Related papers: Largest nearest-neighbour link and connectivity th…

Let $n$ points be placed independently in $d-$dimensional space according to the standard $d-$dimensional normal distribution. Let $d_n$ be the longest edge length for the nearest neighbor graph on these points. We show that \[\lim_{n \rar…

Let $X_1, \ldots, X_n$ be independent random points drawn from an absolutely continuous probability measure with density $f$ in $\mathbb{R}^d$. Under mild conditions on $f$, we derive a Poisson limit theorem for the number of large…

Consider a random uniform sample of $n$ points in a compact region $A$ of Euclidean $d$-space, $d \geq 2$, with a smooth or (when $d=2$) polygonal boundary. Fix $k \in {\bf N}$. Let $T_{n,k}$ be the threshold $r$ at which the geometric…

If we pick $n$ random points uniformly in $[0,1]^d$ and connect each point to its $k-$nearest neighbors, then it is well known that there exists a giant connected component with high probability. We prove that in $[0,1]^d$ it suffices to…

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…

Let $G=G_{n,k}$ denote the graph formed by placing points in a square of area $n$ according to a Poisson process of density 1 and joining each point to its $k$ nearest neighbours. Balister, Bollob\'as, Sarkar and Walters proved that if…

Let $X_1,X_2, \ldots $ be independent random uniform points in a bounded domain $A \subset \mathbb{R}^d$ with smooth boundary. Define the coverage threshold $R_n$ to be the smallest $r$ such that $A$ is covered by the balls of radius $r$…

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 $n$…

Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the inter-point distance is at most $r$. We show that as $n\to\infty$ the probability of full connectivity is…

We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when p = c log n/n. We draw n independent points X_i from a general distribution on a…

In a random key graph (RKG) of $n$ nodes each node is randomly assigned a key ring of $K_n$ cryptographic keys from a pool of $P_n$ keys. Two nodes can communicate directly if they have at least one common key in their key rings. We assume…

In a model of a connected network on random points in the plane, one expects that the mean length of the shortest route between vertices at distance $r$ apart should grow only as $O(r)$ as $r \to \infty$, but this is not always easy to…

Let $X_1,X_2,...$ be an infinite sequence of i.i.d. random vectors distributed exponentially with parameter $\lam .$ For each $y$ and $n\geq 1,$ form a graph $G_n(y)$ with vertex set $V_n = \{X_1,...,X_n\},$ two vertices are connected if…

We consider the typical distance between vertices of the giant component of a random intersection graph having a power law (asymptotic) vertex degree distribution with infinite second moment. Given two vertices from the giant component we…

In this paper, we study the online nearest neighbor random tree in dimension $d\in \mathbb N$ (called $d$-NN tree for short) defined as follows. We fix the torus $\mathbb T^d_n$ of dimension $d$ and area $n$ and equip it with the metric…

One-dimensional geometric random graphs are constructed by distributing $n$ nodes uniformly and independently on a unit interval and then assigning an undirected edge between any two nodes that have a distance at most $r_n$. These graphs…

Let $G_n$ be a random geometric graph with vertex set $[n]$ based on $n$ i.i.d.\ random vectors $X_1,\ldots,X_n$ drawn from an unknown density $f$ on $\R^d$. An edge $(i,j)$ is present when $\|X_i -X_j\| \le r_n$, for a given threshold…

This article discusses random hypergraphs with varying hyperedge sizes, admitting large hyperedges with size tending to infinity, and heavy-tailed limiting hyperedge size distributions. The main result describes a threshold for the random…

Let $\mu$ be a probability distribution on $\mathbb{R}^d$ which assigns measure zero to every hyperplane and $S$ a set of points sampled independently from $\mu$. What can be said about the expected combinatorial structure of the convex…

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…