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

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.

The structured coalescent is a continuous-time Markov chain which describes the genealogy of a sample of homologous genes from a subdivided population. Assuming this model, some results are proved relating to the genealogy of a pair of genes and the extent of subpopulation differentiation, which are valid under certain graph-theoretic symmetry and regularity conditions on the structure of the population. We first review and extend earlier results stating conditions under which the mean time since the most recent common ancestor of a pair of genes from any single subpopulation is independent of the migration rate and equal to that of two genes from an unstructured population of the same total size. Assuming the infinite alleles model of neutral mutation with a small mutation rate, we then prove a simple relationship between the migration rate and the value of Wright's coefficient F ST for a pair of neighbouring subpopulations, which does not depend on the precise structure of the population provided that this is sufficiently symmetric.

It is well known that the maximal displacement of a random walk indexed by an m-ary tree with bounded independent and identically distributed edge weights can reliably yield much larger asymptotics than a classical random walk whose summands are drawn from the same distribution. We show that, if the edge weights are mean-zero, then nonclassical asymptotics arise even when the tree grows much more slowly than exponentially. Our conditions are stated in terms of a Minkowski-type logarithmic dimension of the boundary of the tree.

The binary interval tree is a random structure that underlies interval division and parking problems. Five incomplete one-sided variants of binary interval trees are considered, providing additional flavors and variations on the main applications. The size of each variant is studied, and a Gaussian tendency is proved in each case via an analytic approach. Differential equations on half scale and delayed differential equations arise and can be solved asymptotically by local expansions and Tauberian theorems. Unlike the binary case, in an incomplete interval tree the size determines most other parameters of interest, such as the height or the internal path length.

In this paper, we introduce a compound random mapping model which can be viewed as a generalization of the basic random mapping model considered by Ross and by Jaworski. We investigate a particular example, the Poisson compound random mapping, and compare results for this model with results known for the well-studied uniform random mapping model. We show that, although the structure of the components of the random digraph associated with a Poisson compound mapping differs from the structure of the components of the random digraph associated with the uniform model, the limiting distribution of the normalized order statistics for the sizes of the components is the same as in the uniform case, i.e. the limiting distribution is the Poisson-Dirichlet (½) distribution on the simplex {{x i } : ∑ x i ≤ 1, x i ≥ x i+1 ≥ 0 for every i ≥ 1}.

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of Sbecomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

Uniform sequential tree-building aggregation of n particles is analyzed together with the effect of the avalanche that takes place when a subtree rooted at a uniformly chosen vertex is removed. For large n, the expected subtree size is found to be ≃ logn both for the tree of size n and the tree that remains after an avalanche. Repeated breakage-restoration cycles are seen to give independent avalanches which attain size k(1 ≤ k ≤ n-1) with probability (k(k+1))-1 and restored trees that are recursive.

Following Füredi and Komlós, we develop a graph theory method to study the high moments of large random matrices with independent entries. We apply this method to sparse N × N random matrices A N,p that have, on average, p non-zero elements per row. One of our results is related to the asymptotic behaviour of the spectral norm ∥A N,p ∥ in the limit 1 ≪ p ≪ N. We show that the value pc = log N is the critical one for lim ∥A N,p /√p∥ to be bounded or not. We discuss relations of this result with the Erdős–Rényi limit theorem and properties of large random graphs. In the proof, the principal issue is that the averaged vertex degree of plane rooted trees of k edges remains bounded even when k → ∞. This observation implies fairly precise estimates for the moments of A N,p . They lead to certain generalizations of the results by Sinai and Soshnikov on the universality of local spectral statistics at the border of the limiting spectra of large random matrices.