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Random walks find applications in many areas of science and are the heart of essential network analytic tools. When defined on temporal networks, even basic random walk models may exhibit a rich spectrum of behaviours, due to the…
Real networks are often dynamic. In response to it, analyses of algorithms on {\em dynamic networks} attract more and more attentions in network science and engineering. Random walks on dynamic graphs also have been investigated actively in…
Random walks are a series of up, down, and level steps that enumerate distinct paths from $(0,0)$ to $(2n,0)$, where $n$ is the semi-length of the path. We used these paths to analyze Catalan, Schr\"{o}der, and Motzkin number sequences…
Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we…
We consider a population of $N$ labeled random walkers moving on a substrate, and an excitation jumping among the walkers upon contact. The label $\mathcal{X}(t)$ of the walker carrying the excitation at time $t$ can be viewed as a…
Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover…
Sood and Grassberger studied in [Phys. Rev. Lett. 99, 098701 (2007)] random walks on random graphs that are biased towards a fixed target point. They put forward a critical bias strength b_c such that a random walker on an infinite graph…
The exact formula for the average hitting time (HT, as an abbreviation) of simple random walks from one vertex to any other vertex on the square $C^2_N$ of an $N$-vertex cycle graph $C_N$ was given by N. Chair [\textit{Journal of…
In the context of a random walk on an undirected graph, Kemeny's constant can measure the average travel time for a random walk between two randomly chosen vertices. We are interested in graphs that behave counter-intuitively in regard to…
We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any $\varepsilon>0$ there exists $C>1$ such that the trace of the simple random walk of length $(1+\varepsilon)n\ln{n}$ on the random…
An algorithm observes the trajectories of random walks over an unknown graph $G$, starting from the same vertex $x$, as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges $m$ up…
The number of walks from one vertex to another in a finite graph can be counted by the adjacency matrix. In this paper, we prove two theorems that connect the graph Laplacian with two types of walks in a graph. By defining two types of…
Random walks on expanders play a crucial role in Markov Chain Monte Carlo algorithms, derandomization, graph theory, and distributed computing. A desirable property is that they are rapidly mixing, which is equivalent to having a spectral…
For a finite graph $G=(V,E)$ let $G^*$ be obtained by considering a random perfect matching of $V$ and adding the corresponding edges to $G$ with weight $\varepsilon$, while assigning weight 1 to the original edges of $G$. We consider…
Kemeny's constant measures how fast a random walker moves around in a graph. Expressions for Kemeny's constant can be quite involved, and for this reason, many lines of research focus on graphs with structure that makes them amenable to…
We obtain expected number of arrivals, absorption probabilities and expected time until absorption for an asymmetric discrete random walk on a graph in the presence of multiple function barriers. On each edge of the graph and in each vertex…
The walk distances in graphs are defined as the result of appropriate transformations of the $\sum_{k=0}^\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a graph and $t$ is a sufficiently small positive…
We study a symmetric random walk (RW) in one spatial dimension in environment, formed by several zones of finite width, where the probability of transition between two neighboring points and corresponding diffusion coefficient are…
Network meta-analysis (NMA) is a central tool for evidence synthesis in clinical research. The results of an NMA depend critically on the quality of evidence being pooled. In assessing the validity of an NMA, it is therefore important to…
We study large deviations for random walks on stratified (Carnot) Lie groups. For such groups, there is a natural collection of vectors which generates their Lie algebra, and we consider random walks with increments in only these…