network clustering coefficient without degree correlation biases the clustering coefficient quantifies how well connected are the neighbors of a vertex in a graph in real networks it decreases with the vertex degree which has been taken as a signature of the network hierarchical structure here we show that this signature of hierarchical structure is a consequence of degree correlation biases in the clustering coefficient definition we introduce a definition in which the degree correlation biases are filtered out and provide evidence that in real networks the clustering coefficient is constant or decays logarithmically with vertex degree

closed walks for community detection abstract in this paper we propose a novel measure that integrates both the concept of closed walks and clustering coefficients to replace the edge betweenness in the well known divisive hierarchical clustering algorithm the girvan and newman method gn the edges with the lowest value are removed iteratively until the network is degenerated into isolated nodes the experimental results on computer generated networks and real world networks showed that our method makes a better tradeoff of accuracy and runtime based on the analysis of the results we observe that the nontrivial closed walks of order three and four can be considered as the basic elements in constructing community structures meanwhile we discover that those nontrivial closed walks outperform trivial closed walks in the task of analyzing the structure of networks the double peak structure problem is mentioned in the last part of the article we find that our proposed method is a novel way to solve the double peak structure problem our work can provide us with a new perspective for understanding community structure in complex networks

analysis of directed networks via partial singular value decomposition and gauss quadrature abstract large scale networks arise in many applications it is often of interest to be able to identify the most important nodes of a network or to determine the ease of traveling between them we are interested in carrying out these tasks for directed networks these networks have a nonsymmetric adjacency matrix a benzi et al 6 recently proposed that these tasks can be accomplished by studying certain matrix functions such as hyperbolic cosine and sine of a t a and a a t for small to medium sized networks the required computations can be easily carried out by first computing the singular value decomposition of a however for large networks this is impractical we propose to first compute a partial singular value decomposition of a which allows us to determine a subset of nodes that contains the most important nodes or a subset of nodes between which it is easy to travel we then apply gauss quadrature to rank the nodes in these subsets several computed examples illustrate the performance of the approach proposed

non polynomial spline solution of singularly perturbed boundary value problems a generalized scheme based on quartic non polynomial spline functions is proposed this scheme is designed for numerical solution of singularly perturbed two point boundary value problems arising in the study of science and engineering the scheme leads to a five diagonal linear system of equations convergence analysis of the method is briefly discussed two numerical examples each of constant and variable coefficients are given to show practical usefulness of the method

subgraph centrality in complex networks we introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network smaller subgraphs are given more weight than larger ones which makes this measure appropriate for characterizing network motifs we show that the subgraph centrality c s i can be obtained mathematically from the spectra of the adjacency matrix of the network this measure is better able to discriminate the nodes of a network than alternate measures such as degree closeness betweenness and eigenvector centralities we study eight real world networks for which c s i displays useful and desirable properties such as clear ranking of nodes and scale free characteristics compared with the number of links per node the ranking introduced by c s i for the nodes in the protein interaction network of s cereviciae is more highly correlated with the lethality of individual proteins removed from the proteome

polynomial spline approach for solving second order boundary value problems with neumann conditions abstract in this paper a new difference scheme based on quartic splines is derived for solving linear and nonlinear second order ordinary differential equations subject to neumann type boundary conditions the scheme can achieve sixth order accuracy at the interior nodal points and fourth order accuracy at and near the boundary which is superior to the well known numerov s scheme with the accuracy being fourth order convergence analysis of the present method for linear cases is discussed finally numerical results for both linear and nonlinear cases are given to illustrate the efficiency of our method

membership function as an evaluation abstract after giving an overview of the different kinds of mathematical forms of the membership functions we extract the different demands and determine the rational class of the membership functions from the generalisation of our result we can determine it giving four parameters the internal a b the sharpness and the decision level we show the correspondence of this membership function to the zimmermann zysno model and using their empirical data we also test our mathematical model we also show the connections between the operators of the evaluation and membership functions which gives the generalisation of this concept in a more general form as a special case we can get a linear form

topological generalizations of network motifs biological and technological networks contain patterns termed network motifs which occur far more often than in randomized networks network motifs were suggested to be elementary building blocks that carry out key functions in the network it is of interest to understand how network motifs combine to form larger structures to address this we present a systematic approach to define motif generalizations families of motifs of different sizes that share a common architectural theme to define motif generalizations we first define roles in a subgraph according to structural equivalence for example the feedforward loop triad a motif in transcription neuronal and some electronic networks has three roles an input node an output node and an internal node the roles are used to define possible generalizations of the motif the feedforward loop can have three simple generalizations based on replicating each of the three roles and their connections we present algorithms for efficiently detecting motif generalizations we find that the transcription networks of bacteria and yeast display only one of the three generalizations the multi output feedforward generalization in contrast the neuronal network of c elegans mainly displays the multi input generalization forward logic electronic circuits display a multi input multi output hybrid thus networks which share a common motif can have very different generalizations of that motif using mathematical modeling we describe the information processing functions of the different motif generalizations in transcription neuronal and electronic networks

