Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path.

We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

The random triangle model was recently introduced as a random graph model that captures the property of transitivity that is often found in social networks, i.e. the property that given that two vertices are second neighbors, they are more likely to be neighbors. For parameters p ∊ [0,1] and q ≥ 1, and a finite graph G = (V, E), it assigns to elements η of {0,1}E probabilities which are proportional to where t(η) is the number of triangles in the open subgraph. In this paper the behavior of the random triangle model on the two-dimensional triangular lattice is studied. By mapping the system onto an Ising model with external field on the hexagonal lattice, it is shown that phase transition occurs if and only if p = (q−1)−2/3 and q > q c for a critical value q c which turns out to equal It is furthermore demonstrated that phase transition cannot occur unless p = p c (q), the critical value for percolation of open edges for given q. This implies that for q ≥ q c , p c (q) = (q−1)−2/3.

Let X i : i ≥ 1 be i.i.d. points in ℝd , d ≥ 2, and let T n be a minimal spanning tree on X 1,…,X n . Let L(X 1,…,X n ) be the length of T n and for each strictly positive integer α let N(X 1,…,X n ;α) be the number of vertices of degree α in T n . If the common distribution satisfies certain regularity conditions, then we prove central limit theorems for L(X 1,…,X n ) and N(X 1,…,X n ;α). We also study the rate of convergence for EL(X 1,…,X n ).

In a cubic multigraph certain restrictions on the paths are made to define what is called a railway. On the tracks in the railway (edges in the multigraph) an equivalence relation is defined. The number of equivalence classes induced by this relation is investigated for a random railway achieved from a random cubic multigraph, and the asymptotic distribution of this number is derived as the number of vertices tends to infinity.

We approach sphere of influence graphs (SIGs) from a probabilistic perspective. Ordinary SIGs were first introduced by Toussaint as a type of proximity graph for use in pattern recognition, computer vision and other low-level vision tasks. A random sphere of influence graph (RSIG) is constructed as follows. Consider n points uniformly and independently distributed within the unit square in d dimensions. Around each point,X i , draw an open ball (‘sphere of influence’) with radius equal to the distance to X i 's nearest neighbour. Finally, draw an edge between two points if their spheres of influence intersect. Asymptotically exact values for the expected number of edges in a RSIG are determined for all values of d; previously, just upper and lower bounds were known for this quantity. A modification of the Azuma-Hoeffding exponential inequality is employed to exhibit the sharp concentration of the number of edges around its expected value.

A random mapping (T n ;q) of a finite set V, V = {1,2,…,n}, into itself assigns independently to each i ∊ V its unique image j ∊ V with probability q if i = j and with probability P = (1-q)/(n−1) if i ≠ j. Three versions of epidemic processes on a random digraph G T representing (T n ;q) are studied. The exact probability distributions of the total number of infected elements as well as the threshold functions for these epidemic processes are determined.

Consider the basic location problem in which k locations from among n given points X 1,…,X n are to be chosen so as to minimize the sum M(k; X 1,…,X n ) of the distances of each point to the nearest location. It is assumed that no location can serve more than a fixed finite number D of points. When the X i , i ≥ 1, are i.i.d. random variables with values in [0,1]d and when k = ⌈n/(D+1)⌉ we show that

where α := α(D,d) is a positive constant, f is the density of the absolutely continuous part of the law of X 1, and c.c. denotes complete convergence.

The random triangle model is a Markov random graph model which, for parameters p ∊ (0,1) and q ≥ 1 and a graph G = (V,E), assigns to a subset, η, of E, a probability which is proportional to p |η|(1-p)|E|-|η| q t(η), where t(η) is the number of triangles in η. It is shown that this model has maximum entropy in the class of distributions with given edge and triangle probabilities.

Using an analogue of the correspondence between the Fortuin-Kesteleyn random cluster model and the Potts model, the asymptotic behavior of the random triangle model on the complete graph is examined for p of order n −α, α > 0, and different values of q, where q is written in the form q = 1 + h(n) / n. It is shown that the model exhibits an explosive behavior in the sense that if h(n) ≤ c log n for c < 3α, then the edge probability and the triangle probability are asymptotically the same as for the ordinary G(n,p) model, whereas if h(n) ≥ c' log n for c' > 3α, then these quantities both tend to 1. For critical values, h(n) = 3α log n + o(log n), the probability mass divides between these two extremes.

