link prediction based on path entropy abstract information theory has been taken as a prospective tool for quantifying the complexity of complex networks in this paper first we study the information entropy or uncertainty of a path using the information theory after that we apply the path entropy to the link prediction problem in real world networks specifically we propose a new similarity index namely path entropy pe index which considers the information entropies of shortest paths between node pairs with penalization to long paths empirical experiments demonstrate that pe index outperforms the mainstream of link predictors

variance component testing in generalised linear models with random effects summary there is considerable interest in testing for overdispersion correlation and heterogeneity across groups in biomedical studies in this paper we cast the problem in the framework of generalised linear models with random effects we propose a global score test for the null hypothesis that all the variance components are zero this test is a locally asymptotically most stringent test and is robust in the special sense that the test does not require specifying the joint distribution of the random effects we also propose individual score tests and their approximations for testing the variance components separately both tests can be easily implemented using existing statistical software we illustrate these tests with an application to the study of heterogeneity of mating success across males and females in an experiment on salamander matings and evaluate their performance through simulation

fast computation of von neumann entropy for large scale graphs via quadratic approximations abstract the von neumann graph entropy vnge can be used as a measure of graph complexity which can be the measure of information divergence and distance between graphs however computing vnge is extensively demanding for a large scale graph we propose novel quadratic approximations for fast computing vnge various inequalities for error between the quadratic approximations and the exact vnge are found our methods reduce the cubic complexity of vnge to linear complexity computational simulations on random graph models and various real network datasets demonstrate superior performance

testing the correlation for clustered categorical and censored discrete time to event data when covariates are measured without with errors summary in the analysis of clustered categorical data it is of common interest to test for the correlation within clusters and the heterogeneity across different clusters we address this problem by proposing a class of score tests for the null hypothesis that the variance components are zero in random effects models for clustered nominal and ordinal categorical responses we extend the results to accommodate clustered censored discrete time to event data we next consider such tests in the situation where covariates are measured with errors we propose using the simex method to construct the score tests for the null hypothesis that the variance components are zero key advantages of the proposed score tests are that they can be easily implemented by fitting standard polytomous regression models and discrete failure time models and that they are robust in the sense that no assumptions need to be made regarding the distributions of the random effects and the unobserved covariates the asymptotic properties of the proposed tests are studied we illustrate these tests by analyzing two data sets and evaluate their performance with simulations

levy walk in complex networks an efficient way of mobility abstract to obtain an efficient diffusion process is an intriguing and important issue in the study of dynamical behaviors on real networks most previous studies are mainly focused on the analysis based on the random walk strategy whose entropy rate is bounded by the logarithm of the largest node degree of a given graph in this paper we take into account a novel strategy named levy walk and derive the general expression of entropy rate of levy walk on networks we present numerical evidences for how the levy walk strategy delivers an efficient diffusion process on networks and significantly increases the entropy rate compared with the random walk strategy it is further demonstrated that the capability of levy walk heavily relies on the network topology as well as the amount of information available regarding the network structure specifically the behavior of levy walk is highly sensitive to the distribution of shortest distances of the network and its variation to address this finding we thereby give a theoretical explanation of the relationship between the variation of shortest distances and the entropy rate of levy walk this work may help to enrich our understanding of the behavior of levy walk and further guide us to find an efficient diffusion process on networks

clustering of random scale free networks we derive the finite size dependence of the clustering coefficient of scale free random graphs generated by the configuration model with degree distribution exponent 2l ensuremath gamma l3 degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets we also find that for values of ensuremath gamma ensuremath approx 2 clustering is virtually size independent and at the same time becomes a de facto non self averaging topological property this implies that a single instance network is not representative of the ensemble even for very large network sizes

statistical mechanics of multiedge networks statistical properties of binary complex networks are well understood and recently many attempts have been made to extend this knowledge to weighted ones there are however subtle yet important considerations to be made regarding the nature of the weights used in this generalization weights can be either continuous or discrete magnitudes and in the latter case they can additionally have undistinguishable or distinguishable nature this fact has not been addressed in the literature insofar and has deep implications on the network statistics in this work we face this problem introducing multiedge networks as graphs where multiple distinguishable connections between nodes are considered we develop a statistical mechanics framework where it is possible to get information about the most relevant observables given a large spectrum of linear and nonlinear constraints including those depending both on the number of multiedges per link and their binary projection the latter case is particularly interesting as we show that binary projections can be understood from multiedge processes the implications of these results are important as many real agent based problems mapped onto graphs require this treatment for a proper characterization of their collective behavior

