Configuration Graphs Research Articles

The structure of a configuration graph with a normally distributed parameter of the power series distribution of vertex degrees

We consider configuration graphs with N vertices. The degrees of the vertices are independent random variables identically distributed according to the power law, with a positive parameter τ . They are equal to the number of vertex semiedges that are numbered in an arbitrary order. The graph is constructed by joining all of the semiedges pairwise equiprobably to form edges. 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. Let τ be a random variable following a truncated normal distribution on an arbitrary fixed finite interval. We obtained the limit distributions of the maximum vertex degree and the number of vertices with a given degree for various zones of N and n tendency to infinity.

On clustering of conditional configuration graphs

We consider configuration graphs with N vertices. The degrees of the vertices are independent identically distributed limited random variables. They are equal to the number of vertex semiedges that are numbered in an arbitrary order. The graph is constructed by joining all of the semiedges pairwise equiprobably to form edges. 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. An important characteristic of the topology of a graph is the local clustering coefficient. We obtained the limit distributions of the local clustering coefficient and the number of triangles of each vertex as N and n tend to infinity. We also considered the limit behaviour of their mathematical expectations.

Conditional configuration graphs with discrete power-law distribution of vertex degrees

The paper is concerned with configuration graphs with discrete power-law distribution of vertex degrees. The parameter of this distribution is a random variable, which is unknown except that it imposes relatively weak constraints on the asymptotic behaviour of the probabilities of large values of degrees. For such graphs with the known number of edges, we find the limiting distributions of the maximal degree of a vertex and of the number of vertices of a given degree for various laws of convergence to infinity of the numbers of vertices and edges. The results in the present paper, which are proved using the generalized scheme of allocation of particles to cells, demonstrate the potency of this method in the case of independent random variables with known limiting behaviour of the tail of the distribution. Bibliography: 13 titles.

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

Рассматриваются конфигурационные графы с $N$ вершинами. Степени вершин являются независимыми одинаково распределенными случайными величинами, имеющими степенное распределение с положительным параметром $\tau.$ Изучаются случайные графы при условии, что сумма степеней вершин не превосходит $n$, а параметр $\tau$ есть случайная величина, равномерно распределенная на интервале $[a,b], 0<a<b<\infty$. Найдены предельные распределения числа $\mu_r$ вершин степени $r$ при различных способах изменения $N,n$ и $r$.

Conditional configuration graphs with discrete power-law distribution of vertex degrees

Рассматриваются конфигурационные графы с дискретным степенным законом распределения степеней вершин. Параметр этого распределения является случайной величиной, о распределении которой известно только то, что оно обеспечивает достаточно слабые ограничения на асимптотическое поведение вероятностей больших значений степеней. Для таких графов при условии, что число ребер известно, найдены предельные распределения максимальной степени вершины и числа вершин заданной степени в различных зонах стремления числа вершин и числа ребер к бесконечности. Полученные результаты доказаны с помощью обобщенной схемы размещения частиц по ячейкам и демонстрируют возможность применения этой схемы для случая независимых случайных величин с известным предельным поведением хвоста распределения. Библиография: 13 названий.

A case of limit behaviour of vertex degrees in conditional configuration graphs

We consider configuration graphs with N vertices. The degrees of the vertices are independent identically distributed random variables according to power-law distribution with positive parameter . They are equal to the number of vertex’s semiedges that are numbered in an arbitrary order. The graph is constructed by joining all of the semiedges pairwise equiprobably to form edges. Such models can be used for describing different communication networks and Internet topology. We study the subset of random graphs under the condition that the sum of vertex degrees is known and it is equal to n. The properties of the graph depend on the behaviour of the parameter . We assume that is a random variable following uniform distribution on the interval [a; b]; 0 1

On limit behavior of maximum vertex degree in a conditional configuration graph near critical points

Abstract We consider configuration graphs with N vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with parameter τ > 0. There are two critical values of this parameter: τ = 1 and τ = 2. The properties of a graph change significantly when τ = τ(N) passes these points as N → ∞. Let G N,n be the subset of random graphs under the condition that sum of degrees of its vertices is equal to n. The limit theorem for the maximum vertex degree in G N,n as N,n → ∞ and τ → 1 or τ → 2 is proved.

Random Graphs' Robustness in Random Environment

We consider configuration graphs the vertex degrees of which are independent and follow the power-law distribution. Random graphs dynamics takes place in a random environment with the parameter of vertex degree distribution following uniform distributions on finite fixed intervals. As the number of vertices tends to infinity the limit distributions of the maximum vertex degree and the number of vertices with a given degree were obtained. By computer simulations we study the robustness of those graphs from the viewpoints of link saving and node survival in the two cases of the destruction process: the ``targeted attack'' and the ``random breakdown''. We obtained and compared the results under the conditions that the vertex degree distribution was averaged with respect to the distribution of the power-law parameter or that the values of the parameter were drawn from the uniform distribution separately for each vertex.