Moreover, if h(n) is of higher order than log n, then the probability that η = E tends to 1, whereas if h(n) = o(log n) and α > 2/3, then, with a probability tending to 1, the resulting graph can be coupled with a graph resulting from the G(n,p) model. In particular these facts mean that for values of p in the range critical for the appearance of the giant component and the connectivity of the graph, the way in which triangles are rewarded can only have a degenerate influence.

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are N i (0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, ηi = N i (∞), as a function of N i (0).

We are able to obtain, for some special graphs, the limiting distribution of N i if the total number of particles N → ∞ in such a way that the fraction, N i (0)/S = ξi , at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S 2, the two-leaf star which has three vertices and two edges.

Let n points be placed independently in ν-dimensional space according to the standard ν-dimensional normal distribution. Let M n be the longest edge-length of the minimal spanning tree on these points; equivalently let M n be the infimum of those r such that the union of balls of radius r/2 centred at the points is connected. We show that the distribution of (2 log n)1/2M n - b n converges weakly to the Gumbel (double exponential) distribution, where b n are explicit constants with b n ~ (ν - 1)log log n. We also show the same result holds if M n is the longest edge-length for the nearest neighbour graph on the points.

We develop a technique for establishing statistical tests with precise confidence levels for upper bounds on the critical probability in oriented percolation. We use it to give pc < 0.647 with a 99.999967% confidence. As Monte Carlo simulations suggest that pc ≈ 0.6445, this bound is fairly tight.

This article continues an investigation begun in [2]. A random graph Gn (x) is constructed on independent random points U 1, · ··, Un distributed uniformly on [0, 1]d , d ≧ 1, in which two distinct such points are joined by an edge if the l ∞-distance between them is at most some prescribed value 0 < x < 1.

Almost-sure asymptotic results are obtained for the convergence/divergence of the minimum vertex degree of the random graph, as the number n of points becomes large and the edge distance x is allowed to vary with n. The largest nearest neighbor link dn, the smallest x such that Gn (x) has no vertices of degree zero, is shown to satisfy Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be complete, a.s. These criteria imply a.s. limiting behavior of the diameter of the vertex set.

On independent random points U 1 ,· ··,Un distributed uniformly on [0, 1]d , a random graph Gn (x) is constructed in which two distinct such points are joined by an edge if the l ∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn } are provided which guarantee the random graph to be empty of edges, a.s.

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree Tn with n labeled nodes is a recursive tree if n = 1, or n > 1 and Tn can be constructed by joining node n to a node of some recursive tree Tn– 1. For arbitrary nodes i < n in a random recursive tree we give the exact distribution of Xi,n , the distance between nodes i and n. We characterize this distribution as the convolution of the law of Xi,j+ 1 and n – i – 1 Bernoulli distributions. We further characterize the law of Xi,j+ 1 as a mixture of sums of Bernoullis. For i = in growing as a function of n, we show that is asymptotically normal in several settings.

Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any non-negative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random p-vector, which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of p colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve for example, martingale, Markov chain or various mixing assumptions.

In this paper, we first give a characterization of a regular endomorphism of a graph. Then, we explicitly describe its inverses through idempotents.

The behaviour of the size of the largest complete subgraph contained in an overlap graph of a random acyclic digraph is presented.

Consider a forest of maple trees in autumn, with leaves falling on the ground. Those coming late cover the others below, so eventually the fallen leaves form a statistically homogeneous spatial pattern. In particular, the uncovered leaf boundaries form a mosaic. We formulate a mathematical model to describe this mosaic, firstly in the case where the leaves are polygonal and later for leaves with curved boundaries. Mean values of certain statistics of the mosaic are derived.

A simple identity for the incomplete factorial of sums of zero-one variables is exploited to provide the factorial moments of the number of components and the number of cyclical elements of the random mapping (T, {pi }) considered by Ross (1981).

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertex i to vertex j where i < j, the probability of ‘down arrows' from i to j where i ≥ j, and the probability of bidirectional arrows between i and j. In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.