on the convergence of additive and multiplicative splitting iterations for systems of linear equations we study convergence conditions for the additive and the multiplicative splitting iteration methods i e two generalizations of the additive and the multiplicative schwarz iterations for hermitian and non hermitian systems of linear equations under an algebraic setting theoretical analyses show that when the coefficient and the splitting matrices are hermitian or non hermitian but diagonalizable satisfying mild conditions both additive and multiplicative splitting iteration methods are convergent even if the coefficient matrix is indefinite

average path length in random networks analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found we apply the approach to classical random graphs of erdos and renyi er evolving networks introduced by barabasi and albert as well as random networks with asymptotic scale free connectivity distributions characterized by an arbitrary scaling exponent alpha 2 our result for 2 infinity there is a saturation effect for the average path length

on the convergence of parallel nonstationary multisplitting iteration methods the convergence properties of a variant of the parallel chaotic multisplitting iteration method called the nonstationary multisplitting iteration method for solving large sparse systems of linear equations are further discussed when the coefficient matrix is an h matrix or a positive definite matrix respectively moreover when the coefficient matrix is a monotone matrix the monotone convergence theory and the monotone comparison theorem about this method are established this directly leads to several novel sufficient conditions for guaranteeing the convergence of this parallel nonstationary multisplitting iteration method

weighted max norms splittings and overlapping additive schwarz iterations summary weighted max norm bounds are obtained for algebraic additive schwarz iterations with overlapping blocks for the solution of ax b when the coefficient matrix a is an m matrix the case of inexact local solvers is also covered these bounds are analogous to those that exist using a norms when the matrix a is symmetric positive definite a new theorem concerningp regular splittings is presented which provides a useful tool for the a norm bounds furthermore a theory of splittings is developed to represent algebraic additive schwarz iterations this representation makes a connection with multisplitting methods with this representation and using a comparison theorem it is shown that a coarse grid correction improves the convergence of additive schwarz iterations when measured in weighted max norm

random walks on discrete groups of polynomial volume growth let mu be a probability measure with finite support on a discrete group gamma of polynomial volume growth the main purpose of this paper is to study the asymptotic behavior of the convolution powers mu n of mu if mu is centered then we prove upper and lower gaussian estimates we prove a central limit theorem and we give a generalization of the berry esseen theorem these results also extend to noncentered probability measures we study the associated riesz transform operators the main tool is a parabolic harnack inequality for centered probability measures which is proved by using ideas from homogenization theory and by adapting the method of krylov and safonov this inequality implies that the positive mu harmonic functions are constant finally we give a characterization of the mu harmonic functions which grow polynomially

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

a directed network of greek and roman mythology abstract we construct a directed network using a dictionary of greek and roman mythology in which the nodes represent the entries listed in the dictionary and we make directional links from an entry to other entries that appear in its explanatory part we find that this network is clearly not a random network but a directed scale free network in which the distributions of out degree and in degree follow a power law with exponents out 3 0 and in 2 5 respectively also we measure several quantities which describe the topological properties of the network and compare it to that of other real networks

local world evolving networks with tunable clustering abstract we propose an extended local world evolving network model including a triad formation tf step in the process of network evolution random fluctuation in the number of new edges is involved we derive analytical expressions for degree distribution clustering coefficient and average path length our model can unify the generic properties of real life networks scale free degree distribution high clustering and small inter node separation moreover in our model the clustering coefficient is tunable simply by changing the expected number of tf steps after a single local preferential attachment step

poisson furstenberg boundary and growth of groups we study the poisson furstenberg boundary of random walks on permutational wreath products we give a sufficient condition for a group to admit a symmetric measure of finite first moment with non trivial boundary and show that this criterion is useful to establish exponential word growth of groups we construct groups of exponential growth such that all finitely supported not necessarily symmetric possibly degenerate random walks on these groups have trivial boundary this gives a negative answer to a question of kaimanovich and vershik