depth based complexity traces of graphs in this paper we aim to characterize graphs in terms of a structural measure of complexity our idea is to decompose a graph into layered substructures of increasing size and then to measure the information content of these substructures to locate dominant substructures within a graph we commence by identifying a centroid vertex which has the minimum shortest path length variance to the remaining vertices for each graph a family of centroid expansion subgraphs is derived from the centroid vertex in order to capture dominant structural characteristics of the graph since the centroid vertex is identified through a global analysis of the shortest path length distribution the expansion subgraphs provide a fine representation of a graph structure we then show how to characterize graphs using depth based complexity traces here we explore two different strategies the first strategy is to measure how the entropies on the centroid expansion subgraphs vary with the increasing size of the subgraphs the second strategy is to measure how the entropy differences vary with the increasing size of the subgraphs we perform graph classification in the principal component space of the complexity trace vectors experiments on graph datasets abstracted from some bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph complexity traces our methods are competitive to state of the art methods highlightswe identify a centroid vertex of a graph through its shortest path matrix we establish a family of expansion subgraphs of a graph from its centroid vertex we construct depth based complexity traces of a graph on its expansion subgraphs we evaluate the performance of our complexity traces on standard graph datasets we demonstrate the effectiveness and efficiency of the graph complexity traces

quantum statistics in complex networks in this work we discuss the symmetric construction of bosonic and fermionic networks and we present a case of a network showing a mixed quantum statistics this model takes into account the different nature of nodes described by a random parameter that we call energy and includes rewiring of the links the system described by the mixed statistics is an inhomogeneous system formed by two class of nodes in fact there is a threshold energy epsilon s such that nodes with lower energy epsilon epsilon s decrease their connectivity in time

entropy of spatial network ensembles we analyze complexity in spatial network ensembles through the lens of graph entropy mathematically we model a spatial network as a soft random geometric graph i e a graph with two sources of randomness namely nodes located randomly in space and links formed independently between pairs of nodes with probability given by a specified function the pair connection function of their mutual distance we consider the general case where randomness arises in node positions as well as pairwise connections i e for a given pair distance the corresponding edge state is a random variable classical random geometric graph and exponential graph models can be recovered in certain limits we derive a simple bound for the entropy of a spatial network ensemble and calculate the conditional entropy of an ensemble given the node location distribution for hard and soft probabilistic pair connection functions under this formalism we derive the connection function that yields maximum entropy under general constraints finally we apply our analytical framework to study two practical examples ad hoc wireless networks and the us flight network through the study of these examples we illustrate that both exhibit properties that are indicative of nearly maximally entropic ensembles

symmetric laplacians quantum density matrices and their von neumann entropy abstract we show that the normalized symmetric laplacian of a simple graph can be obtained from the partial trace over a pure bipartite quantum state that resides in a bipartite hilbert space one part corresponding to the vertices the other corresponding to the edges this suggests an interpretation of the symmetric laplacian s von neumann entropy as a measure of bipartite entanglement present between the two parts of the state we then study extreme values for a connected graph s generalized renyi p entropy specifically we show that 1 the complete graph achieves maximum entropy 2 the 2 regular graph a achieves minimum renyi 2 entropy among all k regular graphs b is within log 4 3 of the minimum renyi 2 entropy and log 4 2 3 of the minimum von neumann entropy among all connected graphs c achieves a von neumann entropy less than the star graph point 2 contrasts sharply with similar work applied to normalized combinatorial laplacians where it has been shown that the star graph almost always achieves minimum von neumann entropy in this work we find that the star graph achieves maximum entropy in the limit as the number of vertices grows without bound

entropies of complex networks with hierarchically constrained topologies the entropy of a hierarchical network topology in an ensemble of sparse random networks with hidden variables associated with its nodes is the log likelihood that a given network topology is present in the chosen ensemble we obtain a general formula for this entropy which has a clear interpretation in some simple limiting cases the results provide keys with which to solve the general problem of fitting a given network with an appropriate ensemble of random networks

