Related papers: Limit theorems for the maximal path weight in a di…

We consider a stochastic directed graph on the integers whereby a directed edge between $i$ and a larger integer $j$ exists with probability $p_{j-i}$ depending solely on the distance between the two integers. Under broad conditions, we…

We consider directed last-passage percolation on the random graph G = (V,E) where V = Z and each edge (i,j), for i < j, is present in E independently with some probability 0 < p <= 1. To every present edge (i,j) we attach i.i.d. random…

This paper provides an overview of results, concerning longest or heaviest paths, in the area of random directed graphs on the integers along with some extensions. We study first-order asymptotics of heaviest paths allowing weights both on…

We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex $(i_1,i_2)$ to $(j_1,j_2)$, whenever $i_1 \le j_1$, $i_2 \le j_2$, with probability $p$, independently for each such pair of vertices.…

Let $H_d(n,p)$ signify a random $d$-uniform hypergraph with $n$ vertices in which each of the ${n}\choose{d}$ possible edges is present with probability $p=p(n)$ independently, and let $H_d(n,m)$ denote a uniformly distributed with $n$…

Let $G(n,c/n)$ and $G_r(n)$ be an $n$-node sparse random graph and a sparse random $r$-regular graph, respectively, and let ${\cal I}(n,r)$ and ${\cal I}(n,c)$ be the sizes of the largest independent set in $G(n,c/n)$ and $G_r(n)$. The…

For any fixed integer $R \geq 2$ we characterise the typical structure of undirected graphs with vertices $1, ..., n$ and maximum degree $R$, as $n$ tends to infinity. The information is used to prove that such graphs satisfy a labelled…

We present a new notion of limits of weighted directed graphs of growing size based on convergence of their random quotients. These limits are specified in terms of random exchangeable measures on the unit square. We call our limits…

The on-line nearest-neighbour graph on a sequence of $n$ uniform random points in $(0,1)^d$ ($d \in \N$) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this…

Consider a `dense' Erd\H{o}s--R\'enyi random graph model $G=G_{n,M}$ with $n$ vertices and $M$ edges, where we assume the edge density $M/\binom{n}{2}$ is bounded away from 0 and 1. Fix $k=k(n)$ with $k/n$ bounded away from 0 and~1, and let…

We get central limit type theorems for the total number of edges in the generalized random graphs with random vertex weights under different moment conditions on distributions of the weights.

In the on-line nearest-neighbour graph (ONG), each point after the first in a sequence of points in R^d is joined by an edge to its nearest-neighbour amongst those points that precede it in the sequence. We study the large-sample asymptotic…

In this paper, we study two examples of minimum weight random graphs with edge constraints. First we consider the complete graph on ${n}$ vertices equipped with uniformly heavy edge weights and use iteration methods to obtain deviation…

Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the…

In this paper, we investigate bounds for the following judicious $k$-partitioning problem: Given an edge-weighted graph $G$, find a $k$-partition $(V_1,V_2,\dots ,V_k)$ of $V(G)$ such that the total weight of edges in the heaviest induced…

To each edge (i,j), i<j of the complete directed graph on the integers we assign unit weight with probability p or weight x with probability 1-p, independently from edge to edge, and give to each path weight equal to the sum of its edge…

We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together…

Suppose that under the action of gravity, liquid drains through the unit $d$-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal…

We consider the class of directed graphs with $N\geq 1$ edges and without loops shorter than $k\geq1$. Using the concept of a labelled graph, we determine graphs from this class that maximize the number of all paths of length $k$. Then we…

We prove a central limit theorem for a certain class of functions on sparse rank-one inhomogeneous random graphs endowed with additional i.i.d. edge and vertex weights. Our proof of the central limit theorem uses a perturbative form of…