maximal planar networks with large clustering coefficient and power law degree distribution in this article we propose a simple rule that generates scale free networks with very large clustering coefficient and very small average distance these networks are called random apollonian networks rans as they can be considered as a variation of apollonian networks we obtain the analytic results of power law exponent ensuremath gamma 3 and clustering coefficient c frac 46 3 ensuremath 36 phantom rule 0 2em 0ex mathrm ln phantom rule 0 2em 0ex frac 3 2 ensuremath approx 0 74 which agree with the simulation results very well we prove that the increasing tendency of average distance of rans is a little slower than the logarithm of the number of nodes in rans since most real life networks are both scale free and small world networks rans may perform well in mimicking the reality the rans possess hierarchical structure as c k ensuremath sim k ensuremath 1 that are in accord with the observations of many real life networks in addition we prove that rans are maximal planar networks which are of particular practicability for layout of printed circuits and so on the percolation and epidemic spreading process are also studied and the comparisons between rans and barab asi albert ba as well as newman watts nw networks are shown we find that when the network order n the total number of nodes is relatively small as n ensuremath sim 10 4 the performance of rans under intentional attack is not sensitive to n while that of ba networks is much affected by n and the diseases spread slower in rans than ba networks in the early stage of the suseptible infected process indicating that the large clustering coefficient may slow the spreading velocity especially in the outbreaks

parallel multisplitting methods with optimal weighting matrices for linear systems in this paper the parallel multisplitting iterative methods with optimal weighting matrices are presented to solve a linear system of equations in which the coefficient matrix is a symmetric positive definite matrix the zero pattern in weighting matrices is determined by preset set the non zero entries of weighting matrices in overlap multisplitting methods are determined optimally by finding the optimal point in hyperplane or convex combination of m points generated by parallel multisplitting iterations several schemes of finding the optimal weighting matrices are given especially the nonnegative assumption in the weighting matrices is eliminated the convergence properties are discussed for these parallel multisplitting methods finally our numerical examples show that these parallel multisplitting methods with the optimal weighting matrices are effective

a note on comparison theorems for splittings and multisplittings of hermitian positive definite matrices abstract we discuss iterative methods for the solution of the linear system ax b which are based on a single splitting or a multisplitting of a in order to compare different methods it is common to compare the spectral radius of the iterative matrix for m matrices a and weak regular splittings there exist well known comparison theorems here we give a comparison theorem for splittings of hermitian positive definite matrices furthermore we establish a comparison theorem for multisplittings of a hermitian positive definite matrix

carne varopoulos bounds for centered random walks we extend the carne varopoulos upper bound on the probability transitions of a markov chain to a certain class of nonreversible processes by introducing the definition of a centering measure in the case of random walks on a group we study the connections between different notions of centering

effect of node deleting on network structure abstract the ever increasing knowledge of the structure of various real world networks has uncovered their complex multi mechanism governed evolution processes therefore a better understanding of the structure and evolution of these networked complex systems requires us to describe such processes in a more detailed and realistic manner in this paper we introduce a new type of network growth rule which comprises addition and deletion of nodes and propose an evolving network model to investigate the effect of node deleting on network structure it is found that with the introduction of node deleting network structure is significantly transformed in particular degree distribution of the network undergoes a transition from scale free to exponential forms as the intensity of node deleting increases at the same time nontrivial disassortative degree correlation develops spontaneously as a natural result of network evolution in the model we also demonstrate that node deleting introduced in the model does not destroy the connectedness of a growing network so long as the increasing rate of edges is not excessively small in addition it is found that node deleting will weaken but not eliminate the small world effect of a growing network and generally it will decrease the clustering coefficient in a network