spectral similarity for barabasi albert and chung lu models abstract in the paper we have analyzed spectral similarity between barabasi albert and chung lu models we have shown the similarity of spectral distribution for sufficiently large barabasi albert parameter value contrary extreme eigenvalues and principal eigenvector are not similar for those model we provide applications of obtained results related to the spectral graph theory and efficiency of quantum spatial search

regression analysis of clustered interval censored data with informative cluster size interval censored data are commonly found in studies of diseases that progress without symptoms which require clinical evaluation for detection several techniques have been suggested with independent assumption however the assumption will not be valid if observations come from clusters furthermore when the cluster size relates to response variables commonly used methods can bring biased results for example in a study on lymphatic filariasis a parasitic disease where worms make several nests in the infected person s lymphatic vessels and reside until adulthood the response variable of interest is the nest extinction times as the extinction times of nests are checked by repeated ultrasound examinations exact extinction times are not observed instead data are composed of two examination points the last examination time with living worms and the first examination time with dead worms furthermore as williamson et al statistics in medicine 2008 27 543 555 pointed out larger nests show a tendency for low clearance rates this association has been denoted as an informative cluster size to analyze the relationship between the numbers of nests and interval censored nest extinction times this study proposes a joint model for the relationship between cluster size and clustered interval censored failure data a proportional hazard model with random effect and a mixed ordinal regression model are applied to failure times and cluster size respectively the joint model approach addresses both the association among failure times from the same cluster and the dependency of failure times on cluster size simulation studies are performed to assess the finite sample properties of the estimators and lymphatic filariasis data are analyzed as an illustration

entropy measures for networks toward an information theory of complex topologies the quantification of the complexity of networks is today a fundamental problem in the physics of complex systems a possible roadmap to solve the problem is via extending key concepts of information theory to networks in this rapid communication we propose how to define the shannon entropy of a network ensemble and how it relates to the gibbs and von neumann entropies of network ensembles the quantities we introduce here will play a crucial role for the formulation of null models of networks through maximum entropy arguments and will contribute to inference problems emerging in the field of complex networks

tailored graph ensembles as proxies or null models for real networks i tools for quantifying structure we study the tailoring of structured random graph ensembles to real networks with the objective of generating precise and practical mathematical tools for quantifying and comparing network topologies macroscopically beyond the level of degree statistics our family of ensembles can produce graphs with any prescribed degree distribution and any degree degree correlation function its control parameters can be calculated fully analytically and as a result we can calculate asymptotically formulae for entropies and complexities and for information theoretic distances between networks expressed directly and explicitly in terms of their measured degree distribution and degree correlations

bias reduction of maximum likelihood estimates summary it is shown how in regular parametric problems the first order term is removed from the asymptotic bias of maximum likelihood estimates by a suitable modification of the score function in exponential families with canonical parameterization the effect is to penalize the likelihood by the jeffreys invariant prior in binomial logistic models poisson log linear models and certain other generalized linear models the jeffreys prior penalty function can be imposed in standard regression software using a scheme of iterative adjustments to the data

on taking square roots without quadratic nonresidues over finite fields we present a novel idea to compute square roots over finite fields without being given any quadratic nonresidue and without assuming any unproven hypothesis the algorithm is deterministic and the proof is elementary in some cases the square root algorithm runs in o log 2 q bit operations over finite fields with q elements as an application we construct a deterministic primality proving algorithm which runs in o log 3 n for some integers n

a simple and fast probabilistic algorithm for computing square roots modulo a prime number corresp a probabilistic polynomial time algorithm for computing the square root of a number x in bf z p bf z where p 2 s q 1 q odd s 0 is a prime number is described in contrast to the adleman manders and miller algorithm this algorithm gets faster as s grows as with the berlekamp rabin algorithm the expected running time of the algorithm is independent of x however the algorithm presented here is considerably faster for values of s greater than 2

analysis of clustered and interval censored data from a community based study in asthma many authors in recent years have proposed extensions of familiar survival analysis methodologies to apply in dependent data settings for example when data are clustered or subject to repeated measures however these extensions have been considered largely in the context of right censored data in this paper we discuss a parametric frailty model for the analysis of clustered and interval censored failure time data details are presented for the specific case where the underlying time to event data follow a weibull distribution maximum likelihood estimates will be obtained using commercially available software and the empirical efficiency of these estimators will be explored via a simulation study we also discuss a score test to make inferences about the magnitude and significance of over dispersion in clustered data settings these methods will be illustrated using data from the east boston asthma study

