Configuration Graphs Research Articles

On Connectivity of Conditional Configuration Graphs under Destruction

Configuration graphs with node degrees being independent identically distributed random variables following the power-law distribution with a shift are studied. We consider conditional graphs that consist of only one connected component. Computer simulations are used to study how the connectivity of such graphs changes under the two types of destruction processes: the “target attack” and the “random breakdown”. We propose models for the dependence of the probability of graph connectivity destruction on the graph size, the node degree distribution parameter, and the number of deleted nodes.

Minimum and maximum of the average distance in power-law configuration graphs

The article studies configuration graphs with a discrete fixed-parameter power-law distribution of node degrees. Simulation is applied to estimate the average distance (the arithmetic mean of all distances between all pairs of graph vertices), and two characteristics are considered: the minimum and the maximum average distance in the graph. The dependences of these characteristics on the graph size and the vertex degree distribution parameter are obtained and the empirical confidence interval of the average distance in the power-law configuration graph is constructed

Sizes of Trees in a Random Forest and Configuration Graphs

Sizes of Trees in a Random Forest and Configuration Graphs

Recent Advances in Positive-Instance Driven Graph Searching

Research on the similarity of a graph to being a tree—called the treewidth of the graph—has seen an enormous rise within the last decade, but a practically fast algorithm for this task has been discovered only recently by Tamaki (ESA 2017). It is based on dynamic programming and makes use of the fact that the number of positive subinstances is typically substantially smaller than the number of all subinstances. Algorithms producing only such subinstances are called positive-instance driven (PID). The parameter treedepth has a similar story. It was popularized through the graph sparsity project and is theoretically well understood—but the first practical algorithm was discovered only recently by Trimble (IPEC 2020) and is based on the same paradigm. We give an alternative and unifying view on such algorithms from the perspective of the corresponding configuration graphs in certain two-player games. This results in a single algorithm that can compute a wide range of important graph parameters such as treewidth, pathwidth, and treedepth. We complement this algorithm with a novel randomized data structure that accelerates the enumeration of subproblems in positive-instance driven algorithms.

The maximum tree of a random forest in the configuration graph

Galton-Watson random forests with a given number of root trees and a known number of nonroot vertices are investigated. The distribution of the number of direct offspring of each particle in the forest- generating process is assumed to have infinite variance. Branching processes of this kind are used successfully to study configuration graphs aimed at simulating the structure and development dynamics of complex communication networks, in particular the internet. The known relationship between configuration graphs and random forests reflects the local tree structure of simulated networks. Limit theorems are proved for the maximum size of a tree in a random forest in all basic zones where the number of trees and the number of vertices tend to infinity. Bibliography: 14 titles.

Asymptotics of the number of edges of an internet graph

We consider configuration graphs with N vertices. The degrees of the vertices are independent and identically distributed according to a distribution law that depends on an unknown slowly varying function. The degree of a vertex has a finite expectation and an infinite variance. Such models can be used to describe various communication networks and Internet topologies. The paper proves the local limit theorem for the number of edges in a graph as N → ∞.

Diameter of power-law configuration graphs

We consider random configuration graphs with a power-law distribution of vertex degrees. The distribution parameter takes fixed values. Estimates of the diameter of the graph were found by simulations. Relationships are suggested for the dependence of the graph diameter on its volume and the vertex degree distribution parameter.

On the connectivity of configuration graphs

AbstractWe consider configuration graphs with vertex degrees being independent identically distributed random variables. The distribution of these variables satisfies only relatively weak constraints on the probabilities of large values of degrees. For the case when the number of vertices tends to infinity, the conditions are found under which the graph is asymptotically almost surely connected. We also give estimates of the rate of convergence to zero of the probability that the graph is not connected.

Уровневая периодизация онтогенеза. Становление системы ведущих функций для произвольного уровня организации.