Construction of Directed Assortative Configuration Graphs

Constructions of directed configuration graphs based on a given bi-degree distribution were introduced in random graph theory some years ago. These constructions lead to graphs where the degrees of two nodes belonging to the same edge are independent. However, it is observed that many real-life networks are assortative, meaning that edges tend to connect low degree nodes with high degree nodes, or variations thereof. In this article we provide an explicit algorithm to construct directed assortative configuration graphs based on a given bi-degree distribution and an arbitrary pre-specified assortativity.

Limit distributions of the number of vertices with given degree in a conditional configuration graph

We consider configuration graphs with N vertices. The degrees of the vertices are independent identically distributed random variables according to power-law distribution. Node degrees form semiedges that are numbered in an arbitrary order. The graph is constructed by joining all the stubs pairwise equiprobably to form edges. Such models can be used for describing different communication networks and Internet topology. We study the subset of random graphs under the condition that the sum of vertex degrees is equal to n. The properties of the graph depend on the value of the parameter τ of the vertex degree distribution. Let µr be the number of vertices with degree r. We obtained the limit distributions of µr as N, n → ∞ with all possible values of r and τ . Also in our model the parameter τ can be changed together with N, n.

On conditional configuration graphs with random distribution of vertex degrees

We consider a configuration graph with N vertices. The degrees of the vertices are drawn independently from a discrete power-law distribution with positive parameter τ . They are equal to the number of each vertex’s numbered semiedges. The graph is constructed by joining all of the semiedges pairwise equiprobably to form edges. Research in the last years showed that configuration power-law random graphs with τ ∈ (1, 2) are deemed to be a good implementation of Internet topology. Such graphs could be used also for modeling forest fires as well as banking system defaults. But in these cases usually τ > 2. Parameter τ may depend on N and even be random. In the paper we consider configuration random graphs under the condition that the sum of vertex degrees is equal to n. Random graph dynamics as N → ∞ is assumed to take place in a random environment, where τ is a random variable following uniform distribution on the interval [a, b], 0 < a < b < ∞. We obtained the limit distributions of the maximum vertex degree and the number of vertices with a given degree as N, n → ∞.

On the limit distributions of the degrees of vertices in configuration graphs with a bounded number of edges

A model of a configuration graph on vertices is considered in which the degrees of the vertices are distributed identically and independently according to the law , , , and the number of edges is no greater than . Limit theorems for the number of vertices of a particular degree and for the maximum vertex degree as are established. Bibliography: 18 titles.

Configraphics: Graph Theoretical Methods for Design and Analysis of Spatial Configurations

This dissertation reports a PhD research on mathematical-computational models, methods, and techniques for analysis, synthesis, and evaluation of spatial configurations in architecture and urban design. Spatial configuration is a technical term that refers to the particular way in which a set of spaces are connected to one another as a network. Spatial configuration affects safety, security, and efficiency of functioning of complex buildings by facilitating certain patterns of movement and/or impeding other patterns. In cities and suburban built environments, spatial configuration affects accessibilities and influences travel behavioural patterns, e.g. choosing walking and cycling for short trips instead of travelling by cars. As such, spatial configuration effectively influences the social, economic, and environmental functioning of cities and complex buildings, by conducting human movement patterns. In this research, graph theory is used to mathematically model spatial configurations in order to provide intuitive ways of studying and designing spatial arrangements for architects and urban designers. The methods and tools presented in this dissertation are applicable in: arranging spatial layouts based on configuration graphs, e.g. by using bubble diagrams to ensure certain spatial requirements and qualities in complex buildings; and analysing the potential effects of decisions on the likely spatial performance of buildings and on mobility patterns in built environments for systematic comparison of designs or plans, e.g. as to their aptitude for pedestrians and cyclists. The dissertation reports two parallel tracks of work on architectural and urban configurations. The core concept of the architectural configuration track is the ‘bubble diagram’ and the core concept of the urban configuration track is the ‘easiest paths’ for walking and cycling. Walking and cycling have been chosen as the foci of this theme as they involve active physical, cognitive, and social encounter of people with built environments, all of which are influenced by spatial configuration. The methodologies presented in this dissertation have been implemented in design toolkits and made publicly available as freeware applications.

On the limit distributions of the degrees of vertices in configuration graphs with a bounded number of edges

Рассмотрена модель конфигурационного графа c $N$ вершинами, в котором степени вершин независимы и одинаково распределены по закону $\mathbf P\{\xi=k\}=k^{-\tau}-(k+1)^{-\tau}$, $k=1,2,…$, $\tau>0$, а число ребер не превосходит $n$. Доказаны предельные теоремы для числа вершин заданной степени и для максимальной степени вершины при $N,n\to\infty$. Библиография: 18 названий.

