Related papers: Vacant sets and vacant nets: Component structures …

We construct a bounded degree graph $G$, such that a simple random walk on it is transient but the random walk path (i.e., the subgraph of all the edges the random walk has crossed) has only finitely many cutpoints, almost surely. We also…

In this work, Transition Probability Matrix (TPM) is proposed as a new method for extracting the features of nodes in the graph. The proposed method uses random walks to capture the connectivity structure of a node's close neighborhood. The…

Random walks on discrete lattices are fundamental models that form the basis for our understanding of transport and diffusion processes. For a single random walker on complex networks, many properties such as the mean first passage time and…

It was recently shown that continuous-time quantum walks on dynamic graphs, i.e., sequences of static graphs whose edges change at specific times, can implement a universal set of quantum gates. This result treated all isolated vertices as…

Message-passing graph neural networks (GNNs) excel at capturing local relationships but struggle with long-range dependencies in graphs. In contrast, graph transformers (GTs) enable global information exchange but often oversimplify the…

The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a…

We show that the edges crossed by a random walk in a network form a recurrent graph a.s. In fact, the same is true when those edges are weighted by the number of crossings.

We consider a simple random walk on a discrete torus (Z/NZ)^d with dimension d at least 3 and large side length N. For a fixed constant u > 0, we study the percolative properties of the vacant set, consisting of the set of vertices not…

Various graph algorithms have been developed with multiple random walks, the movement of several independent random walkers on a graph. Designing an efficient graph algorithm based on multiple random walks requires investigating multiple…

Many network analysis and graph learning techniques are based on models of random walks which require to infer transition matrices that formalize the underlying stochastic process in an observed graph. For weighted graphs, it is common to…

Following the recent work of Sznitman (arXiv:0805.4516), we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G_N x Z, where G_N is a large finite connected weighted graph, and relate it to…

We investigate a randomly evolving process of subgraphs in an underlying host graph using the spectral theory of semigroups related to the Tsetlin library and hyperplane arrangements. Starting with some initial subgraph, at each iteration,…

In the study of dynamical processes on networks, there has been intense focus on network structure -- i.e., the arrangement of edges and their associated weights -- but the effects of the temporal patterns of edges remains poorly…

This work studies the typical structure of sparse $H$-free graphs, that is, graphs that do not contain a subgraph isomorphic to a given graph $H$. Extending the seminal result of Osthus, Pr\"omel, and Taraz that addressed the case where $H$…

A $\delta$ once-reinforced random walk ($\delta$-ORRW) on connected graph is a self-interacting random walk which moves to its neighbors at each step according to the weights of the edges at that time, where the weights are $1$ on edges…

Random walk neural networks (RWNNs) have emerged as a promising approach for graph representation learning, leveraging recent advances in sequence models to process random walks. However, under realistic sampling constraints, RWNNs often…

We study the entropy of the distribution of the set R_n of vertices visited by a simple random walk on a graph with bounded degrees in its first n steps. It is shown that this quantity grows linearly in the expected size of R_n if the graph…

We consider a random object that is associated with both random walks and random media, specifically, the superposition of a configuration of subcritical Bernoulli percolation on an infinite connected graph and the trace of the simple…

We revisit a simple model class for machine learning on graphs, where a random walk on a graph produces a machine-readable record, and this record is processed by a deep neural network to directly make vertex-level or graph-level…

Dynamic graphs have emerged as an appropriate model to capture the changing nature of many modern networks, such as peer-to-peer overlays and mobile ad hoc networks. Most of the recent research on dynamic networks has only addressed the…