If we pick random points uniformly in [0, 1] and connect each point to its log -nearest neighbors, where ≥ 2 is the dimension and is a constant depending on the dimension, then it is well known that the graph is connected with high probability. We prove that it suffices to connect every point to log log points chosen randomly among its log -nearest neighbors to ensure a giant component of size - () with high probability. This construction yields a much sparser random graph with ~ log log instead of ~ log edges that has comparable connectivity properties. This result has nontrivial implications for problems in data science where an affinity matrix is constructed: instead of connecting each point to its nearest neighbors, one can often pick ' ≪ random points out of the nearest neighbors and only connect to those without sacrificing quality of results. This approach can simplify and accelerate computation; we illustrate this with experimental results in spectral clustering of large-scale datasets.

We consider a class of Sevastyanov branching processes with non-homogeneous Poisson immigration. These processes relax the assumption required by the Bellman-Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper, we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include novel LLN and CLT which emerge from the non-homogeneity of the immigration process.

"Let X(t) be a non-homogeneous birth and death process. In this paper we develop a general method of estimating bounds for the state probabilities for X(t), based on inequalities for the solutions of the forward Kolmogorov equations."

"A Malthusian parameter for the generation-dependent general branching process is introduced and some asymptotics of the expected population size, counted by a general non-negative characteristic, are discussed. Processes both with and without immigration are considered."

"Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections...are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations."

"The kin number problem in its simplest form is that of the relationship between sibship sizes and offspring numbers.... Further studies have been made, concerning relatives of other degrees of affinity than siblings, but these did not usually yield joint distributions. Recently this aspect of the problem has been studied in the framework of a Galton-Watson process.... In these studies the population is treated as monotype. Applications such as pedigree studies of diseases require a multitype approach (in the example, two types: victims and others). In this paper such a study is undertaken." Problems and questions involved in this type of approach are discussed, and joint distributions are obtained under a number of sampling schemes.

"In the familiar immigration-birth-death process the events of immigration, birth and death relate to the individual. There are processes in which the whole family and not just an individual migrates. Such population growth models are studied in some detail." Both human and animal populations are considered.

"The distribution of the maximum and the extinction probability for a Markovian population is derived. Asymptotic growth is described, using the sequence of sojourn times. A regularity criterion for the processes under consideration exists under certain assumptions. For a class of processes with specific population-dependent transition rates the asymptotic behaviour is given explicitly."

"A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process."