We introduce and study a semi-random multigraph process, which forms no-replacement variant of the process that was introduced in [3]. The starts with an empty graph on vertex set [n]. For every positive integers q \(1\le r\le n\), \(((q-1)n+r)\)th round decision-maker, called Builder, is offered \(\pi _q(r)\), where _1, \pi _2, \ldots \) sequence permutations \(S_n\), chosen independently unif...

Gaussian processes have been successful in both supervised and unsupervised machine learning tasks, but their computational complexity has constrained practical applications. We introduce a new approximation for large-scale Gaussian processes, the Gaussian Process Random Field (GPRF), in which local GPs are coupled via pairwise potentials. The GPRF likelihood is a simple, tractable, and paralle...

The parabolic or forward scattering approximation to the equation describing wave propagation in a random medium leads to a stochastic partial differential equation which has the form of a random SchrOdinger equation. Existence, uniqueness and continuity of solutions to this equation are established. The resulting process is a Markov diffusion process on the unit sphere in complex Hilbert space...

Vertices of an affiliation network are linked to features and two vertices are declared adjacent whenever they share a common feature. We introduce a random intersection graph process aimed at modeling sparse evolving affiliation networks. We establish the asymptotic degree distribution and provide explicit asymptotic formulas for assortativity and clustering coefficients showing how these edge...

Given a random 3-uniform hypergraph H = H(n, p) on n vertices where each triple independently appears with probability p, consider the following graph process. We start with the star G0 on the same vertex set, containing all the edges incident to some vertex v0, and repeatedly add an edge xy if there is a vertex z such that xz and zy are already in the graph and xzy ∈ H. We say that the process...