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

relationship between the in degree and out degree of www abstract in this paper the relationship between the in degree and out degree of world wide web is studies at each time step a new node with out degree k out is added where k out obeys the power law distribution and its mean value is m the analytical and simulation results suggest that the exponent of in degree distribution would be i 2 1 m depending on the average out degree this finding is supported by the empirical data which has not been emphasized by the previous studies on directed networks

symmetric multisplitting of a symmetric positive definite matrix abstract a parallel symmetric multisplitting method for solving a symmetric positive system ax b is presented here the s p d symmetric positive definite matrix a need not be assumed in a special form e g the dissection form r e white siam j matrix anal appl 11 1990 69 82 the main tool for deriving our method is the diagonally compensated reduction cf o axelsson l kolotilina numer linear algebra appl 1 1994 155 177 the convergence of the presented parallel symmetric multisplitting method is also discussed by using this tool

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

analytical results for bond percolation and k core sizes on clustered networks an analytical approach to calculating bond percolation thresholds sizes of k cores and sizes of giant connected components on structured random networks with nonzero clustering is presented the networks are generated using a generalization of trapman s p trapman theor popul biol 71 160 2007 model of cliques embedded in treelike random graphs the resulting networks have arbitrary degree distributions and tunable degree dependent clustering the effect of clustering on the bond percolation thresholds for networks of this type is examined and contrasted with some recent results in the literature for very high levels of clustering the percolation threshold in these generalized trapman networks is increased above the value it takes in a randomly wired unclustered network of the same degree distribution in assortative scale free networks where the variance of the degree distribution is infinite this clustering effect can lead to a nonzero percolation epidemic threshold

convergence of non stationary parallel multisplitting methods for hermitian positive definite matrices non stationary multisplitting algorithms for the solution of linear systems are studied convergence of these algorithms is analyzed when the coefficient matrix of the linear system is hermitian positive definite asynchronous versions of these algorithms are considered and their convergence investigated

scale free properties of weighted networks with connectivity driven topology abstract the rate equations are used to study the scale free behavior of the weight distribution in evolving networks whose topology is determined only by degrees of preexisting vertices an analysis of these equations shows that the degree distribution and thereby the weight distribution remain unchanged when the probability rate of attaching new nodes is replaced with an unnormalized rate determined by the ratio of the degree of a randomly selected old node to the maximal node degree at the current stage of the network evolution such a modification of the attachment rule is argued to accelerate considerable numerical simulations of both unweighted and weighted networks belonging to the class of investigated evolving systems it is also proved that the studied rate equations have a solution corresponding to the total weight concentrated at individual vertices distribution displaying the power law behavior for asymptotically large weights

epidemic outbreaks in growing scale free networks with local structure abstract the class of generative models has already attracted considerable interest from researchers in recent years and much expanded the original ideas described in ba model most of these models assume that only one node per time step joins the network in this paper we grow the network by adding n interconnected nodes as a local structure into the network at each time step with each new node emanating m new edges linking the node to the preexisting network by preferential attachment this successfully generates key features observed in social networks these include power law degree distribution p k k 3 where n 1 m is a tuning parameter defined as the modularity strength of the network nontrivial clustering assortative mixing and modular structure moreover all these features are dependent in a similar way on the parameter we then study the susceptible infected epidemics on this network with identical infectivity and find that the initial epidemic behavior is governed by both of the infection scheme and the network structure especially the modularity strength the modularity of the network makes the spreading velocity much lower than that of the ba model on the other hand increasing the modularity strength will accelerate the propagation velocity

exploring complex networks by walking on them we carry out a comparative study of the problem of a walker searching several typical complex networks the search efficiency is evaluated for various strategies having no knowledge of the global properties of the underlying networks and the optimal path between any two given nodes it is found that the best search strategy is the self avoiding random walk the preferentially self avoiding random walk does not help in improving the search efficiency further in return topological information of the underlying networks may be drawn by comparing the results of the different search strategies