Configraphics

This dissertation reports a PhD research on mathematical-computational models, methods, and techniques for analysis, synthesis, and evaluation of spatial configurations in architecture and urban design. Spatial configuration is a technical term that refers to the particular way in which a set of spaces are connected to one another as a network. Spatial configuration affects safety, security, and efficiency of functioning of complex buildings by facilitating certain patterns of movement and/or impeding other patterns. In cities and suburban built environments, spatial configuration affects accessibilities and influences travel behavioural patterns, e.g. choosing walking and cycling for short trips instead of travelling by cars. As such, spatial configuration effectively influences the social, economic, and environmental functioning of cities and complex buildings, by conducting human movement patterns. In this research, graph theory is used to mathematically model spatial configurations in order to provide intuitive ways of studying and designing spatial arrangements for architects and urban designers. The methods and tools presented in this dissertation are applicable in: • arranging spatial layouts based on configuration graphs, e.g. by using bubble diagrams to ensure certain spatial requirements and qualities in complex buildings; and • analysing the potential effects of decisions on the likely spatial performance of buildings and on mobility patterns in built environments for systematic comparison of designs or plans, e.g. as to their aptitude for pedestrians and cyclists. The dissertation reports two parallel tracks of work on architectural and urban configurations. The core concept of the architectural configuration track is the ‘bubble diagram’ and the core concept of the urban configuration track is the ‘easiest paths’ for walking and cycling. Walking and cycling have been chosen as the foci of this theme as they involve active physical, cognitive, and social encounter of people with built environments, all of which are influenced by spatial configuration. The methodologies presented in this dissertation have been implemented in design toolkits and made publicly available as freeware applications.

On First-Order Logic and CPDA Graphs

Higher-order pushdown automata (n-PDA) are abstract machines equipped with a nested `stack of stacks of stacks'. Collapsible pushdown automata (n-CPDA) extend these devices by adding `links' to the stack and are equi-expressive for tree generation with simply typed ?Y terms. Whilst the configuration graphs of HOPDA are well understood, relatively little is known about the CPDA graphs. The order-2 CPDA graphs already have undecidable MSO theories but it was only recently shown by Kartzow (Log. Methods Comput. Sci. 9(1), 2013) that first-order logic is decidable at the second level. In this paper we show the surprising result that first-order logic ceases to be decidable at order-3 and above. We delimit the fragments of the decision problem to which our undecidability result applies in terms of quantifer alternation and the orders of CPDA links used. Additionally we exhibit a natural sub-hierarchy enjoying limited decidability.

Adaptive Exponential Synchronization of Coupled Complex Networks on General Graphs

We investigate the synchronization in complex dynamical networks, where the coupling configuration corresponds to a weighted graph. An adaptive synchronization method on general coupling configuration graphs is given. The networks may synchronize at an arbitrarily given exponential rate by enhancing the updated law of the variable coupling strength and achieve synchronization more quickly by adding edges to original graphs. Finally, numerical simulations are provided to illustrate the effectiveness of our theoretical results.

On the energy of singular graphs

The nullity, �(G), of a graph G is the algebraic multiplicity of the eigenvalue zero in the graph's spectrum. If �(G) > 0, then the graph G is said to be singular. The energy of a graph, E(G), was first defined by I. Gutman (1985) as the sum of the absolute values of the eigenvalues of the graph's adjacency matrix A(G). This paper considers the energy concept for singular graphs. In particular, it is proved that the change in energy upon the simple act of deleting a vertex is related to the type of vertices of the singular graph. Certain upper bounds are improved for the energy of the induced subgraph, G u, which is obtained by deleting vertex u, with the aid of a parameter known as the null spread of u, �u(G) = �(G) �(G u). Also, some new bounds are given for the energy of the minimal configuration graphs, an important class of singular graphs of nullity one that are related to the graph's core. Furthermore, certain graphs that increase their energy when an edge is deleted are considered, such as the complete multipartite graphs and the hypercubes of even dimensions.

Unambiguous Functions in Logarithmic Space

We investigate different variants of unambiguity in the context of computing multi-valued functions. We propose a modification to the standard computation models of Turing machines and configuration graphs, which allows for unambiguity-preserving composition. We define a notion of reductions (based on function composition), which allows nondeterminism but controls its level of ambiguity. In light of this framework we establish reductions between different variants of path counting problems. We obtain improvements of results related to inductive counting.

Probabilistic regular graphs

Deterministic graph grammars generate regular graphs, that form a structural extension of configuration graphs of pushdown systems. In this paper, we study a probabilistic extension of regular graphs obtained by labelling the terminal arcs of the graph grammars by probabilities. Stochastic properties of these graphs are expressed using PCTL, a probabilistic extension of computation tree logic. We present here an algorithm to perform approximate verification of PCTL formulae. Moreover, we prove that the exact model-checking problem for PCTL on probabilistic regular graphs is undecidable, unless restricting to qualitative properties. Our results generalise those of EKM06, on probabilistic pushdown automata, using similar methods combined with graph grammars techniques.