Tatyana S. Turova

Diffusion Approximation for the Components in Critical Inhomogeneous Random Graphs of Rank 1.

Consider the random graph on n vertices 1,...,n. Each vertex i is assigned a type x(i) with x(1),...,x(n) being independent identically distributed as a nonnegative random variable X. We assume that EX3 < infinity. Given types of all vertices, an edge exists between vertices i and j independent of anything else and with probability min{1, x(i)x(j)/n (1 + a/n(1/3))}. We study the critical phase, which is known to take place when EX2 = 1. We prove that normalized by n(-2/3) the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion with diffusion coefficient root EXEX3 and drift a - EX3/EX s. In particular, we conclude that the size of the largest connected component is of order n(2/3). (c) 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 486-539, 2013 (Less)

Weak Convergence of the Hill Estimator Process

Let X 1 X 2,…, be a sequence of nonnegative i. i. d. random variables and for each n ≥ 1 let X 1, n ≤… ≤ Xn, n denote the order statistics based on the first n of these X’s. The Hill estimator is the sum of extreme values Σi≤kn )/k n , where k n → ∞ and k/ n →0, as n→ ∞. A weak convergence result is established for a process motivated by the Hill estimator, which we call the Hill estimator process.

On a phase diagram for random neural networks with embedded spike timing dependent plasticity

This paper presents an original mathematical framework based on graph theory which is a first attempt to investigate the dynamics of a model of neural networks with embedded spike timing dependent plasticity. The neurons correspond to integrate-and-fire units located at the vertices of a finite subset of 2D lattice. There are two types of vertices, corresponding to the inhibitory and the excitatory neurons. The edges are directed and labelled by the discrete values of the synaptic strength. We assume that there is an initial firing pattern corresponding to a subset of units that generate a spike. The number of activated externally vertices is a small fraction of the entire network. The model presented here describes how such pattern propagates throughout the network as a random walk on graph. Several results are compared with computational simulations and new data are presented for identifying critical parameters of the model.

Long Paths and Cycles in Dynamical Graphs

We study the large-time dynamics of a Markov process whose states are finite directed graphs. The number of the vertices is described by a supercritical branching process, and the edges follow a certain mean-field dynamics determined by the rates of appending and deleting. We find sufficient conditions under which asymptotically a.s. the order of the largest component is proportional to the order of the graph. A lower bound for the length of the longest directed path in the graph is provided as well. We derive an explicit formula for the limit as time goes to infinity, of the expected number of cycles of a given finite length. Finally, we study the phase diagram.

Synchronization of firing times in a stochastic neural network model with excitatory connections

We investigate a finite, stochastic, completely neural network model with excitatory couplings. The dynamics of the moments of firing in the net is described by a Markov chain. We derive exponential bounds for its transition probabilities. Moreover, the exponential fast synchronization of the moments of firing is proved. The results are illustrated by computer simulations.

A dynamic network in a dynamic population: asymptotic properties

We derive asymptotic properties for a stochastic dynamic network model in a stochastic dynamic population. In the model, nodes give birth to new nodes until they die, each node being equipped with ...

The largest component in subcritical inhomogeneous random graphs

We study the ‘rank 1 case’ of the inhomogeneous random graph model. In the subcritical case we derive an exact formula for the asymptotic size of the largest connected component scaled to log n. This result complements the corresponding known result in the supercritical case. We provide some examples of applications of the derived formula.

Bootstrap Percolation on a Graph with Random and Local Connections

Let

The Ising Model on the Random Planar Causal Triangulation: Bounds on the Critical Line and Magnetization Properties

G_{n,p}^1