orderings for conjugate gradient preconditionings many preconditioners e g ssor ilu for the conjugate gradient method require the solution of sparse triangular systems of equations for elliptic boundary value problems one approach to obtaining additional parallelism in the solution of these systems is the use of red black or multicolor orderings there has been increasing evidence however that these orderings degrade the rate of convergence compared with the natural ordering an alternative is the diagonal ordering which maintains the rate of convergence of the natural ordering but has less parallelism than multicolor orderings this paper reviews these as well as other orderings and then gives some results that help to explain why the red black ordering gives an inferior rate of convergence

convergence analysis without regularity assumptions for multigrid algorithms based on sor smoothing convergence estimates for the standard multigrid algorithms based on sor smoothing are established without regularity assumptions for symmetric and positive definite variational problems the results are applied to second order elliptic equations discretized by using finite element methods it is shown that the convergence is deteriorated no worse than 1 c sigma j 1 for the multigrid algorithms with sor nonsymmetric smoothing where j is the number of levels and sigma the relaxation parameter the results can be regarded as a generalization of the estimates in 10 where only the jacobi smoothing was considered this extension is not trivial since sor smoothing yields a nonsymmetric operator while the jacobi gives a symmetric smoothing operator the estimates are well suited to difficult problems with rough solutions as no regularity is assumed in the analysis

models of social networks based on social distance attachment we propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space we derive analytical results corroborated by extensive numerical simulations showing that the model reproduces the main statistical characteristics of real social networks large clustering coefficient positive degree correlations and the emergence of a hierarchy of communities the model is confronted with the social network formed by people that shares confidential information using the pretty good privacy pgp encryption algorithm the so called web of trust of pgp

detect overlapping and hierarchical community structure in networks abstract clustering and community structure is crucial for many network systems and the related dynamic processes it has been shown that communities are usually overlapping and hierarchical however previous methods investigate these two properties of community structure separately this paper proposes an algorithm eagle to detect both the overlapping and hierarchical properties of complex community structure together this algorithm deals with the set of maximal cliques and adopts an agglomerative framework the quality function of modularity is extended to evaluate the goodness of a cover the examples of application to real world networks give excellent results

local identifiability of the factor analysis and measurement error model parameter abstract measurement error regression models are factor analysis models the latent correct regressors are the factors there is however no common statistical method between the factor analysis and the regression model because the covariance elements that are known identifying constituents of the former model are unknown in the latter instead the idea that the data come from the same regression model as with panel data but can be grouped in two or more groups each group having its own different regrressor generating process is shown to supply credible restrictions we generalize and compare relevant identifiability criteria and corresponding asymptotically efficient estimators that are recursive in the number of overidentifying restrictions

new parallel sor method by domain partitioning in this paper we propose and analyze a new parallel sor method the psor method formulated by using domain partitioning and interprocessor data communication techniques we prove that the psor method has the same asymptotic rate of convergence as the red black r b sor method for the five point stencil on both strip and block partitions and as the four color r b g o sor method for the nine point stencil on strip partitions we also demonstrate the parallel performance of the psor method on four different mimd multiprocessors a ksr1 an intel delta a paragon and an ibm sp2 finally we compare the parallel performance of psor r b sor and r b g o sor numerical results on the paragon indicate that psor is more efficient than r b sor and r b g o sor in both computation and interprocessor data communication

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

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

diagonally striped matrices and approximate inverse preconditioners the inverse of a banded matrix is in general dense if the structure of the original banded matrix is striped that is the non zero diagonals are separated by one or more zero diagonals the inverse may exhibit a similar striped structure the motivation for studying inverses of striped matrices is to obtain efficient preconditioners for systems arising from radiation transport equations whose matrices include dominant values along diagonal stripes linear systems whose system matrix has a striped inverse lend themselves to the use of a sparse approximate inverse spai preconditioner whose structure is derived from that of the actual inverse

factor analysis by instrumental variables methods three alternative estimation procedures for factor analysis based on the instrumental variables method are presented these procedures are justified by the method of least squares formulas for asymptotic standard errors of factor loadings are derived the procedures are empirically compared to the method of maximum likelihood the conclusion based on the data used in this study is that two of the procedures seem to work well

analysis of weighted networks the connections in many networks are not merely binary entities either present or not but have associated weights that record their strengths relative to one another recent studies of networks have by and large steered clear of such weighted networks which are often perceived as being harder to analyze than their unweighted counterparts here we point out that weighted networks can in many cases be analyzed using a simple mapping from a weighted network to an unweighted multigraph allowing us to apply standard techniques for unweighted graphs to weighted ones as well we give a number of examples of the method including an algorithm for detecting community structure in weighted networks and a simple proof of the maximum flow minimum cut theorem