The article discusses the multi-level periodisation of ontogeny, namely the becoming of a system of leading functions for an arbitrary level of organisation. The adjective arbitrary is understood as a level in which everything that is repeated in other levels is detailed, but any of their specifics is quite omitted. As for the concept of becoming, its content is enhanced by several intra-level layers added to the upper one. And finally, becoming itself is derived from the configuration of cycles, which organizes the interaction of intra-level layers and determines the level system of leading functions. The proposed analysis scheme is declared as a basis for describing each level of ontogeny organisation. The previously published part of the article includes two sections. In the first one, “Arbitrary scheme of level periodization”, definitions and descriptions of its basic concepts are given: a) the level of organisation, the formation of level and the initial function, b) layers of level and the system of leading functions, c) the hierarchy of layers and cycles of interaction, d) leading functions and developmental periods. The second section, “Configuration of interaction cycles”, expands the cycles’ configuration graphs and considers the layers of organisation that constitute them. This second part of the article completes the work presented in the first one. This is indicated by the title of its only section “Becoming of a system of leading functions”, declared as key expression in the title of both parts of the article. Therefore, the second of them begins with the classical (L. Vygotsky) introduction “Structure and dynamics of development”, which is further turned into an analysis of the interaction cycles of intra-level layers with a description of the corresponding leading function.

On conditions of configuration graphs’ connectivity

We consider two types of random configuration graphs: with the power-law and with the Poisson vertex degree distributions. The parameters of these distributions are fixed. By simulations we estimate the probabilities of graph connectivity (when all graph vertices are joined into one connected component) in different graph types and their dependence on the graph size and the vertex degree distribution parameter.

On the maximum vertex degree in a conditional configuration graph

We consider configuration graphs under the condition that the sum of vertex degrees is bounded from above by n. The vertex degrees are independent identically distributed random variables. The distribution of the vertex degree ξ is unknown and has the only limiting condition that P{ξ=k} пропорционально d/( k g ln h k), where d>0, g>1, h≥0. We obtained the limit distributions of the maximum vertex degree as the number of graph vertices and n tends to infinity.

On the asymptotics of the degree structure of conditional internet graphs

We consider configuration graphs with N vertices, whose degrees are independent and identically distributed according to an unknown distribution law that depends on an arbitrary slowly varying function. The degree of each vertex is equal to the number of incident numbered semi-edges. The graph is constructed by joining all of the semi-edges pairwise equiprobably to form edges. Such models can be used to adequately describe various communication networks and Internet topologies. We study the subset of such random graphs under the condition that the sum of vertex degrees is known and it is equal to n. The paper finds the limit distributions of the maximum vertex degree and the number vertices with a given degree as N, n → ∞ in such a way that 0 < C 1 ≤ n/N ≤ C 2 < ∞.

CABE: A Cloud-Based Acoustic Beamforming Emulator for FPGA-Based Sound Source Localization.

Microphone arrays are gaining in popularity thanks to the availability of low-cost microphones. Applications including sonar, binaural hearing aid devices, acoustic indoor localization techniques and speech recognition are proposed by several research groups and companies. In most of the available implementations, the microphones utilized are assumed to offer an ideal response in a given frequency domain. Several toolboxes and software can be used to obtain a theoretical response of a microphone array with a given beamforming algorithm. However, a tool facilitating the design of a microphone array taking into account the non-ideal characteristics could not be found. Moreover, generating packages facilitating the implementation on Field Programmable Gate Arrays has, to our knowledge, not been carried out yet. Visualizing the responses in 2D and 3D also poses an engineering challenge. To alleviate these shortcomings, a scalable Cloud-based Acoustic Beamforming Emulator (CABE) is proposed. The non-ideal characteristics of microphones are considered during the computations and results are validated with acoustic data captured from microphones. It is also possible to generate hardware description language packages containing delay tables facilitating the implementation of Delay-and-Sum beamformers in embedded hardware. Truncation error analysis can also be carried out for fixed-point signal processing. The effects of disabling a given group of microphones within the microphone array can also be calculated. Results and packages can be visualized with a dedicated client application. Users can create and configure several parameters of an emulation, including sound source placement, the shape of the microphone array and the required signal processing flow. Depending on the user configuration, 2D and 3D graphs showing the beamforming results, waterfall diagrams and performance metrics can be generated by the client application. The emulations are also validated with captured data from existing microphone arrays.

