Related papers: Random Walks, Equidistribution and Graphical Desig…
Approximate random $k$-colouring of a graph $G$ is a well studied problem in computer science and statistical physics. It amounts to constructing a $k$-colouring of $G$ which is distributed close to {\em Gibbs distribution} in polynomial…
This is the second in a series of articles devoted to showing that a typical covering map of large degree to a fixed, regular graph has its new adjacency eigenvalues within the bound conjectured by Alon for random regular graphs. The first…
In Chung-Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution, and two vertices form an edge independently with probability proportional to the product of their…
Simple random walks are a basic staple of the foundation of probability theory and form the building block of many useful and complex stochastic processes. In this paper we study a natural generalization of the random walk to a process in…
In this paper we address the problem of understanding the success of algorithms that organize patches according to graph-based metrics. Algorithms that analyze patches extracted from images or time series have led to state-of-the art…
$M$-Lipschitz mappings of graphs (or equivalently graph-indexed random walks) are a generalization of standard random walk on $\mathbb{Z}$. For $M \in \N$, an \emph{$M$-Lipschitz mapping} of a connected rooted graph $G = (V,E)$ is a mapping…
A matrix random walk is a stochastic process of the form $B_k = (I+A_1)\cdots(I+A_k)$ where $A_j$ are independent ``step'' matrices in $\mathrm{M}_N(\mathbb{C})$. With the right entry-covariance, a rescaled matrix random walk converges to…
Approximate random k-colouring of a graph G=(V,E) is a very well studied problem in computer science and statistical physics. It amounts to constructing a k-colouring of G which is distributed close to Gibbs distribution, i.e. the uniform…
We analyze greedy routing in a random graph G_n constructed on the vertex set V = {1, 2, ..., n} embedded in Z. Vertices are inserted according to a uniform random permutation pi, and each newly inserted vertex connects to its nearest…
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…
Based on studies on four specific networks, we conjecture a general relation between the walk dimensions $d_{w}$ of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that $d_{w}$ of the…
We extend the pseudorandomness of random walks on expander graphs using the sticky random walk. Building on prior works, it was recently shown that expander random walks can fool all symmetric functions in total variation distance (TVD)…
We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ defined by this action for some fixed generating set. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the…
We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability $r$, and with…
We present analytical results for the distribution of first return (FR) times of random walks (RWs) on random regular graphs (RRGs) consisting of $N$ nodes of degree $c \ge 3$. Starting from a random initial node $i$ at time $t=0$, at each…
Let G be a graph with set of vertices 1,...,n and adjacency matrix A of size nxn. Let d(i,j)=d, we say that f_d:N->N is a d-function on G if for every pair of vertices i,j and k>=d, we have a_ij^(k)=f_d(k). If this function f_d exists on G…
Let $G(N,p)=(V,E)$ be an Erd\"os-R\'enyi random graph and $(X_n)_{n \in \mathbb{N}}$ be a simple random walk on it. We study the the order of magnitude of $\sum_{i \in V} \pi_ih_{ij} $ where $\pi_i=d_i / 2|E|$ for $d_i$ the number of…
Consider a measure $\mu_\lambda = \sum_x \xi_x \delta_x$ where the sum is over points $x$ of a Poisson point process of intensity $\lambda$ on a bounded region in $d$-space, and $\xi_x$ is a functional determined by the Poisson points near…
We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle…
Random walks are at the heart of many existing deep learning algorithms for graph data. However, such algorithms have many limitations that arise from the use of random walks, e.g., the features resulting from these methods are unable to…