Viewpoint Of Graph Theory Research Articles

Vulnerability analysis of multiprocessor system based on burnt pancake networks

With the large scale network’s popularity in different fields, such as cloud computing, privacy protection, social networks and so on, the robustness analysis of networks against faulty processors has recently attracted more attention. When some processors are attacked and lose function, how to characterize the communication ability and efficiency of the surviving network is very crucial. From the viewpoint of graph theory, the size of the maximal component after removing certain faulty vertices can be viewed as a metric of fault tolerability, which is reckoned as an important dimension for the robustness analysis of a network. The h-component connectivity of a network G, denoted by cκh(G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least h components. In this paper, we show that when removing any subset X with order of at most 5n−9 in the burnt pancake network BPn(n≥5), BPn∖X has the largest component C with |C|≥|V(BPn)|−|X|−4 vertices, and all small components present the following structures: (a) an empty graph; (b) one 3-path, one K1,3, one K1,2, or an edge, or an isolated vertex; (c) one K1,2 and an edge, one K1,2 and an isolated vertex, two edges, one edge and an isolated vertex, or two isolated vertices; (d) an edge and two isolated vertices, or three isolated vertices; (e) four isolated vertices. Besides, we determine that the 6-component connectivity of BPn is cκ6(BPn)=5n−6 for n≥5.

GIS-Analysis Of The Ural Power Grid Vulnerability To The Impact Of Sleet And Wind

In this paper, we describe an experiment of complex power grid structure and wind and sleet mapping of territory using two different network indices: standard edge betweenness centrality and new author’s index – electrical grid centrality. Such analysis of the network allows to identify power lines with high load which could be vulnerable elements of the power grid. It is very important for strategic planning of power grids to reduce the risk of accidents by distributing loads across several lines so that they will be able to reserve each other. As a case territory for this research, we took the Ural united power system in Russia which is greatly exposed to different sleet and wind according to the statistics of the power grid operator. The degree of natural hazard consequences could be compensated by the network structure through alternative paths of energy supply or vice versa – increased if they are absent. At the same time, in this paper we consider that power grids have their own features from the graph theory point of view, for example multiple (parallel) edges, branches, different types of vertices. The existing index of edge betweenness centrality does not perfectly cope with them. We compare two indices characterizing power line importance within the system – betweenness centrality and electrical grid centrality and analyze the network structure features together with the spatial distribution of sleet and wind. As a result, we could identify bottlenecks in the study network. According to this study the most vulnerable power lines were detected, for example 500 kV Iriklinskaya CHP – Gazovaya and 500 kV Yuzhnouralskaya CHP-2 – Shagol power lines, that supply big cities such as Chelyabinsk and Orenburg and a bunch of industries around them.

Yablo’s Paradox, the Liar, and Referential Contradictions from a Graph Theory Point of View

F -systems are useful digraphs to model sentences that predicate the falsity of other sentences. Paradoxes like the Liar and the one of Yablo can be analyzed with that tool to find graph-theoretic patterns. In this paper we studied this general model consisting of a set of sentences and the binary relation ‘... affirms the falsity of...’ among them. The possible existence of non-referential sentences was also considered. To model the sets of all the sentences that can jointly be valued as true we introduced the notion of conglomerate, the existence of which guarantees the absence of paradox. Conglomerates also enabled us to characterize referential contradictions, i.e., sentences that can only be false under a classical valuation due to the interactions with other sentences in the model. A Kripke-style fixed-point characterization of groundedness was offered, and complete (meaning that every sentence is deemed either true or false) and consistent (meaning that no sentence is deemed true and false) fixed points were put in correspondence with conglomerates. Furthermore, argumentation frameworks are special cases of F -systems. We showed the relation between local conglomerates and admissible sets of arguments and argued about the usefulness of the concept for the argumentation theory.

Controllability and observability of multi-agent systems with general linear dynamics under switching topologies

This paper investigates controllability and observability of multi-agent systems, in which all the agents adopt identical general linear dynamics and the interconnection topologies are switching. For controllability, criteria are established by virtue of the switching sequence and the constructed subspace sequence, respectively. Furthermore, controllability is considered from the viewpoint of graph theory, and distance partition and almost equitable partition are introduced into switching topologies to quantitatively analyse the controllable state set of the system. For observability, sufficient and/or necessary conditions are presented in terms of the system matrices and the associated invariant subspace. Finally, some numerical simulations are worked out to illustrate the theoretical results.