processor efficient sparse matrix vector multiplication abstract the matrix vector multiplication operation is the kernel of most numerical algorithms typically matrix vector multiplication algorithms exploit the sparsity of a matrix either to reduce the time taken or the memory used in the case of parallel implementations of numerical algorithms the sparsity of a matrix can also be used to reduce the number of processing elements pes required for example in leiserson s systolic array algorithm for matrix vector multiplication the nonzeros in the matrix are partitioned into bands and each band is fed into a different pe similarly in melhem s algorithm the nonzeros are partitioned into stripes and each stripe is fed into a distinct pe however in many practical applications such as the numerical solution of partial differential equations a matrix is much sparser than is indicated by its bandwidth or by its stripe width a new measure of sparsity of a matrix called staircase width is introduced a staircase in a matrix m mi j n n is a set of matrix positions s i j mi j 0 such that no position in s is strictly to the northeast of another position in s the staircase cover of a matrix is a partition of the positions of nonzero entries in the matrix into staircases and the staircase width of a matrix is the size of a smallest staircase cover of the matrix the staircase width of a matrix is never larger than the bandwidth or the stripe width of a matrix moreover it is significantly smaller than either for some commonly occurring types of matrices such as arrowhead matrices experiments with sparse matrices derived from a variety of engineering problems suggest that in practice the staircase width of a matrix is about half the stripe width of the matrix an efficient algorithm to compute a minimum width staircase cover of a matrix is presented staircase covers are used as the basis of a new parallel algorithm to compute a matrix vector product on a linear systolic array of pes this algorithm uses as many pes as the staircase width of the matrix and therefore saves on the number of pes relative to algorithms of leiserson and melhem it is shown that despite using fewer pes the algorithm uses no more time steps than leiserson s or melhem s algorithm

on red black sor smoothing in multigrid optimal relaxation parameters are obtained for red black gauss seidel relaxation in multigrid solvers of a family of elliptic equations the resulting relaxation schemes are found to retain very high efficiency over an appreciable range of coefficients of the elliptic differential operator yielding simple inexpensive and fully parallelizable smoothers in many situations where less cost effective block and alternating direction schemes are commonly used

benchmark graphs for testing community detection algorithms community structure is one of the most important features of real networks and reveals the internal organization of the nodes many algorithms have been proposed but the crucial issue of testing i e the question of how good an algorithm is with respect to others is still open standard tests include the analysis of simple artificial graphs with a built in community structure that the algorithm has to recover however the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks here we introduce a class of benchmark graphs that account for the heterogeneity in the distributions of node degrees and of community sizes we use this benchmark to test two popular methods of community detection modularity optimization and potts model clustering the results show that the benchmark poses a much more severe test to algorithms than standard benchmarks revealing limits that may not be apparent at a first analysis

the asymptotic covariance matrix of maximum likelihood estimates in factor analysis the case of nearly singular matrix of estimates of unique variances this paper is concerned with the asymptotic covariance matrix acm of maximum likelihood estimates mles of factor loadings and unique variances when one element of mles of unique variances is nearly zero i e the matrix of mles of unique variances is nearly singular in this situation standard formulas break down we give explicit formulas for the acm of mles of factor loadings and unique variances that could be used even when an element of mles of unique variances is very close to zero we also discuss an alternative approach using the augmented information matrix under a nearly singular matrix of mles of unique variances and derive the partial derivatives of the alternative constraint functions with respect to the elements of factor loadings and unique variances 2000 elsevier science inc all rights reserved ams classification 62h25 62f12

a new parallel gauss seidel method based on alternating group explicit method and domain decomposition method a new parallel gauss seidel method is presented for solution of system of linear equations related to finite difference discretization of partial differential equations this method is based on domain decomposition method and local coupling between interfaces of neighbor sub domains same as alternating group explicit method this method is convergent and number of iterations for achieving convergence criteria is near the original gauss seidel method sometimes better and sometimes worse but difference is very small the convergence theory is discussed in details numerical results are given to justify the convergence and performance of the proposed iterative method

