On maximal entries in the principal eigenvector of graphs

For an undirected, connected graph it is well known that an eigenvector belonging to the principal eigenvalue of G can be given such that all entries are positive. We ask whether this vector carries information on the structure of the graph and approach this question by investigating the values that can occur in its maximal entry.     

A note on the Estrada-Hatano communicability algorithm for detecting community structure in complex networks

Recently Estrada and Hatano proposed an algorithm for the detection of community structure in complex networks using the concept of network communicability. Here we amend this algorithm by eliminating the subjectivity of choosing degree of overlapping and including an additional check of the fitness of detected communities. We show that this amendment can detect some communities which remain undetected by Estrada and Hatano’s algorithm. For instance, let G(p, q) be a graph obtained from two cliques, K_p and K_q(p ? q ? 3), joined by a single edge. It is apparent that this graph contains two communities, namely the two cliques. However, Estrada and Hatano’s algorithm detects only K_q as a community when p is sufficiently larger than q. Our algorithm correctly detects both communities. Also, our method also finds the correct community structure in one of the classic benchmark networks, the Zachary karate club.     

Communicability graph and community structures in complex networks

We use the concept of the network communicability [E. Estrada, N. Hatano, Communicability in complex networks, Phys. Rev. E 77 (2008) 036111] to define communities in a complex network. The communities are defined as the cliques of a “communicability graph”, which has the same set of nodes as the complex network and links determined by the communicability function. Then, the problem of finding the network communities is transformed to an all-clique problem of the communicability graph. We discuss the efficiency of this algorithm of community detection. In addition, we extend here the concept of the communicability to account for the strength of the interactions between the nodes by using the concept of inverse temperature of the network. Finally, we develop an algorithm to manage the different degrees of overlapping between the communities in a complex network. We then analyze the USA airport network, for which we successfully detect two big communities of the eastern airports and of the western/central airports as well as two bridging central communities. In striking contrast, a well-known algorithm groups all but two of the continental airports into one community.     

Graphs for small multiprocessor interconnection networks

Let D be the diameter of a graph G and let λ₁ be the largest eigenvalue of its (0, 1)-adjacency matrix. We give a proof of the fact that there are exactly 69 non-trivial connected graphs with (D + 1)λ₁ ? 9. These 69 graphs all have up to 10 vertices and were recently found to be suitable models for small multiprocessor interconnection networks. We also examine the suitability of integral graphs to model multiprocessor interconnection networks, especially with respect to the load balancing problem. In addition, we classify integral graphs with small values of (D + 1)λ₁ in connection with the load balancing problem for multiprocessor systems.     

The graph coloring problem: A neuronal network approach

Solution of an optimization problem with linear constraints through the continuous Hopfield network (CHN) is based on an energy or Lyapunov function that decreases as the system evolves until a local minimum value is attained. This approach is extended in to optimization problems with quadratic constraints. As a particular case, the graph coloring problem (GCP) is analyzed. The mapping procedure and an appropriate parameter-setting procedure are detailed. To test the theoretical results, some computational experiments solving the GCP are shown.     

Fourth-generation warfare: Jihadist networks and percolation

This paper studies the approach to the fourth-generation warfare (4GW) paradigm from the perspective of physical and mathematical disciplines, through the interdisciplinary bridge offered by the analysis of complex networks. The study is within an emerging multidisciplinary field, Sociophysics, which attempts to apply statistical mechanics and the science of complex systems to predict human social behavior. The fourth-generation warfare concept is reviewed, and the war of the Jihadist Islam against the West will be contextualized as 4GW. The paradigm of complex systems has in diverse branches of science changed how collective phenomena are processed. The jihadist networks phenomenon in particular is appropriate for study from the standpoint of complex networks. We present an empirical study of the 9/11 and 11M networks, implemented from public information, and we give a comparison of both networks from the standpoint of complex networks. Several authors have made use of the phenomenon of percolation in complex physical systems to analyse complex networks, particularly jihadist actions like 9/11. The relationship between jihadist networks and percolation is considered. The percolation concept is reviewed and related to 4GW, and the definition of memetic dimension is introduced.     

WDM网络中的一个改进的最优半光通道路由算法

本文在一个限定的条件下，提出了一个WDM网络中的寻找最优半光通道算法，使时间复杂度从O（k^2n km knlog(kn))提高到O（k^2n km nlogn)。     

Distances in random graphs with infinite mean degrees

We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function is regularly varying with exponent . In particular, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed. The minimal number of edges between two arbitrary nodes, also called the graph distance or the hopcount, is investigated when the size of the graph tends to infinity. The paper is part of a sequel of three papers. The other two papers study the case where , and , respectively. The main result of this paper is that the graph distance for converges in distribution to a random variable with probability mass exclusively on the points and . We also consider the case where we condition the degrees to be at most for some , where is the size of the graph. For fixed and such that , the hopcount converges to in probability, while for , the hopcount converges to the same limit as for the unconditioned degrees. The proofs use extreme value theory. AMS 2000 Subject Classifications Primary—60G70; Secondary—05C80     

Relational dominant strategy equilibrium as a generalization of dominant strategy equilibrium in terms of a social psychological aspect of decision making

In this paper, incorporating a social psychological aspect of decision making, called attitudes, into analysis of games in normal form, the author proposes a new equilibrium concept, called relational dominant strategy equilibrium (RDSE). The author also analyzes relationships between RDSE and dominant strategy equilibrium (DSE), and shows the following fact: RDSE coincides with DSE under a condition on attitudes of players. The story of “The Gift of the Magi” by Henry is analyzed and given a convincing interpretation using the proposed equilibrium concept.     

A new closure concept preserving graph Hamiltonicity and based on neighborhood equivalence

A graph is Hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is Hamiltonian is known as an NP-complete problem, and no satisfactory characterization for these graphs has been found.In 1976, Bondy and Chvàtal introduced a way to get round the Hamiltonicity problem complexity by using a closure of the graph. This closure is a supergraph of G which is Hamiltonian iff G is. In particular, if the closure is the complete graph, then G is Hamiltonian. Since this seminal work, several closure concepts preserving Hamiltonicity have been introduced. In particular, in 1997, Ryjá?ek defined a closure concept for claw-free graphs based on local completion.Following a different approach, in 1974, Goodman and Hedetniemi gave a sufficient condition for Hamiltonicity based on the existence of a clique covering of the graph. This condition was recently generalized using the notion of Eulerian clique covering. In this context, closure concepts based on local completion are interesting since the closure of a graph contains more simplicial vertices than the graph itself, making the search for a clique covering easier.In this article, we introduce a new closure concept based on local completion which preserves the Hamiltonicity for every graph. Note that, moreover, the closure may be claw free even when the graph is not.     

基于复杂网络理论的含分布式发电的电力网络脆弱度评估

基于复杂网络理论研究含分布式发电(DG, Distributed Generation)的电力网络脆弱度评估问题,有针对性地提出三类脆弱度评估指标,其中基于结构的脆弱度指标能够体现网络拓扑和节点功率对系统供电效率的影响;攻击脆弱度指标可用于评估系统抵御节点和线路移除的能力;基于运行方式的脆弱度指标能够反映整个电网有功功率在传输距离上的均衡度.仿真算例验证了所提指标的有效性和DG对于改善系统功率传输性能与提高抗干扰能力方面的作用.     

A theory of decomposition into prime factors of layered interconnection networks

The cross-product technique, introduced by Even and Litman (1992) [8], is extended into a full decomposition theory enabling a unique (up to isomorphism) and polynomial factorization of layered interconnection networks (including many well-known networks) into a product of prime factors. A polynomial algorithm is provided for checking whether a given layered interconnection network is isomorphic to a network that is uniquely decomposable into prime factors.