a multiple imputation approach for clustered interval censored survival data multivariate interval censored failure time data arise commonly in many studies of epidemiology and biomedicine analysis of these type of data is more challenging than the right censored data we propose a simple multiple imputation strategy to recover the order of occurrences based on the interval censored event times using a conditional predictive distribution function derived from a parametric gamma random effects model by imputing the interval censored failure times the estimation of the regression and dependence parameters in the context of a gamma frailty proportional hazards model using the well developed em algorithm is made possible a robust estimator for the covariance matrix is suggested to adjust for the possible misspecification of the parametric baseline hazard function the finite sample properties of the proposed method are investigated via simulation the performance of the proposed method is highly satisfactory whereas the computation burden is minimal the proposed method is also applied to the diabetic retinopathy study drs data for illustration purpose and the estimates are compared with those based on other existing methods for bivariate grouped survival data

a note on square roots in finite fields a simple method showing the quadratic character in a finite field of odd order q can be computed in o log sup 2 q steps is presented it is also shown how sequences generated deterministically from a random seed can be used reliably in a recent randomized algorithm of r peralta 1986 for computing square roots in finite fields

approximate von neumann entropy for directed graphs in this paper we develop an entropy measure for assessing the structural complexity of directed graphs although there are many existing alternative measures for quantifying the structural properties of undirected graphs there are relatively few corresponding measures for directed graphs to fill this gap in the literature we explore an alternative technique that is applicable to directed graphs we commence by using chung s generalization of the laplacian of a directed graph to extend the computation of von neumann entropy from undirected to directed graphs we provide a simplified form of the entropy which can be expressed in terms of simple node in degree and out degree statistics moreover we find approximate forms of the von neumann entropy that apply to both weakly and strongly directed graphs and that can be used to characterize network structure we illustrate the usefulness of these simplified entropy forms defined in this paper on both artificial and real world data sets including structures from protein databases and high energy physics theory citation networks

solution of the two star model of a network the p star model or exponential random graph is among the oldest and best known of network models here we give an analytic solution for the particular case of the two star model which is one of the most fundamental of exponential random graphs we derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry broken phase separated from the normal phase of the model by a conventional continuous phase transition

network ensemble comparisons with stochastic rewiring and von neumann entropy assessing whether a given network is typical or atypical for a random network ensemble i e network ensemble comparison has widespread applications ranging from null model selection and hypothesis testing to clustering and classifying networks we develop a framework for network ensemble comparison by subjecting the network to stochastic rewiring we study two rewiring processes uniform and degree preserved rewiring which yield random network ensembles that converge to the erdos renyi and configuration model ensembles respectively we study convergence through von neumann entropy vne a network summary statistic measuring information content based on the spectra of a laplacian matrix and develop a perturbation analysis for the expected effect of rewiring on vne our analysis yields an estimate for how many rewires are required for a given network to resemble a typical network from an ensemble offering a computationally efficient quantity for network ensemble comparison that does not require

structural information and dynamical complexity of networks in 1953 shannon proposed the question of quantification of structural information to analyze communication systems the question has become one of the longest great challenges in information science and computer science here we propose the first metric for structural information given a graph g we define the k dimensional structural information of g or structure entropy of g denoted by mathcal h k g to be the minimum overall number of bits required to determine the k dimensional code of the node that is accessible from random walk in g the k dimensional structural information provides the principle for completely detecting the natural or true structure which consists of the rules regulations and orders of the graphs for fully distinguishing the order from disorder in structured noisy data and for analyzing communication systems solving the shannon s problem and opening up new directions the k dimensional structural information is also the first metric of dynamical complexity of networks measuring the complexity of interactions communications operations and even evolution of networks the metric satisfies a number of fundamental properties including additivity locality robustness local and incremental computability and so on we establish the fundamental theorems of the one and two dimensional structural information of networks including both lower and upper bounds of the metrics of classic data structures general graphs the networks of models and the networks of natural evolution we propose algorithms to approximate the k dimensional structural information of graphs by finding the k dimensional structure of the graphs that minimizes the k dimensional structure entropy we find that the k dimensional structure entropy minimization is the principle for detecting the natural or true structures in real world networks consequently our structural information provides the foundation for knowledge discovering from noisy data we establish a black hole principle by using the two dimensional structure information of graphs we propose the natural rank of locally listing algorithms by the structure entropy minimization principle providing the basis for a next generation search engine