On the asymptotics of degree structure of configuration graphs with bounded number of edges

Abstract We consider configuration graphs with N vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with positive parameter $\tau .$We study properties of random graphs such that the sum of vertex degrees does not exceed n and the parameter is a random variable uniformly distributed on the interval $\left[ a,\,\,b \right],0&lt;a&lt;b&lt;\infty .$We find limit distributions of the number ${{\mu }_{r}}$of vertices with degree r for various types of variation of N, n and r.

ON ASSORTATIVITY AND CLUSTERING OF A POWER-LAW CONFIGURATION GRAPH MODEL

The paper deals with configuration graphs with vertex degrees distributed independently according to the power law. The distribution parameter is either fixed or randomly distributed according to either uniform or truncated normal distribution on a predefined interval (a, b). The assortativity and clustering of these graphs are studied by computer simulation. The estimates of numerical structural characteristics are found, such as the assortativity and clustering coefficients. The obtained results are compared with those known from the literature for real world complex networks.

ON THE DISTRIBUTION OF THE SECOND DEGREES OF CONFIGURATION GRAPHS VERTICES

The object is configuration graphs with N vertices, numbered from 1 to N, whose vertex degrees are independent identically distributed random variables. The second degree η(2) of an arbitrary vertex A of a configuration graph is equal to the sum of the degrees of the vertices adjacent to the vertex A excluding the edges going to A. As N →∞, the form of the generating function and the distribution of the random variable η(2) are found for graphs whose vertex degrees have the Poisson distribution. Also, as N →∞, the form of the generating function of a random variable η(2) is obtained for graphs with the vertex degree distribution pk &gt; 0, k= 1, 2, . . . , such that pk ∼ d/kg( lnk)h, h 0, g&gt;7/3, d&gt;0, as k→∞.

ON THE MAXIMUM OF THE MODULARITY OF RANDOM CONFIGURATION GRAPHS

Configuration graphs with random independent identically distributed vertex degrees are considered. The degrees are equal to the number of vertex semiedges that are numbered in an arbitrary order. The graph is constructed by joining all of semiedges pairwise equiprobably to form edges. Such models can be used to adequately describe the topology of transport, electricity, social networks and the Internet. An important characteristic of the structure of a graph is its modularity. It is a measure for graph clustering in the case vertices are divided into groups (clusters). Graphs with high modularity have dense edges between the vertices within clusters but sparse connections between vertices of different clusters. The notion of modularity and its properties in random configuration graphs are discussed. The maximum modularity of a graph is used to describe the level of graph clustering and to find the best division of vertices. The limit theorem for the maximum modularity as the number of vertices tends to infinity is proved.

On the connectivity of configuration graphs

Рассматриваются конфигурационные графы, степени вершин которых являются независимыми одинаково распределенными случайными величинами. Распределение этих величин неизвестно, за исключением того, что заданы достаточно слабые ограничения на предельное поведение вероятностей больших значений степеней. Найдены условия, при выполнении которых граф асимптотически достоверно связан при стремлении числа вершин к бесконечности. Даны оценки скорости сходимости к нулю вероятности того, что граф не связен.

Limit distributions of maximum vertex degree in a conditional configuration graph

We consider configuration graphs with N vertices. The degrees of the vertices are independent identically distributed random variables following the power-law distribution with positive parameter τ. We study the random graphs under the conditions that the sum of vertex degrees does not exceed n and the parameter τ is a random variable uniformly distributed on the interval [a,b], 1≤a<b<∞. We obtain the limit distributions of maximum vertex degree for for different relations between the parameters N and n tending to infinity.

On comparing configuration graphs robustness in a random environment

We consider configuration graphs with vertex degrees distributed independently according to the power law, with a truncated parameter τ normally distributed on the interval (a, b). By computer simulation we study the robustness of such graphs to both random and targeted destructions in the cases of preserving graph connectivity and «forest fire». In the first case, the models of the dependence of the size of the giant component and the probability of graph destruction on the initial graph size N and the percentage of removed vertices were obtained. In the case of the forest fire, the main interest was in the number of vertices remaining after the fire. Along with the two «fire starts» (random ignition and targeted lightning-up of the node with the highest degree), we considered the two fire propagation processes: with a fixed probability of fire transition through a graph edge and a random one. For both types of external destructive impacts, we compared the new results with the ones obtained previously for the configuration graphs with a fixed parameter of node degree distribution and the one uniformly distributed on the interval [a, b].