Many classic networks grow by hooking small components via vertices. We introduce a class of networks that grows by fusing the edges of a small graph to an edge chosen uniformly at random from the network. For this random edge-hooking network, we study the local degree profile, that is, the evolution of the average degree of a vertex over time. For a special subclass, we further determine the exact distribution and an asymptotic gamma-type distribution. We also study the “core,” which consists of the well-anchored edges that experience fusing. A central limit theorem emerges for the size of the core.

At the end, we look at an alternative model of randomness attained by preferential hooking, favoring edges that experience more fusing. Under preferential hooking, the core still follows a Gaussian law but with different parameters. Throughout, Pólya urns are systematically used as a method of proof.

We investigate vertex levels of containment in a random hypergraph grown in the spirit of a recursive tree. We consider a local profile tracking the evolution of the containment of a particular vertex over time, and a global profile concerned with counts of the number of vertices of a particular containment level.

For the local containment profile, we obtain the exact mean, variance, and probability distribution in terms of standard combinatorial quantities such as generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: the early vertices have an asymptotically normal distribution, intermediate vertices have a Poisson distribution, and late vertices have a degenerate distribution.

As for the global containment profile, we establish an asymptotically normal distribution for the number of vertices at the smallest containment level as well as their covariances with the number of vertices at the second smallest containment level and the variances of these numbers. The orders in the variance–covariance matrix establish concentration laws.

A hooking network is built by stringing together components randomly chosen from a set of building blocks (graphs with hooks). The vertices are endowed with “affinities” which dictate the attachment mechanism. We study the distance from the master hook to a node in the network chosen according to its affinity after many steps of growth. Such a distance is commonly called the depth of the chosen node. We present an exact average result and a rather general central limit theorem for the depth. The affinity model covers a wide range of attachment mechanisms, such as uniform attachment and preferential attachment, among others. Naturally, the limiting normal distribution is parametrized by the structure of the building blocks and their probabilities. We also take the point of view of a visitor uninformed about the affinity mechanism by which the network is built. To explore the network, such a visitor chooses the nodes uniformly at random. We show that the distance distribution under such a uniform choice is similar to the one under random choice according to affinities.

We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

For integers l ≥ 1, d ≥ 0 we study (undirected) graphs with vertices 1, …, n such that the vertices can be partitioned into l parts such that every vertex has at most d neighbours in its own part. The set of all such graphs is denoted Pn (l, d). We prove a labelled first-order limit law, i.e., for every first-order sentence φ, the proportion of graphs in Pn (l, d) that satisfy φ converges as n → ∞. By combining this result with a result of Hundack, Prömel and Steger [12] we also prove that if 1 ≤ s1 ≤ … ≤ sl are integers, then Forb() has a labelled first-order limit law, where Forb() denotes the set of all graphs with vertices 1, …, n, for some n, in which there is no subgraph isomorphic to the complete (l + 1 )-partite graph with parts of sizes 1, s1, …, sl. In the course of doing this we also prove that there exists a first-order formula ξ, depending only on l and d, such that the proportion of ∈ Pn (l, d) with the following property approaches 1 as n → ∞: there is a unique partition of {1, …, n} into l parts such that every vertex has at most d neighbours in its own part, and this partition, viewed as an equivalence relation, is defined by ξ.

In a stationary ergodic process, clustering is defined as the tendency of events to appear in series of increasedfrequency separated by longer breaks. Such behavior, contradicting the theoretical “unbiased behavior” with exponential distribution of the gaps between appearances, is commonly observed in experimental processes and often difficult to explain. In the last section we relate one such empirical example of clustering, in the area of marine technology. In the theoretical part of the paper we prove, using ergodic theory and the notion of category, that clustering (even very strong) is in fact typical for “rare events” defined as long cylinder sets in processes generated by a finite partition of an arbitrary (infinite aperiodic) ergodic measure preserving transformation.

The problem of estimating an unknown compact convex set K in the plane, from a sample (X1,···,Xn) of points independently and uniformly distributed over K, is considered. Let Kn be the convex hull of the sample, Δ be the Hausdorff distance, and Δn := Δ (K, Kn). Under mild conditions, limit laws for Δn are obtained. We find sequences (an), (bn) such that

(Δn - bn)/an → Λ (n → ∞), where Λ is the Gumbel (double-exponential) law from extreme-value theory. As expected, the directions of maximum curvature play a decisive role. Our results apply, for instance, to discs and to the interiors of ellipses, although for eccentricity e < 1 the first case cannot be obtained from the second by continuity. The polygonal case is also considered.

Assume a population is distributed in an infinite lattice of colonies in a migration and random mating model in which all individuals are selectively equivalent. In one and two dimensions, the population tends in time to consolidate into larger and larger blocks, each composed of the descendants of a single initial individual. Let N(t) be the (random) size of a block intersecting a fixed colony at time t. Then E[N(t)] grows like √t in one dimension, t/log t in two, and t in three or more dimensions. On the other hand, each block by itself eventually becomes extinct. In two or more dimensions, we prove that N(t)/E[N(t)] has a limiting gamma distribution, and thus the mortality of blocks does not make the limiting distribution of N(t) singular. Results are proven for discrete time and sketched for continuous time.

If a mutation rate u > 0 is imposed, the ‘block structure' has an equilibrium distribution. If N(u) is the size of a block intersecting a fixed colony at equilibrium, then as u → 0 N(u)/E[N(u)] has a limiting exponential distribution in two or more dimensions. In biological systems u ≈ 10–6 is usually quite small.

The proofs are by using multiple kinship coefficients for a stepping stone population.

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.