Extinction scenarios in evolutionary processes: a multinomial Wright-Fisher approach.

We study a discrete-time multi-type Wright-Fisher population process. The mean-field dynamics of the stochastic process is induced by a general replicator difference equation. We prove several results regarding the asymptotic behavior of the model, focusing on the impact of the mean-field dynamics on it.

SHIFTING POWERS IN SPIVEY'S BELL NUMBER FORMULA.

In this paper, we consider extensions of Spivey's Bell number formula wherein the argument of the polynomial factor is translated by an arbitrary amount. This idea is applied more generally to the -Whitney numbers of the second kind, denoted by (), where some new identities are found by means of algebraic and combinatorial arguments. The former makes use of infinite series manipulations and Dobinski-like formulas satisfied by (), whereas the latter considers distributions of certain statistics on the underlying enumerated class of set partitions.

AVALANCHES IN A SHORT-MEMORY EXCITABLE NETWORK.

We study propagation of avalanches in a certain excitable network. The model is a particular case of the one introduced in [24], and is mathematically equivalent to an endemic variation of the Reed-Frost epidemic model introduced in [28]. Two types of heuristic approximation are frequently used for models of this type in applications, a branching process for avalanches of a small size at the beginning of the process and a deterministic dynamical system once the avalanche spreads to a significant fraction of a large network.

Horizontal visibility graph of a random restricted growth sequence.

We study the distributional properties of horizontal visibility graphs associated with random restrictive growth sequences and random set partitions of size . Our main results are formulas expressing the expected degree of graph nodes in terms of simple explicit functions of a finite collection of Stirling and Bernoulli numbers.

Staircase patterns in words: subsequences, subwords, and separation number.

We revisit staircases for words and prove several exact as well as asymptotic results for longest left-most staircase subsequences and subwords and staircase separation number. The latter is defined as the number of consecutive maximal staircase subwords packed in a word. We study asymptotic properties of the sequence (), the number of -array words with separations over alphabet [] and show that for any ≥ 0, the growth sequence ( ,()) converges to a characterized limit, independent of .

Finite automata, probabilistic method, and occurrence enumeration of a pattern in words and permutations.

The main theme of this paper is the enumeration of the order-isomorphic occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence , the number of -array -ary words that contain a given pattern exactly times. In addition, we study the asymptotic behavior of the random variable , the number of pattern occurrences in a random -array word.

Random walk on the Poincaré disk induced by a group of Möbius transformations.

We consider a discrete-time random motion, Markov chain on the Poincaré disk. In the basic variant of the model a particle moves along certain circular arcs within the disk, its location is determined by a composition of random Möbius transformations. We exploit an isomorphism between the underlying group of Möbius transformations and to study the random motion through its relation to a one-dimensional random walk.

On ballistic deposition process on a strip.

We revisit the model of the ballistic deposition studied in [5] and prove several combinatorial properties of the random tree structure formed by the underlying stochastic process. Our results include limit theorems for the number of roots and the empirical average of the distance between two successive roots of the underlying tree-like structure as well as certain intricate moments calculations.

On the optimal convergence probability of univariate estimation of distribution algorithms.

In this paper we obtain bounds on the probability of convergence to the optimal solution for the compact genetic algorithm (cGA) and the population based incremental learning (PBIL). Moreover, we give a sufficient condition for convergence of these algorithms to the optimal solution and compute a range of possible values for algorithm parameters at which there is convergence to the optimal solution with a predefined confidence level.

A step forward in studying the compact genetic algorithm.

The compact Genetic Algorithm (cGA) is an Estimation of Distribution Algorithm that generates offspring population according to the estimated probabilistic model of the parent population instead of using traditional recombination and mutation operators. The cGA only needs a small amount of memory; therefore, it may be quite useful in memory-constrained applications. This paper introduces a theoretical framework for studying the cGA from the convergence point of view in which, we model the cGA by a Markov process and approximate its behavior using an Ordinary Differential Equation (ODE).