riemannian geometric entropy for measuring network complexity a central issue in the science of complex systems is the quantitative characterization of complexity in the present work we address this issue by resorting to information geometry actually we propose a constructive way to associate with a in principle any network a differentiable object a riemannian manifold whose volume is used to define the entropy the effectiveness of the latter in measuring network complexity is successfully proved through its capability of detecting a classical phase transition occurring in both random graphs and scale free networks as well as of characterizing small exponential random graphs configuration models and real networks

depth based hypergraph complexity traces from directed line graphs in this paper we aim to characterize the structure of hypergraphs in terms of structural complexity measure measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders we thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis to locate the dominant substructure within a line graph we identify a centroid vertex by computing the minimum variance of its shortest path lengths a family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex we compute the depth based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs the resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs we perform hyper graph classification in the principal component space of the complexity trace vectors experiments on hyper graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces highlightswe aim to characterize hypergraphs in terms of structural complexity measures straightforwardly measuring the complexity of a hypergraph tends to be elusive we transform a hypergraph into a line graph for measuring the complexities we compute depth based complexity traces for the hypergraph on line graphs the complexity traces provide a flexible framework for hypergraphs and graphs

a note on the von neumann entropy of random graphs abstract in this note we consider the von neumann entropy of a density matrix obtained by normalizing the combinatorial laplacian of a graph by its degree sum we prove that the von neumann entropy of the typical erdos renyi random graph saturates its upper bound since connected regular graphs saturate this bound as well our result highlights a connection between randomness and regularity a general interpretation of the von neumann entropy of a graph is an open problem

square root computation over even extension fields this paper presents a comprehensive study of the computation of square roots over finite extension fields we propose two novel algorithms for computing square roots over even field extensions of the form bbf q 2 with q p n p an odd prime and n geq 1 both algorithms have an associate computational cost roughly equivalent to one exponentiation in bbf q 2 the first algorithm is devoted to the case when q equiv 1 rm mod 4 whereas the second one handles the case when q equiv 3 rm mod 4 numerical comparisons show that the two algorithms presented in this paper are competitive and in some cases more efficient than the square root methods previously known

gibbs entropy of network ensembles by cavity methods the gibbs entropy of a microcanonical network ensemble is the logarithm of the number of network configurations compatible with a set of hard constraints this quantity characterizes the level of order and randomness encoded in features of a given real network here we show how to relate this entropy to large deviations of conjugated canonical ensembles we derive exact expression for this correspondence using the cavity methods for the configuration model for the ensembles with constraint degree sequence and community structure and for the ensemble with constraint degree sequence and number of links at a given distance

note on taking square roots modulo n in this article it is shown how gauss 1981 famous cyclotomic sum formula can be used for extracting square roots modulo n

fast depth based subgraph kernels for unattributed graphs in this paper we investigate two fast subgraph kernels based on a depth based representation of graph structure both methods gauge depth information through a family of k layer expansion subgraphs rooted at a vertex 1 the first method commences by computing a centroid based complexity trace for each graph using a depth based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices 2 this subgraph kernel is computed by measuring the jensen shannon divergence between centroid based complexity entropy traces the second method on the other hand computes a depth based representation around each vertex in turn the corresponding subgraph kernel is computed using isomorphisms tests to compare the depth based representation rooted at each vertex in turn for graphs with n vertices the time complexities for the two new kernels are o n 2 and o n 3 in contrast to o n 6 for the classic gartner graph kernel 3 key to achieving this efficiency is that we compute the required shannon entropy of the random walk for our kernels with o n 2 operations this computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state of the art graph kernels experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels highlightswe investigate two fast subgraph kernels based on a depth based representation our kernels gauge depth information rooted at a vertex for a graph the time complexities for the two kernels are o n 2 and o n 3 we evaluate the performance of our subgraph kernels on standard graph datasets we demonstrate the effectiveness of the proposed subgraph kernels