Kernel-Based Sensor Fusion With Application to Audio-Visual Voice Activity Detection

In this paper, we address the problem of multiple view data fusion in the presence of noise and interferences. Recent studies have approached this problem using kernel methods, by relying particularly on a product of kernels constructed separately for each view. From a graph theory point of view, we analyze this fusion approach in a discrete setting. More specifically, based on a statistical model for the connectivity between data points, we propose an algorithm for the selection of the kernel bandwidth, a parameter that, as we show, has important implications on the robustness of this fusion approach to interferences. Then, we consider the fusion of audio-visual speech signals measured by a single microphone and by a video camera pointed to the face of the speaker. Specifically, we address the task of voice activity detection, i.e., the detection of speech and nonspeech segments, in the presence of structured interferences such as keyboard taps and office noise. We propose an algorithm for voice activity detection based on the audio-visual signal. Simulation results show that the proposed algorithm outperforms competing fusion and voice activity detection approaches. In addition, we demonstrate that a proper selection of the kernel bandwidth indeed leads to improved performance.

Classification of real Bott manifolds and acyclic digraphs

We completely characterize real Bott manifolds up to affine diffeomorphism in terms of three simple matrix operations on square binary matrices obtained from strictly upper triangular matrices by permuting rows and columns simultaneously. We also prove that any graded ring isomorphism between the cohomology rings of real Bott manifolds with $\mathbb Z/2$ coefficients is induced by an affine diffeomorphism between the real Bott manifolds. Our characterization can also be described in terms of graph operations on directed acyclic graphs. Using this combinatorial interpretation, we prove that the decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. Finally, we produce some numerical invariants of real Bott manifolds from the viewpoint of graph theory and discuss their topological meaning. As a by-product, we prove that the toral rank conjecture holds for real Bott manifolds.

Protocols Design and Uncontrollable Topologies Construction for Multi-Agent Networks

This note investigates the controllability issues of multi-agent systems, where each node contains generic linear dynamics. First, a neighbor-based control protocol is proposed, under which it is shown that the controllability of a multi-agent system is solely decided by its communication topology structure. We then further consider the effects of communication topology on the controllability from a graph theory point of view by explicitly constructing topologies. The construction exhibits a partition of the designed graph with identified leaders under which the system is shown to be uncontrollable.

Structural Observability Analysis of Large Scale Systems Using Modelica and Python

State observability of dynamic systems is a notion which determines how well the states can be inferred from input-output data. For small-scale systems, observability analysis can be done manually, while for large-scale systems an automated systematic approach is advantageous. Here we present an approach based on the concept of structural observability analysis, using graph theory. This approach can be automated and applied to large-scale, complex dynamic systems modeled using Modelica. Modelica models are imported into Python via the JModelica.org-CasADi interface, and the Python packages NetworkX (for graph-theoretic analysis) and PyGraphviz (for graph layout and visualization) are used to analyze the structural observability of the systems. The method is demonstrated with a Modelica model created for the Copper production plant at Glencore Nikkelverk, Kristiansand, Norway. The Copper plant model has 39 states, 11 disturbances and 5 uncertain parameters. The possibility of estimating disturbances and parameters in addition to estimating the states are also discussed from the graph-theory point of view. All the software tools used on the analysis are freely available.

A Review on Applications of Graph Theory in Network Analysis of Biological Processes

Promising steps have been taken to shift the research focus from individual molecules to groups of molecules. Thanks to advances in high-throughput technologies, large-scale biological datasets have become available at an unprecedented rate. As a result, a systematic approach is needed to reveal design principles of biological processes. This is achievable through modeling molecular activities as a network. Researchers have focused on the structural and statistical properties of such networks to investigate features of biological processes. In this transition, fundamental concepts in graph theory help researchers in the analysis of biological networks. From the viewpoint of graph theory, network construction methods in conjunction with popular visualization techniques are discussed. We also study the modularity properties of biological networks using complex clustering and community detection algorithms. This article aims to provide a comprehensive review about numerous applications of graph theory concep...

Matroidal Structure of Rough Sets from the Viewpoint of Graph Theory

Constructing structures with other mathematical theories is an important research field of rough sets. As one mathematical theory on sets, matroids possess a sophisticated structure. This paper builds a bridge between rough sets and matroids and establishes the matroidal structure of rough sets. In order to understand intuitively the relationships between these two theories, we study this problem from the viewpoint of graph theory. Therefore, any partition of the universe can be represented by a family of complete graphs or cycles. Then two different kinds of matroids are constructed and some matroidal characteristics of them are discussed, respectively. The lower and the upper approximations are formulated with these matroidal characteristics. Some new properties, which have not been found in rough sets, are obtained. Furthermore, by defining the concept of lower approximation number, the rank function of some subset of the universe and the approximations of the subset are connected. Finally, the relationships between the two types of matroids are discussed, and the result shows that they are just dual matroids.