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

cycles and clustering in bipartite networks we investigate the clustering coefficient in bipartite networks where cycles of size three are absent and therefore the standard definition of clustering coefficient cannot be used instead we use another coefficient given by the fraction of cycles with size four showing that both coefficients yield the same clustering properties the new coefficient is computed for two networks of sexual contacts one bipartite and another where no distinction between the nodes is made monopartite in both cases the clustering coefficient is similar furthermore combining both clustering coefficients we deduce an expression for estimating cycles of larger size which improves previous estimations and is suitable for either monopartite and multipartite networks and discuss the applicability of such analytical estimations

a dirichlet principle for non reversible markov chains and some recurrence theorems we extend the dirichlet principle to non reversible markov processes on countable state spaces we present two variational formulas for the solution of the poisson equation or equivalently for the capacity between two disjoint sets as an application we prove a some recurrence theorems in particular we show the recurrence of two dimensional cycle random walks under a second moment condition on the winding numbers

generation of scale free networks using a simple preferential rewiring dynamics abstract we propose a simple dynamical process for non growing networks where steady states in the long time limit exhibit power law degree distributions with the exponent 2 at each time step two nodes i and j are randomly selected and one incoming link to i is redirected to j with the rewiring probability r determined only by degrees of two nodes k i and k j while higher degree nodes are preferred to get another link this is an application of the general model introduced earlier s ree phys rev e 73 2006 026115 to take the structure of networks into account we also consider three possible distinctions for the model i how we choose a rewiring link out of all incoming links to i three cases ii whether links are directed or not two cases iii types of networks considering the existence of self loops and multiple links two cases as a result we specify the total of 12 different cases of the model we then observe numerically that networks will evolve to steady states with power law degree distributions when parameters of the model satisfy certain conditions this work is from an effort to find a simple model of the network dynamics generating scale free networks and has a potential to become an underlying mechanism for wide range of scale free non growing networks

markov chain approximations to nonsymmetric diffusions with bounded coefficients we consider a certain class of nonsymmetric markov chains and obtain heat kernel bounds and parabolic harnack inequalities using the heat kernel estimates we establish a sufficient condition for the family of markov chains to converge to nonsymmetric diffusions as an application we approximate nonsymmetric diffusions in divergence form with bounded coefficients by nonsymmetric markov chains this extends the results by stroock and zheng to the nonsymmetric divergence forms 2012 wiley periodicals inc

voter model dynamics in complex networks role of dimensionality disorder and degree distribution we analyze the ordering dynamics of the voter model in different classes of complex networks we observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks we also find that when there is no ordering in the system the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity the existence of hubs i e highly connected nodes in the network modifies the linear system size scaling law of the survival time the size of an ordered domain is sensitive to the network disorder and the average degree decreasing with both however it seems not to depend on network size and on the heterogeneity of the degree distribution

effect of aging on network structure in network evolution the effect of aging is universal in scientific collaboration network scientists have a finite time span of being active in movie actors network once popular stars are retiring from stage devices on the internet may become outmoded with techniques developing so rapidly here we find in citation networks that this effect can be represented by an exponential decay factor e ensuremath ensuremath beta ensuremath tau where ensuremath tau is the node age while other evolving networks the internet for instance may have different types of aging for example a power law decay factor which is also studied and compared it has been found that as soon as such a factor is introduced to the barabasi albert scale free model the network will be significantly transformed the network will be clustered even with infinitely large size and the clustering coefficient varies greatly with the intensity of the aging effect i e it increases linearly with ensuremath beta for small values of ensuremath beta and decays exponentially for large values of ensuremath beta at the same time the aging effect may also result in a hierarchical structure and a disassortative degree degree correlation generally the aging effect will increase the average distance between nodes but the result depends on the type of the decay factor the network appears like a one dimensional chain when exponential decay is chosen but with power law decay a transformation process is observed i e from a small world network to a hypercubic lattice and to a one dimensional chain finally the disparities observed for different choices of the decay factor in clustering average node distance and probably other aspects not yet identified are believed to bear significant meaning on empirical data acquisition

