Since the paper of Kleinberg and Kleinberg, SODA'05, where it was proven that the preferential attachment random graph with degeneracy at least 3 does not obey the first order 0-1 law, no general methods were developed to study logical limit laws for recursive random graph models with arbitrary degeneracy. Even in the (possibly) simplest case of the uniform attachment, it is still not known whether the first order convergence law holds in this model. We prove that the uniform attachment random graph with bounded degrees obeys the first order convergence law. To prove the law, we describe dynamics of first order equivalence classes of the random graph using Markov chains. The convergence law follows from the existence of a limit distribution of the considered Markov chain. To show the latter convergence, we use stochastic approximation processes.

We study the asymptotic behavior of the maximum degree in the preferential attachment model with a choice-based edge step. We add vertex type to the model and prove, among others types of behavior, the effect of condensation on multiple vertices with different types.

We study the asymptotic behavior of the maximum degree in the geometric graph model with a preferential attachment choice-based edge step. Geometric graphs are natural models that describe some nanoscale systems, while preferential attachment provides a good description of complex networks, particularly different neural networks. The model is a recursively built sequence of graphs. We start with the initial graph on a single vertex and we add a new vertex and draw a few edges on each step. Each vertex is assigned a parameter that represents its location. The recursion step consists of two parts. First, we introduce a new vertex and draw edges to close enough vertices. This step represents the geometric part of the model. Then, we draw edges between vertices by preferential attachment with the choice rule. We prove that dependent on model parameters, the maximum degree could exhibit sublinear (similar to the standard preferential attachment) and linear (representing concentration effect) behavior.

In this paper, we prove the first-order convergence law for the uniform attachment random graph with almost all vertices having the same degree. In the considered model, vertices and edges are introduced recursively: at time m+1 we start with a complete graph on m+1 vertices. At step n+1 the vertex n+1 is introduced together with m edges joining the new vertex with m vertices chosen uniformly from those vertices of 1,…,n, whom degree is less then d=2m. To prove the law, we describe the dynamics of the logical equivalence class of the random graph using Markov chains. The convergence law follows from the existence of a limit distribution of the considered Markov chain.

We study logical limit laws for uniform attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time n+1, the vertex n+1 is introduced together with m edges joining the new vertex with m different vertices chosen uniformly at random from 1,…,n. We prove that this random graph obeys convergence law for first-order sentences with at most m−2 variables.

Исследуется асимптотическое поведение максимальной степени вершины в графе предпочтительного присоединения с выбором вершины, основанном как на ее степени, так и на дополнительном параметре (пригодности). Модели предпочтительного присоединения широко используются для моделирования сложных сетей (таких как нейронные сети и т.д.). Они строятся следующим образом. Мы начинаем с двух вершин и ребра между ними. Затем на каждом шаге мы рассматриваем выборку из уже существующих вершин, выбранных с вероятностями, пропорциональными их степеням плюс некоторый параметр β>- 1. Затем мы добавляем новую вершину и соединяем ее ребром с вершиной из выборки, на которой достигается максимум произведения ее степени на ее пригодность. Мы доказали, что в зависимости от параметров модели возможны три типа поведения максимальной степени вершины - сублинейное, линейное и порядка /ln , где n - число вершин в графе. We study the asymptotic behavior of the maximum degree in the preferential attachment tree model with a choice based on both the degree and fitness of a vertex. The preferential attachment models are natural models for complex networks (like neural networks, etc.) and constructed in the following recursive way. To each vertex is assigned a parameter that is called a fitness of a vertex. We start from two vertices and an edge between them. On each step, we consider a sample with repetition of d vertices, chosen with probabilities proportional to their degrees plus some parameter β>-1. Then we add a new vertex and draw an edge from it to the vertex from the sample with the highest product of fitness and degree. We prove that the maximum degree, dependent on parameters of the model, could exhibit three types of asymptotic behavior: sublinear, linear, and of /ln order, where n is the number of edges in the graph.

We prove the monadic second-order 0-1 law for two recursive tree models: uniform attachment tree and preferential attachment tree. We also show that the first order 0-1 law does not hold for non-tree uniform attachment models.

We prove almost sure convergence of the maximum degree in an evolving graph model combining a growing number of local choices with sublinear preferential attachment. At each step in the growth of the graph, a new vertex is introduced. Then we draw a random number of edges from it to existing vertices, chosen independently by the following rule. For each edge, we consider a sample of the growing size of vertices chosen with probabilities proportional to the sublinear function of their degrees. Then new vertex attaches to the vertex with the highest degree from the sample. Depending on the growth rate of the sample and the sublinear function, the maximum degree could be of the sublinear order, of the linear order or having almost all edges drawing to it. The prove using various stochastic approximation processes and a large deviation approach.

Samples of ceramics based on barium ferrite (FB) and sodium potassium niobate (KNN) are obtained and studied. The elemental composition of the obtained composite is analyzed. A study of pyroelectric properties and magnetic hysteresis loops showed that the composite based on barium ferrite and sodium potassium niobate ceramics obtained in this work had magnetic and ferroelectric properties.

We introduce a new type of preferential attachment tree that includes choices in its evolution, like with Achlioptas processes. At each step in the growth of the graph, a new vertex is introduced. Two possible neighbor vertices are selected independently and with probability proportional to degree. Between the two, the vertex with smaller degree is chosen, and a new edge is created. We determine with high probability the largest degree of this graph up to some additive error term.

We prove almost sure convergence of the maximum degree in an evolving tree model combining local choice and preferential attachment. At each step in the growth of the graph, a new vertex is introduced. A fixed, finite number of possible neighbors are sampled from the existing vertices with probability proportional to degree. Of these possibilities, the new vertex attaches to the vertex from the sample that has the highest degree. The maximal degree in this model has linear or near-linear behavior. This behavior contrasts sharply with the behavior in the same choice model with uniform attachment as well as the preferential attachment model without choice. The proof is based on showing the tree has a persistent hub by comparison with the standard preferential attachment model, as well as martingale and stochastic approximation arguments.

Optically transparent single crystals of strontium-barium niobate, SrxBa1-xNb2O6, of different compositions (x = 0.26...0.7) were examined with the aid of conoscopic light interference figures. A regular change of the isochrome concentric ring number and diameters consistent with the temperature variation of the value of birefringence is demonstrated by direct observations of polar cuts of optically uniaxial samples. Anomalous violations of the conventional (uniaxial) interference patterns occur occasionaly in some samples being indicative of the existence of biaxial trait in their behaviour even though no voltage is applied. These features may depend on annealing treatments at elevated temperatures. The results of the study show that conoscopic images may serve as a sensitive indicator of the structural state of SBN crystals related to the effects of stress?induced change of optical anisotropy and temperature dependent birefringence parameters.