subgraphs and network motifs in geometric networks many real world networks describe systems in which interactions decay with the distance between nodes examples include systems constrained in real space such as transportation and communication networks as well as systems constrained in abstract spaces such as multivariate biological or economic data sets and models of social networks these networks often display network motifs subgraphs that recur in the network much more often than in randomized networks to understand the origin of the network motifs in these networks it is important to study the subgraphs and network motifs that arise solely from geometric constraints to address this we analyze geometric network models in which nodes are arranged on a lattice and edges are formed with a probability that decays with the distance between nodes we present analytical solutions for the numbers of all three and four node subgraphs in both directed and nondirected geometric networks we also analyze geometric networks with arbitrary degree sequences and models with a bias for directed edges in one direction scaling rules for scaling of subgraph numbers with system size lattice dimension and interaction range are given several invariant measures are found such as the ratio of feedback and feed forward loops which do not depend on system size dimension or connectivity function we find that network motifs in many real world networks including social networks and neuronal networks are not captured solely by these geometric models this is in line with recent evidence that biological network motifs were selected as basic circuit elements with defined information processing functions

a model for the measurement of membership and the consequences of its empirical implementation representation and uniqueness theorems are presented for the fundamental measurement of the membership of a fuzzy set when the domain of discourse is order dense the conclusion that membership is on an interval scale is justified by the inapplicability of extensive measurement to fuzziness and the lack of a natural origin for membership some descriptive statistics are also presented from an empirical study currently in progress to construct membership functions they suggest that subjects differ in their perception of fuzziness and that the models in the literature for certain linguistic operators such as not very and antonym are not sufficiently general to account for the range of individual perceptions of vague attributes a more general mathematical model is proposed the question of the meaningfulness of statements about membership functions is also discussed in the light of the assertion that membership can be fundamentally measured at best on an interval scale some of the difficulties which arise in fuzzy set theory due to this property are illustrated and a solution is suggested involving the use of a function which is derived from membership but is on an absolute scale

high order accuracy nonpolynomial spline solutions for 2μth order two point boundary value problems sixth order accurate method based on quintic nonpolynomial spline function for the numerical solution of 2 th order two point bvps where is a positive integer is presented by transforming the problem into a system of second order two point bvps the nonpolynomial spline function is used to drive some consistency relations for computing approximation to the solution of this problem the proposed approach gives better approximations than existing polynomial spline and finite difference methods and has a lower computational cost convergence analysis of the proposed method is discussed numerical examples are included to illustrate the practical usefulness of our method

clustering of correlated networks we obtain the clustering coefficient the degree dependent local clustering and the mean clustering of networks with arbitrary correlations between the degrees of the nearest neighbor vertices the resulting formulas allow one to determine the nature of the clustering of a network

community detection in hypernetwork via density ordered tree partition hypernetwork as a useful representation of natural and social systems has received increasing interests from researchers community is crucial to understand the structural and functional properties of the hypernetworks here we propose a new method to uncover the communities of hypernetworks we construct a density ordered tree dot to represent original data by combining density and distance and the community detection in hypernetwork is converted to a dot partition problem then an anomaly detection strategy using box plot rule is applied to partition dot and judge whether there is a significant community structure in the hypernetwork moreover visual inspection as a complementary approach of box plot rule can effectively improve the effectiveness of community detection finally the method is compared with existing methods in both synthetic and real world networks

exploring the evolutionary mechanism of complex supply chain systems using evolving hypergraphs abstract a new evolutionary model is proposed to describe the characteristics and evolution pattern of supply chain systems using evolving hypergraphs in which nodes represent enterprise entities while hyperedges represent the relationships among diverse trades the nodes arrive at the system in accordance with a poisson process with the evolving process incorporating the addition of new nodes linking of old nodes and rewiring of links grounded in the poisson process theory and continuum theory the stationary average hyperdegree distribution is shown to follow a shifted power law spl and the theoretical predictions are consistent with the results of numerical simulations testing the impact of parameters on the model yields a positive correlation between hyperdegree and degree the model also uncovers macro characteristics of the relationships among enterprises due to the microscopic interactions among individuals

mean field theory for clustering coefficients in barabasi albert networks we applied a mean field approach to study clustering coefficients in barabasi albert ba networks we found that the local clustering in ba networks depends on the node degree analytic results have been compared to extensive numerical simulations finding a very good agreement for nodes with low degrees clustering coefficient of a whole network calculated from our approach perfectly fits numerical data