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

from regular to growing small world networks abstract we propose a growing model which interpolates between one dimensional regular lattice and small world networks the model undergoes an interesting phase transition from large to small worlds we investigate the structural properties by both theoretical predictions and numerical simulations our growing model is a complementarity for the important static watts strogatz network model

scale free networks with a large to hypersmall world transition abstract recently there has been a tremendous interest in models of networks with a power law distribution of degree so called scale free networks it has been observed that such networks normally have extremely short path lengths scaling logarithmically or slower with system size as an exotic and counterintuitive example we propose a simple stochastic model capable of generating scale free networks with linearly scaling distances furthermore by tuning a parameter the model undergoes a phase transition to a regime with extremely short average distances apparently slower than log log n which we call a hypersmall world regime we characterize the degree degree correlation and clustering properties of this class of networks

a unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations abstract a unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods suitable to the simd and mimd multiprocessor systems respectively is presented and its convergence theory is established under rather weak conditions these afford general method models and systematical convergence criterions for studying the parallel iterations in the sense of matrix multisplitting in addition how the known parallel matrix multisplitting iterative methods can be classified into this new framework and what novel ones can be generated by it are shown in detail

cyclic topology in complex networks we introduce a cyclic coefficient r which characterizes the degree of circulation in complex networks if a network has a perfect treelike structure then r becomes zero the larger value of r represents that the network has more cyclic structure we measure both the cyclic coefficients and the distributions of local cyclic coefficients for various networks and discuss the cyclic structures of them

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

statistical analysis of 22 public transport networks in poland public transport systems in 22 polish cities have been analyzed the sizes of these networks range from n 152 to 2881 depending on the assumed definition of network topology the degree distribution can follow a power law or can be described by an exponential function distributions of path lengths in all considered networks are given by asymmetric unimodal functions clustering assortativity and betweenness are studied all considered networks exhibit small world behavior and are hierarchically organized a transition between dissortative small networks n approximately or 500 is observed

on the convergence of nonstationary multisplitting two stage iteration methods for hermitian positive definite linear systems convergence properties of the nonstationary multisplitting two stage iteration methods for solving large sparse system of linear equations are further studied when the coefficient matrices are hermitian positive definite matrices

topology and correlations in structured scale free networks we study a recently introduced class of scale free networks showing a high clustering coefficient and nontrivial connectivity correlations we find that the connectivity probability distribution strongly depends on the fine details of the model we solve exactly the case of low average connectivity providing also exact expressions for the clustering and degree correlation functions the model also exhibits a lack of small world properties in the whole parameter range we discuss the physical properties of these networks in the light of the present detailed analysis

network clustering and community detection using modulus of families of loops we study the structure of loops in networks using the notion of modulus of loop families we introduce an alternate measure of network clustering by quantifying the richness of families of simple loops modulus tries to minimize the expected overlap among loops by spreading the expected link usage optimally we propose weighting networks using these expected link usages to improve classical community detection algorithms we show that the proposed method enhances the performance of certain algorithms such as spectral partitioning and modularity maximization heuristics on standard benchmarks

convergence theorems for block splitting iterative methods for linear systems in this paper we will present the block splitting iterative methods with general weighting matrices for solving linear systems of algebraic equations ax b when the coefficient matrix a is symmetric positive definite of block form and establish the convergence theories with respect to the general weighting matrices but special splittings finally a numerical example shows the advantage of this method

a new deterministic complex network model with hierarchical structure abstract we introduce a new simple pseudo tree like network model deterministic complex network dcn the proposed dcn model may simulate the hierarchical structure nature of real networks appropriately and have the unique property of skipping the levels which is ubiquitous in social networks our results indicate that the dcn model has a rather small average path length and large clustering coefficient leading to the small world effect strikingly our dcn model obeys a discrete power law degree distribution p k k with exponent approaching 1 0 we also discover that the relationship between the clustering coefficient and degree follows the scaling law c k k 1 which quantitatively determines the dcn s hierarchical structure

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved