{X,Y}-Cyclacene Graphs with Next Nearest Neighbor Interactions

Eigensolutions of {X( = C,B,N),Y( = C,B,N)}-cyclacene graphs with next nearest neighbor (nnn) interactions have been obtained in analytical forms by adapting n-fold rotational symmetry followed by two-fold rotational symmetry (or a plane of symmetry). Expressions of eigensolution indicate the subspectral relationship among such cyclacenes with an even number of hexagonal rings e.g., eigenvalues of {X,Y}-di-cyclacene are found in the eigenspectra of all such even cyclacenes. Total π-electron energies and highest occupied molecular orbital and lowest unoccupied molecular orbital (HOMO–LUMO) gaps are calculated using the analytical expressions obtained and are found to vary negligibly with the variation of nnn interactions in such cyclacenes. Total π-electron energy is found to increase due to increase in restriction intensity of nnn interactions, whereas the HOMO–LUMO gap of polyacenecs having the even number of hexagonal rings and with one electron at each site (atom) decreases with increase in the restriction intensity since such systems contain degenerate half-filled HOMO (bonding or nonbonding) that are much more vulnerable for perturbations imposed through nnn interactions.     

Stochastic graph as a model for social networks

Social networks are usually modeled and represented as deterministic graphs with a set of nodes as users and edges as connection between users of networks. Due to the uncertain and dynamic nature of user behavior and human activities in social networks, their structural and behavioral parameters are time varying parameters and for this reason using deterministic graphs for modeling and analysis of behavior of users may not be appropriate. In this paper, we propose that stochastic graphs, in which weights associated with edges are random variables, may be a better candidate as a graph model for social network analysis. Thus, we first propose generalization of some network measures for stochastic graphs and then propose six learning automata based algorithms for calculating these measures under the situation that the probability distribution functions of the edge weights of the graph are unknown. Simulations on different synthetic stochastic graphs for calculating the network measures using the proposed algorithms show that in order to obtain good estimates for the network measures, the required number of samples taken from edges of the graph is significantly lower than that of standard sampling method aims to analysis of human behavior in online social networks.     

Another look at the phenomenon of phase transition

This study is motivated by the question whether there exists a characteristic feature common to the phenomenon of phase transition of properties taking place in the course of evolution of various sets of objects of different nature. A novel measure model of phase transition is presented, providing an insight into the process underlying different phase transitions. First, it is shown to conform to several famous and extensively investigated cases of phase transition. Then a few close consequences of applying the measure model are presented. It has been proved that monotone properties undergo a sharp phase transition. The measure model provides a sufficient condition of a sharp phase transition common to monotone and non-monotone properties as well. The measure model reveals that if a property Q undergoes a phase transition in the course of evolution of objects of a certain type, then there exist other secondary properties related to the measure of Q that also undergo a phase transition during the evolution.     

Elementary Statistics

Fractional two-level factorial designs are often used in the early stages of an investigation to screen for important factors. Traditionally, 2 ^n-k fractional factorial designs of resolution III, IV, or V have been used for this purpose. When the investigator is able to specify the set of nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design with fewer runs than a standard textbook design by searching within a wider class of designs called parallel-flats designs. The run sizes in this class of designs do not necessarily need to be powers of 2. We discuss an algorithm for constructing orthogonal parallel-flats designs to meet user specifications. Several examples illustrate the use of the algorithm.     

Conditional matching preclusion for the alternating group graphs and split-stars

The matching preclusion number of a graph is the minimum number of edges the deletion of which results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges the deletion of which results in a graph with no isolated vertices that has neither perfect matchings nor almost-perfect matchings. In this article, we find this number and classify all optimal sets for the alternating group graphs, one of the most popular interconnection networks, and their companion graphs, the split-stars. Moreover, some general results on the conditional matching preclusion problems are also presented.     

On parallel push–relabel based algorithms for bipartite maximum matching

We study multithreaded push–relabel based algorithms for computing maximum cardinality matching in bipartite graphs. Matching is a fundamental combinatorial problem with applications in a wide variety of problems in science and engineering. We are motivated by its use in the context of sparse linear solvers for computing the maximum transversal of a matrix. Other applications can be found in many fields such as bioinformatics (Azad et al., 2010) [4], scheduling (Timmer and Jess, 1995) [27], and chemical structure analysis (John, 1995) [14]. We implement and test our algorithms on several multi-socket multicore systems and compare their performance to state-of-the-art augmenting path-based serial and parallel algorithms using a test set comprised of a wide range of real-world instances.     

Graph-balancing algorithms for average consensus over directed networks

Consensus strategies find extensive applications in coordination of robot groups and decision-making of agents. Since balanced graph plays an important role in the average consensus problem and many other coordination problems for directed communication networks, this work explores the conditions and algorithms for the digraph balancing problem. Based on the analysis of graph cycles, we prove that a digraph can be balanced if and only if the null space of its incidence matrix contains positive vectors. Then, based on this result and the corresponding analysis, two weight balance algorithms have been proposed, and the conditions for obtaining a unique balanced solution and a set of analytical results on weight balance problems have been introduced. Then, we point out the relationship between the weight balance problem and the features of the corresponding underlying Markov chain. Finally, two numerical examples are presented to verify the proposed algorithms.     

偏斜交圆柱圆锥表面展开图计算机绘制数模原理

分析了轴线偏斜交的圆柱与圆锥相对位置关系,建立了偏斜交圆柱、圆锥表面展开图及相贯曲线绘制的数学模型,并编制了通用计算机绘图程序,给出计算机绘制的展开图样例。     

Optimal search-relocation trade-off in Markovian-target searching

In this study, a standard moving-target search model was extended with a multiple-search-speed option, whereby a trade-off is enabled between the increased detection chances owing to the searcher's better location and the increased uncertainty of the target's location resulting from the diminished search performance incurred in the relocation. This enhances the detection probability of the output search path and, thereby, the model's practicality. However, the scalability of the solution method is essential to its implementation, as the basic model is already NP-hard. We developed an efficient heuristic by combining the idea of approximate nondetection probability minimization and a hybridized shortest-path heuristic that exploits the fast-mixing property of the Markov chain. According to the results of an intensive experiment, the heuristic achieves a near-optimal trade-off within a very reasonable computation time.     

Node discovery in networks

This paper abstracts the problem of network nodes discovering one another in a network of unknown size using all-to-all gossip. The problem is studied in terms of evolving directed graphs where each vertex represents a participating node and each edge represents one node’s knowledge about another. Ideally, such a graph has diameter one, i.e., each node knows all others. Nodes share their knowledge by sending gossip messages. Gossip among the nodes allows them to discover one another, decreasing the diameter of the graph. Here this problem is considered in several synchronous settings under different assumptions about the ability of the participating nodes to communicate. Specifically, the following aspects of communication are considered: (1) the ability of the nodes to multicast gossip messages, and (2) the size of the messages. The results describe the lower and upper bounds on the number of synchronous rounds required for the participants to discover each other. A particular question of interest is: if the network size is unknown, how does a node know that it has discovered all other nodes? Given a weakly-connected graph describing the initial knowledge of the nodes, every node in our algorithm can stop the discovery process knowing that there are no unknown nodes—this is done without any prior knowledge of the total number of nodes participating in the computation.