measurement of membership functions and their acquisition abstract three basic views of the representation of membership functions are reviewed together with fundamental measurement of linguistic terms of linguistic variables the conclusion is that such measurements are either on an ordinal or an interval scale based on whether the appropriate axioms are validated by the empirical data with an allowance for stochastic variation the conjoint measurement is introduced for the case of multi dimensional linguistic variables whose linguistic terms are compositions of two or finitely many more component linguistic terms of distinct independent linguistic variables it is shown that any composition of distinct independent component linguistic terms of component linguistic variables by any t norm or s norm or any finite convext linear combination preserves the monotonic weak order property of the components in the composite once the scale properties of the measurement values of particular terms are validated composition procedures may be applied to the experimental data to obtain compound membership functions of fuzzy sets induced by meaningful representations of compositions of linguistic terms of linguistic variables finally four methods of membership acquisition and construction are reviewed from the perspective of fundamental measurement

higher order clustering coefficients in barabasi albert networks abstract higher order clustering coefficients c x are introduced for random networks the coefficients express probabilities that the shortest distance between any two nearest neighbours of a certain vertex i equals x when one neglects all paths crossing the node i using c x we found that in the barabasi albert ba model the average shortest path length in a node s neighbourhood is smaller than the equivalent quantity of the whole network and the remainder depends only on the network parameter m our results show that small values of the standard clustering coefficient in large ba networks are due to random character of the nearest neighbourhood of vertices in such networks

parametric spline method for a class of singular two point boundary value problems we develop a three point formula based on cubic spline in compression for a class of singular two point boundary value problems for different values of parameters we obtain the classes of methods these methods are illustrated by three numerical examples two linear and one non linear

quantifying vagueness in decision models abstract when modelling specific decision situations the decision maker often feels overstrained when he is asked for precise numerical quotations concerning the objectives or the constraints whereas qualitative statements are easily given in the recent past the theory of fuzzy sets has proven to be very useful for representing this type of information though it is quite advanced formally the practical determination of its core elements i e membership functions and operators has only been explored to a very limited extent this paper presents results of empirical research which focused on the problem of modelling vagueness i e determining membership functions of fuzzy sets which are considered as quantitative representations of vague concepts such as young man long sticks high profits etc

communicability betweenness in complex networks abstract betweenness measures provide quantitative tools to pick out fine details from the massive amount of interaction data that is available from large complex networks they allow us to study the extent to which a node takes part when information is passed around the network nodes with high betweenness may be regarded as key players that have a highly active role at one extreme betweenness has been defined by considering information passing only through the shortest paths between pairs of nodes at the other extreme an alternative type of betweenness has been defined by considering all possible walks of any length in this work we propose a betweenness measure that lies between these two opposing viewpoints we allow information to pass through all possible routes but introduce a scaling so that longer walks carry less importance this new definition shares a similar philosophy to that of communicability for pairs of nodes in a network which was introduced by estrada and hatano e estrada n hatano phys rev e 77 2008 036111 having defined this new communicability betweenness measure we show that it can be characterized neatly in terms of the exponential of the adjacency matrix we also show that this measure is closely related to a frechet derivative of the matrix exponential this allows us to conclude that it also describes network sensitivity when the edges of a given node are subject to infinitesimally small perturbations using illustrative synthetic and real life networks we show that the new betweenness measure behaves differently to existing versions and in particular we show that it recovers meaningful biological information from a protein protein interaction network

on the convergence of a finite difference method for a class of singular boundary value problems arising in physiology using chawla s identity bit 29 1989 566 a finite difference method based on uniform mesh is described for a class of singular boundary value problems p x y p x f x y 0 x 1 y 0 a y 1 y 1 or y 0 0 y 1 y 1 with p x xb0 g x b0 0 and it is shown that the method is of second order accuracy under quite general conditions on p x and f x y this work also extends the method developed by chawla et al bit 26 1986 326 for p x xb0 b0 1 to a general class of function p x xb0 g x b0 0 numerical examples for general function p x verify the order of the convergence of the method and two physiological problems have also been solved

cubic spline solution of two point boundary value problems with significant first derivatives a cubic spline function has been applied to the solution of the second order nonlinear two point boundary value problems with significant first derivatives the method obtained involves some parameters these parameters may be suitably chosen to obtain an oscillation free solution of the test equation y ky k 1 depending upon the choice of the parameters methods of order two three and four may be obtained the methods have been tested on both linear and nonlinear problems

tage iterative algorithm and nonpolynomial spline basis for the solution of nonlinear singular second order ordinary differential equations abstract in the present paper we discuss three point difference method based on nonpolynomial spline basis for the second order ordinary differential equation difference schemes are derived for linear and nonlinear case and are used to solve via two parameter alternating group explicit iterative algorithm the schemes have a fourth and second order of uniform convergence for the choice of the parameters involved in the method computational results are presented comparing the two methods in terms of accuracy and execution times the results indicate the advantage of using parallel implementation of the new method

