The variable generalised stochastic epidemic model, which allows for variability in both the susceptibilities and infectivities of individuals, is analysed. A very different epidemic model which exhibits variable susceptibility and infectivity is the random-graph epidemic model. A suitable coupling of the two epidemic models is derived which enables us to show that, whilst the epidemics are very different in appearance, they have the same asymptotic final size distribution. The coupling provides a novel approach to studying random-graph epidemic models.

We consider growing random recursive trees in random environments, in which at each step a new vertex is attached (by an edge of random length) to an existing tree vertex according to a probability distribution that assigns the tree vertices masses proportional to their random weights. The main aim of the paper is to study the asymptotic behaviour of the distance from the newly inserted vertex to the tree's root and that of the mean numbers of outgoing vertices as the number of steps tends to ∞. Most of the results are obtained under the assumption that the random weights have a product form with independent, identically distributed factors.

An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.

In a recent paper, Kleinberg (2000) considered a small-world network model consisting of a d-dimensional lattice augmented with shortcuts. The probability of a shortcut being present between two points decays as a power, r -α, of the distance, r, between them. Kleinberg showed that greedy routeing is efficient if α = d and that there is no efficient decentralised routeing algorithm if α ≠ d. The results were extended to a continuum model by Franceschetti and Meester (2003). In our work, we extend the result to more realistic models constructed from a Poisson point process wherein each point is connected to all its neighbours within some fixed radius, and possesses random shortcuts to more distant nodes as described above.

We consider a family of long-range percolation models (G p )p>0 on ℤd that allow dependence between edges and have the following connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in G p has a power-law distribution; (ii) the graph distance between points x and y is bounded by a multiple of logpd logpd |x - y | with probability 1 - o(1); and (iii) an adversary can delete a relatively small number of nodes from G p (ℤd ∩ [0, n]d ), resulting in two large, disconnected subgraphs.

In Bhatt and Roy's minimal directed spanning tree construction for a random, partially ordered set of points in the unit square, all edges must respect the ‘coordinatewise’ partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power-weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary plus a contribution introduced by the boundary effects, which can be characterized by a fixed-point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects being dominant. In the critical case in which the weight is simple Euclidean length, both effects contribute significantly to the limit law.

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

The trie is a sort of digital tree. Ideally, to achieve balance, the trie should grow from an unbiased source generating keys of bits with equal likelihoods. In practice, the lack of bias is not always guaranteed. We investigate the distance between randomly selected pairs of nodes among the keys in a biased trie. This research complements that of Christophi and Mahmoud (2005); however, the results and some of the methodology are strikingly different. Analytical techniques are still useful for moments calculation. Both mean and variance are of polynomial order. It is demonstrated that the standardized distance approaches a normal limiting random variable. This is proved by the contraction method, whereby the limit distribution is shown to approach the fixed-point solution of a distributional equation in the Wasserstein metric space.

Random (pseudo)graphs G N with the following structure are studied: first, independent and identically distributed capacities Λi are drawn for vertices i = 1, …, N; then, each pair of vertices (i, j) is connected, independently of the other pairs, with E(i, j) edges, where E(i, j) has distribution Poisson(Λi Λj / ∑k=1 N Λk ). The main result of the paper is that when P(Λ1 > x) ≥ x −τ+1, where τ ∈ (2, 3), then, asymptotically almost surely, G N has a giant component, and the distance between two randomly selected vertices of the giant component is less than (2 + o(N))(log log N)/(-log (τ − 2)). It is also shown that the cases τ > 3, τ ∈ (2, 3), and τ ∈ (1, 2) present three qualitatively different connectivity architectures.

For n independent, identically distributed uniform points in [0, 1]d , d ≥ 2, let L n be the total distance from the origin to all the minimal points under the coordinatewise partial order (this is also the total length of the rooted edges of a minimal directed spanning tree on the given random points). For d ≥ 3, we establish the asymptotics of the mean and the variance of L n , and show that L n satisfies a central limit theorem, unlike in the case d = 2.

Two results are proved involving the quantitative illumination parameter B(d) of the unit ball of a d-dimensional normed space introduced by Bezdek (1992). The first is that B(d) = O(2dd2 log d). The second involves Steiner minimal trees. Let v(d) be the maximum degree of a vertex, and s(d) that of a Steiner point, in a Steiner minimal tree in a d-dimensional normed space, where both maxima are over all norms. Morgan (1992) conjectured that s(d) ≤ 2d, and Cieslik (1990) conjectured that v(d) ≤ 2(2d − 1). It is proved that s(d) ≤ v(d) ≤ B(d) which, combined with the above estimate of B(d), improves the previously best known upper bound v(d) < 3d.

Consider the random graph model of Barabási and Albert, where we add a new vertex in every step and connect it to some old vertices with probabilities proportional to their degrees. If we connect it to only one of the old vertices then this will be a tree. These graphs have been shown to have a power-law degree distribution, the same as that observed in some large real-world networks. We are interested in the width of the tree and we show that it is at the nth step; this also holds for a slight generalization of the model with another constant. We then see how this theoretical result can be applied to directory trees.

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

Let 𝓅 be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞.

In Bhatt and Roy's minimal directed spanning tree construction for n random points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for large n) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickman's function, and their properties are discussed in some detail.

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper we show that there exists a one-regular cubic graph of order 2p or 2p2 where p is a prime if and only if 3 is a divisor of p – 1 and the graph has order greater than 25. All of those one-regular cubic graphs are Cayley graphs on dihedral groups and there is only one such graph for each fixed order. Surprisingly, it can be shown that there is no one-regular cubic graph of order 4p or 4p2.

We study the asymptotic properties of a minimal spanning tree formed by n points uniformly distributed in the unit square, where the minimality is amongst all rooted spanning trees with a direction of growth. We show that the number of branches from the root of this tree, the total length of these branches, and the length of the longest branch each converges weakly. This model is related to the study of record values in the theory of extreme-value statistics and this relation is used to obtain our results. The results also hold when the tree is formed from a Poisson point process of intensity n in the unit square.

We consider a generalized stochastic epidemic on a Bernoulli random graph. By constructing the epidemic and graph in unison, the epidemic is shown to be a randomized Reed–Frost epidemic. Hence, the exact final-size distribution and extensive asymptotic results can be derived.

We consider a stochastic graph generated by a continuous-time birth-and-death process with exponentially distributed waiting times. The vertices are the living particles, directed edges go from mothers to daughters. The size and the structure of the connected components are investigated. Furthermore, the number of connected components is determined.