

There are presently two models for quantum walks on graphs. The "coined" walk uses discrete time steps, and contains, besides the particle making the walk, a second quantum system, the coin, that determines the direction in which the…

We prove the expected duration of a game of hide-and-seek played on a Riemannian manifold under the laws of Brownian Motion is a spectral invariant: it is a zeta-regularized version of the `trace' of the Laplacian. An analogous…

The node2vec random walk is a non-Markovian random walk on the vertex set of a graph, widely used for network embedding and exploration. This random walk model is defined in terms of three parameters which control the probability of,…