nonpolynomial spline approach to the solution of a system of second order boundary value problems we use a cubic spline equivalent nonpolynomial spline functions to develop a numerical method for computing approximations to the solution of a system of second order boundary value problems associated with obstacle unilateral and contact problems we show that the present method gives approximations which are better than those produced by other collocation finite difference and spline methods convergence analysis of the method is discussed a numerical example is given to illustrate practical usefulness of our method

on the r uniform linear hypertrees with extremal estrada indices abstract let h be a hypergraph with n vertices the estrada index of h is defined as the sum of e 1 e n where 1 n are the eigenvalues of the adjacency matrix of h let t n r be the set of r uniform linear hypertrees with n vertices and t n r the subset of t n r in which each hypergraph has a largest vertex degree where r 3 and 2 n 1 r 1 several new transformations for studying the estrada indices of hypergraphs are introduced by these transformations the hypertree with the smallest estrada index is determined in t n r and the hypertrees with the smallest the second smallest the largest and the second largest estrada indices are derived among t n r

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

subgraph centrality and clustering in complex hyper networks abstract the representation of complex systems as networks is inappropriate for the study of certain problems we show several examples of social biological ecological and technological systems where the use of complex networks gives very limited information about the structure of the system consequently we extend the concepts of subgraph centrality and clustering for complex networks represented by hypergraphs complex hyper networks the first parameter characterizes the node participation in different sub hypergraphs and the second one characterizes the transitivity in the hyper network through the proportion of hyper triangles to paths of length two another measure characterizing the formation of triples of mutually adjacent groups in the hyper network is also introduced all of these characteristics are studied in three different hyper networks a scientific collaboration hyper network an ecological competition hyper network and the hyper network formed by the american corporate elite in 1999

an efficient and flexible mechanism for constructing membership functions abstract this paper introduces a bezier curve based mechanism for constructing membership functions of convex normal fuzzy sets the mechanism can fit any given data set with a minimum level of discrepancy in the absence of data the mechanism can be intuitively manipulated by the user to construct membership functions with the desired shape some numerical experiments are included to compare the performance of the proposed mechanism with conventional methods

l percolations of complex networks given a complex network its l paths correspond to sequences of l 1 distinct nodes connected through l distinct edges the l conditional expansion of a complex network can be obtained by connecting all its pairs of nodes which are linked through at least one l path and the respective conditional l expansion of the original network is defined as the intersection between the original network and its l expansion such expansions are verified to act as filters enhancing the network connectivity consequently contributing to the identification of communities in small world models it is shown in this paper for l 2 and 3 in both analytical and experimental fashions that an evolving complex network with a fixed number of nodes undergoes successive phase transitions the so called l percolations giving rise to eulerian giant clusters it is also shown that the critical values of such percolations are a function of the network size and that the network percolates for l 3 before l 2

a fifth order accurate geometric mesh finite difference method for general nonlinear two point boundary value problems second order boundary value problems are treated using fifth order accurate geometric mesh finite difference approximations the method uses evaluation at two off steps three neighbouring knots and it reduces to simple five terms recurrence relations the formal convergence analysis and error estimate reveals that the truncation errors of the new discretizations are fifth order accurate some physical problems are solved to demonstrate efficiency and reliability of the proposed numerical method the root mean square errors for different values of mesh ratio parameter have been given in tables to support the utility of the method

the numerical solution of non linear singular boundary value problems arising in physiology a class of non linear singular ordinary differential equations is solved by a new method based on non polynomial cubic spline we use the quesilinearization technique to reduce the given non linear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using non polynomial cubic spline approximation the resulting system of algebraic equations is solved by using a tri diagonal solver computational results are provided to demonstrate the viability of the new method

subgraphs in random networks understanding the subgraph distribution in random networks is important for modeling complex systems in classic erdos networks which exhibit a poissonian degree distribution the number of appearances of a subgraph g with n nodes and g edges scales with network size as approximately n n g however many natural networks have a non poissonian degree distribution here we present approximate equations for the average number of subgraphs in an ensemble of random sparse directed networks characterized by an arbitrary degree sequence we find scaling rules for the commonly occurring case of directed scale free networks in which the outgoing degree distribution scales as p k approximately k gamma considering the power exponent of the degree distribution gamma as a control parameter we show that random networks exhibit transitions between three regimes in each regime the subgraph number of appearances follows a different scaling law approximately nalpha where alpha n g s 1 for gamma gamma c where s is the maximal outdegree in the subgraph and gamma c s 1 we find that certain subgraphs appear much more frequently than in erdos networks these results are in very good agreement with numerical simulations this has implications for detecting network motifs subgraphs that occur in natural networks significantly more than in their randomized counterparts