The structure of electrical networks: a graph theory based analysis

We study the vulnerability of electrical networks through structural analysis from a graph theory point of view. We measure and compare several important structural properties of different electrical networks, including a real power grid and several synthetic grids, as well as other infrastructural networks. The properties we consider include the minimum dominating set size, the degree distribution and the shortest path distribution. We also study the network vulnerability under attacks in terms of maximum component size, number of components and flow vulnerability. Our results suggest that all grids are more vulnerable to targeted attacks than to random attacks. We also observe that the electrical networks have low treewidth, which explains some of the vulnerability. We prove that with a small treewidth, a few important structural properties can be computed more efficiently.

TOPOLOGICAL PROPERTIES OF THE (n,k)-STAR GRAPH

The star graph, though an attractive alternative to the hypercube, has a major drawback in that the number of nodes for an n-star graph must be n!, and thus considerably limits the choice of the number of nodes in the graph. In order to alleviate this drawback, the arrangement graph was recently proposed as a generalization of the star graph topology. The arrangement graph provides more flexibility than the star graph in choosing the number of nodes, but the degree of the resulting network may be very high. To overcome that disadvantage, this paper presents another generalization of the star graph, called the (n,k)-star graph. We examine some topological properties of the (n,k)-star graph from the graph-theory point of view. It is shown that two different types of edges in the (n,k)-star prevent it from being edge-symmetric, but edges in each class are essentially symmetric with respect to each other. Also, the diameter and the exact average distance of the (n,k)-star graph are derived. In addition, the fault-diameter for the (n,k)-star graph is shown to be at most the fault-free diameter plus three.

A comparative study of topological properties of hypercubes and star graphs

Undertakes a comparative study of two important interconnection network topologies: the star graph and the hypercube, from the graph theory point of view. Topological properties are derived for the star graph and are compared with the corresponding properties of the hypercube. Among other results, the authors determine necessary and sufficient conditions for shortest path routing and characterize maximum-sized families of parallel paths between any two nodes of the star graph. These parallel paths are proven of minimum length within a small additive constant. They also define greedy and asymptotically balanced spanning trees to support broadcasting and personalized communication on the star graph. These results confirm the already claimed topological superiority of the star graph over the hypercube.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Topological properties of star graphs

Our purpose in the present paper is to investigate different topological properties of the recently introduced star graphs which are being viewed as attractive alternatives to n-cubes or hypercubes. These properties are interesting by themselves from a graph theory point of view as well as they have direct in generating vertex disjoint paths and minimal paths in a star graph. They can also be readily utilized to design routing algorithms and to compute contention and traffic congestion in networks that use star graphs as the underlying topology.

Harmonic measure of the planar Cantor set from the viewpoint of graph theory

This paper outlines a graph-theoretical approach to the study of the harmonic measure on the two-dimensional Cantor set. The Cantor set is regarded as the space of ends of a (nonplanar) graph with a tree-like structure. The method is based upon the combinatorics of the random walk with internal states induced on this graph by Brownian motion, and it could be used for numerical approximation.

Graphical properties of polyhexes: Perfect matching vector and forcing

From the viewpoint of graph theory and its applications, subgraphs of the tiling of the plane with unit squares have long been studied in statistical mechanics, In organic chemistry, a much more relevant case concerns subgraphs of the tiling with unit hexagons. Our purpose here is to take a mathematical view of such polyhex graphsG and study two novel concepts concerning perfect matchingsM. First, the forcing number ofM is the smallest number of edges ofM which are not contained in any other perfect matching ofG. Second, the perfect matching vector ofM is written (n3,n2,n1,n0), wherenk is the number of hexagons with exactlyk edges inM. We establish some initial results involving these two concepts and pose some questions.

Topological properties of supercube

The N-node Supercube is a new generalized version of Hypercube topology. Unlike the Binary Hypercube, the Supercube can be constructed for any number of nodes N. In addition, it maintains the connectivity and diameter properties of the corresponding hypercube. In this paper, we examine some topological properties of the Supercube from the graph-theory point of view.

Topological properties of hypercubes

The n-dimensional hypercube is a highly concurrent loosely coupled multiprocessor based on the binary n-cube topology. Machines based on the hypercube topology have been advocated as ideal parallel architectures for their powerful interconnection features. The authors examine the hypercube from the graph-theory point of view and consider those features that make its connectivity so appealing. Among other things, they propose a theoretical characterization of the n-cube as a graph and and show how to map various other topologies into a hypercube.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A type of centrality functions for a directed graph

AbstractIn an undirected graph obtained by abstracting traffic and communication networks, evaluation of the centrality of a vertex is interesting from the viewpoint of graph theory. the purpose of discussion of the centrality function is to provide a theoretical basis for solution of the problem. This paper is an attempt to extend the notion of the centrality function defined for the ujidirected graph to the directed graph. the existence conditions of the quasi‐stable and unstable graphs for the centrality functions with general coefficients are derived from the viewpoint of structural stability of the undirected graphs, for which the classification of the graphs is made by the shift of the center by adding an edge with the central vertex as an end point. Furthermore, the stability of the directed graphs is discussed.