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

spectral properties of random reactance networks and random matrix pencils our goal is to study the statistical properies of dielectric resonances which are poles of conductance of a large random lc network such poles are a particular example of eigenvalues n of matrix pencils h w with w being a positive definite matrix and h a random real symmetric one we first consider spectra of the matrix pencils with independent identically distributed entries of h then we concentrate on an infinite range full connectivity version of a random lc network in all cases we calculate the mean eigenvalue density and the two point correlation function in the framework of efetov s supersymmetry approach fluctuations in spectra turn out to be the same as those provided by the wigner dyson theory of usual random matrices

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

duality between equilibrium and growing networks in statistical physics any given system can be either at an equilibrium or away from it networks are not an exception most network models can be classified as either equilibrium or growing here we show that under certain conditions there exists an equilibrium formulation for any growing network model and vice versa the equivalence between the equilibrium and nonequilibrium formulations is exact not only asymptotically but even for any finite system size the required conditions are satisfied in random geometric graphs in general and causal sets in particular and to a large extent in some real networks

diffusion on hierarchical systems of weakly coupled networks abstract we analyzed diffusion dynamics on weakly coupled networks interconnected networks by means of separation of time scales using an adiabatic approximation we reduced the system dynamics to a markov chain with aggregated variables and derived a transport equation that is analogous to fick s first law and includes a driving force entropy production is a sum of microscopic entropy transport which results from the particle s migration between networks of different topologies and macroscopic entropy production of the markov chain equilibrium particles partition between different sub networks depends only on internal sub network parameters by changing structure of networks one can not only modify diffusion constants but can also induce or reverse the direction of the particles flow between different networks our framework confirmed by numerical simulations is also useful for considering diffusion in nested systems corresponding to hierarchical networks with several different time scales thus it can serve to uncover hidden hierarchy levels from observations of diffusion processes

entropy of network ensembles in this paper we generalize the concept of random networks to describe network ensembles with nontrivial features by a statistical mechanics approach this framework is able to describe undirected and directed network ensembles as well as weighted network ensembles these networks might have nontrivial community structure or in the case of networks embedded in a given space they might have a link probability with a nontrivial dependence on the distance between the nodes these ensembles are characterized by their entropy which evaluates the cardinality of networks in the ensemble in particular in this paper we define and evaluate the structural entropy i e the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence we stress the apparent paradox that scale free degree distributions are characterized by having small structural entropy while they are so widely encountered in natural social and technological complex systems we propose a solution to the paradox by proving that scale free degree distributions are the most likely degree distribution with the corresponding value of the structural entropy finally the general framework we present in this paper is able to describe microcanonical ensembles of networks as well as canonical or hidden variable network ensembles with significant implications for the formulation of network constructing algorithms

research on the search ability of brownian particles on networks with an adaptive mechanism abstract in this paper we focus on the search ability of brownian particles with an adaptive mechanism in the adaptive mechanism nodes are allowed to be able to change their own accepting probability according to their congestion states two searching traffic models the static one in which nodes have fixed accepting probability to the incoming particles and the adaptive one in which nodes have adaptive accepting probability to the incoming particles are presented for testing the adaptive mechanism instead of number of hops we use the traveling time which includes not only the number of hops for a particle to jump from the source node to the destination but also the time that the particle stays in the queues of nodes to evaluate the search ability of brownian particles we apply two models to different networks the experiment results show that the adaptive mechanism can decrease the network congestion and the traveling time of the first arriving particle furthermore we investigate the influence of network topologies on the congestion of networks by addressing several main properties degree distribution average path length and clustering coefficient we show the reason why random topologies are more able to deal with congested traffic states than others we also propose an absorption strategy to deal with the additional brownian particles in networks the experiment results on barabasi albert ba scale free networks show that the absorption strategy can increase the probability of a successful search and decrease the average per node particles overhead for our models

entropy of tournament digraphs abstract the renyi entropy h of complete antisymmetric directed graphs i e tournaments is explored we optimize h when 2 and 3 and find that as increases h s sensitivity to what we refer to as regularity increases as well a regular tournament on n vertices is one with each vertex having out degree n 1 2 but there is a lot of diversity in terms of structure among the regular tournaments for example a regular tournament may be such that each vertex s out set induces a regular tournament a doubly regular tournament or a transitive tournament a rotational tournament as increases on the set of regular tournaments h has maximum value on doubly regular tournaments and minimum value on rotational tournaments the more regular the higher the entropy we show however that among all tournaments on a fixed number of vertices h 2 and h 3 are maximized by any regular tournament on that number of vertices we also provide a calculation that is equivalent to the von neumann entropy but may be applied to any directed or undirected graph and shows that the von neumann entropy is a measure of how quickly a random walk on the graph or directed graph settles

