Related papers: Minimum stationary values of sparse random directe…
Given a branching random walk$\{Z_n\}_{n\geq0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)$ at generation $n$. It is known from \cite{Biggins1977} that under some mild conditions, $n^{-1}\log…
Analytical work probability distributions for open classical systems are scarce; they can only be calculated in a few examples. In this work, I present a new method to derive such quantities for weakly driven processes in the overdamped…
We establish the asymptotic degree distribution of the typical vertex of inhomogeneous and passive random intersection graphs under the minimal moment conditions.
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…
We prove results for first-passage percolation on the configuration model with i.i.d. degrees having finite mean, infinite variance and i.i.d. weights with strictly positive support of the form Y=a+X, where a is a positive constant. We…
We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$…
We show bounds on total variation and $L^{\infty}$ mixing times, spectral gap and magnitudes of the complex valued eigenvalues of a general (non-reversible non-lazy) Markov chain with a minor expansion property. This leads to the first…
We consider a sparse Erd\H{o}s--R\'{e}nyi graph $\mathcal{G}(n,\lambda/n)$ where each edge is independently assigned a random signed weight. For two uniformly chosen vertices, we study the joint distribution of the total weights and…
Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) independently distributed in the unit square~\(S,\) each according to a distribution~\(f.\) Nodes~\(X_i\) and~\(X_j\) are joined by an edge if the Euclidean distance~\(d(X_i,X_j)\) is less…
Random walks constitute a fundamental mechanism for a large set of dynamics taking place on networks. In this article, we study random walks on weighted networks with an arbitrary degree distribution, where the weight of an edge between two…
We propose an approximation for the first return time distribution of random walks on undirected networks. We combine a message-passing solution with a mean-field approximation, to account for the short- and long-term behaviours…
We obtain upper bounds (in most cases, sharp) for the hitting times of random walks on finite undirected graphs expressed as functions of the graph's number of edges. In particular, we show that the maximum hitting time for a simple random…
In this paper, we derive the moderate deviation principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are…
A classical theorem of Ghouila-Houri from 1960 asserts that every directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$ contains a directed Hamilton cycle. In this paper we extend this theorem to a random…
We find Gaussian cutoff profiles for the total variation distance to stationarity of a random walk on a multiplex network: a finite number of directed configuration models sharing a vertex set, each with its own bounded degree distribution…
In this paper we find an integer $h=h(n)$ such that the minimum number of variables of a first order sentence that distinguishes between two independent uniformly distributed random graphs of size $n$ with the asymptotically largest…
We introduce a new random graph model motivated by biological questions relating to speciation. This random graph is defined as the stationary distribution of a Markov chain on the space of graphs on $\{1, \ldots, n\}$. The dynamics of this…
We consider the quantity $P(G)$ associated with a graph $G$ that is defined as the probability that a randomly chosen subtree of $G$ is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this…
We propose the following conjecture extending Dirac's theorem: if $G$ is a graph with $n\ge 3$ vertices and minimum degree $\delta(G)\ge n/2$, then in every orientation of $G$ there is a Hamilton cycle with at least $\delta(G)$ edges…
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…