shannon and von neumann entropy of random networks with heterogeneous expected degree entropic measures of complexity are able to quantify the information encoded in complex network structures several entropic measures have been proposed in this respect here we study the relation between the shannon entropy and the von neumann entropy of networks with given expected degree sequence we find in different examples of network topologies that when the degree distribution contains some heterogeneity an intriguing correlation emerges between the two entropic quantities this results seems to suggest that heterogeneity in the expected degree distribution is implying an equivalence between a quantum and a classical description of networks which respectively corresponds to the von neumann and the shannon entropy

entropic origin of disassortativity in complex networks why are most empirical networks with the prominent exception of social ones generically degree degree anticorrelated to answer this long standing question we define the ensemble of correlated networks and obtain the associated shannon entropy maximum entropy can correspond to either assortative correlated or disassortative anticorrelated configurations but in the case of highly heterogeneous scale free networks a certain disassortativity is predicted offering a parsimonious explanation for the question above our approach provides a neutral model from which in the absence of further knowledge regarding network evolution one can obtain the expected value of correlations when empirical observations deviate from the neutral predictions as happens for social networks one can then infer that there are specific correlating mechanisms at work

class of correlated random networks with hidden variables we study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices we find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices the expressions obtained are checked by means of numerical simulations in a particular example the general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure we also present an extension of the class to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system

identifying the borders of mathematical knowledge based on a divide and conquer approach knowledge about nature has been organized into a set of interrelated facts allowing a natural representation in terms of graphs each chunk of knowledge corresponds to a node while relationships between such chunks are expressed as edges this organization becomes particularly clear in the case of mathematical theorems with their intense cross implications and relationships we have derived a web of mathematical theorems from wikipedia and thanks to the powerful concept of entropy identified its more central and frontier elements our results also suggest that the central nodes are the oldest theorems while the frontier nodes are those recently added to the network the network communities have also been identified allowing further insights about the organization of this network such as its highly modular structure

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

entropy rate of diffusion processes on complex networks we introduce the concept of entropy rate to characterize a diffusion process on a complex network the entropy rate represents the minimal amount of information necessary to describe the diffusion on the network and is a quantity extremely sensitive to the network topology and dynamics by opportunely tuning the kind of diffusion the entropy rate allows one to extract different properties of the network structure moreover entropy maximization indicates how to design optimal diffusion processes providing a new theoretical tool with applications to social technological and communication systems

a method for analyzing clustered interval censored data based on cox s model methods for analyzing interval censored data are well established unfortunately these methods are inappropriate for the studies with correlated data in this paper we focus on developing a method for analyzing clustered interval censored data our method is based on cox s proportional hazard model with piecewise constant baseline hazard function the correlation structure of the data can be modeled by using clayton s copula or independence model with proper adjustment in the covariance estimation we establish estimating equations for the regression parameters and baseline hazards and a parameter in copula simultaneously simulation results confirm that the point estimators follow a multivariate normal distribution and our proposed variance estimations are reliable in particular we found that the approach with independence model worked well even when the true correlation model was derived from clayton s copula we applied our method to a family based cohort study of pandemic h1n1 influenza in taiwan during 2009 2010 using the proposed method we investigate the impact of vaccination and family contacts on the incidence of ph1n1 influenza

mixture cure model with random effects for clustered interval censored survival data the mixture cure model is an effective tool for analysis of survival data with a cure fraction this approach integrates the logistic regression model for the proportion of cured subjects and the survival model either the cox proportional hazards or accelerated failure time model for uncured subjects methods based on the mixture cure model have been extensively investigated in the literature for data with exact failure censoring times in this paper we propose a mixture cure modeling procedure for analyzing clustered and interval censored survival time data by incorporating random effects in both the logistic regression and ph regression components under the generalized linear mixed model framework we develop the reml estimation for the parameters as well as an iterative algorithm for estimation of the survival function for interval censored data the estimation procedure is implemented via an em algorithm a simulation study is conducted to evaluate the performance of the proposed method in various practical situations to demonstrate its usefulness we apply the proposed method to analyze the interval censored relapse time data from a smoking cessation study whose subjects were recruited from 51 zip code regions in the southeastern corner of minnesota