analysis of the sor iteration for the 9 point laplacian the sor iteration for solving linear systems of equations depends upon an overrelaxation factor omega a theory for determining omega was given by young iterative methods for solving partial differential equations of elliptic types trans amer math soc 76 1954 pp 92 111 for consistently ordered matrices here we determine the optimal omega for the 9 point stencil for the model problem of laplace s equation on a square we consider several orderings of the equations including the natural rowwise and multicolor orderings all of which lead to nonconsistently ordered matrices and find two equivalence classes of orderings with different convergence behavior and optimal omega s we compare our results for the natural rowwise ordering to those of garabedian estimation of the relaxation factor for small mesh size math comp 10 1956 pp 183 185 and explain why both results are in a sense correct even though they differ we also analyze a pseudo sor method for the model p

determination of stripe structures for finite element matrics stripe structures were introduced by the author parallel computing to appear as a means for the inclusion of the nonzero elements of a sparse matrix into a regular pattern which allows for efficient parallel manipulation in this paper the stripe structures of stiffness matrices resulting from irregular domains covered by regular grids are analyzed it is proved that the nonzero elements in these matrices may be covered by very few stripes and that these stripes may be nonoverlapping if the nodes of the grids are numbered appropriately the exact number of stripes which is independent of the size of the problem is derived for different types of grids and different numbering schemes the stripe structures of some irregular grids are also examined

orderings for incomplete factorization preconditioning of nonsymmetric problems numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned krylov subspace methods for the solution of nonsymmetric linear systems is shown the preconditioners used in this study are different variants of incomplete factorizations it is shown that certain reorderings for direct methods such as reverse cuthill mckee can be very beneficial the benefit can be seen in the reduction of the number of iterations and also in measuring the deviation of the preconditioned operator from the identity

noniterative estimation and the choice of the number of factors in exploratory factor analysis based on the usual factor analysis model this paper investigates the relationship between improper solutions and the number of factors and discusses the properties of the noniterative estimation method of ihara and kano in exploratory factor analysis the consistency of the ihara and kano estimator is shown to hold even for an overestimated number of factors which provides a theoretical basis for the rare occurrence of improper solutions and for a new method of choosing the number of factors the comparative study of their estimator and that based on maximum likelihood is carried out by a monte carlo experiment

parallel domain decomposition method with non blocking communication for flow through porous media this paper introduces a domain decomposition method for numerically solving the stokes equation for very large complex geometries examples arise from realistic porous media the computational method is based on the simple semi implicit method for pressure linked equations algorithm which uses a finite differences approach for discretizing the underlying equations it achieves comparable speed and efficiency as lattice boltzmann methods the domain decomposition method splits a large three dimensional region into slices that can be processed in parallel on multi processor computation environments with only minimal communication between the computation nodes with this method the flow through a porous medium with grid sizes up to 20483 voxel has been calculated

image recovery using partitioned separable paraboloidal surrogate coordinate ascent algorithms iterative coordinate ascent algorithms have been shown to be useful for image recovery but are poorly suited to parallel computing due to their sequential nature this paper presents a new fast converging parallelizable algorithm for image recovery that can be applied to a very broad class of objective functions this method is based on paraboloidal surrogate functions and a concavity technique the paraboloidal surrogates simplify the optimization problem the idea of the concavity technique is to partition pixels into subsets that can be updated in parallel to reduce the computation time for fast convergence pixels within each subset are updated sequentially using a coordinate ascent algorithm the proposed algorithm is guaranteed to monotonically increase the objective function and intrinsically accommodates nonnegativity constraints a global convergence proof is summarized simulation results show that the proposed algorithm requires less elapsed time for convergence than iterative coordinate ascent algorithms with four parallel processors the proposed algorithm yields a speedup factor of 3 77 relative to single processor coordinate ascent algorithms for a three dimensional 3 d confocal image restoration problem

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

detecting community structure via the maximal sub graphs and belonging degrees in complex networks abstract community structure is a common phenomenon in complex networks and it has been shown that some communities in complex networks often overlap each other so in this paper we propose a new algorithm to detect overlapping community structure in complex networks to identify the overlapping community structure our algorithm firstly extracts fully connected sub graphs which are maximal sub graphs from original networks then two maximal sub graphs having the key pair vertices can be merged into a new larger sub graph using some belonging degree functions furthermore we extend the modularity function to evaluate the proposed algorithm in addition overlapping nodes between communities are founded successfully finally we report the comparison between the modularity and the computational complexity of the proposed algorithm with some other existing algorithms the experimental results show that the proposed algorithm gives satisfactory results

em algorithms for ml factor analysis the details of em algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models the algorithm is essentially the same for both cases and involves only simple least squares regression operations the largest matrix inversion required is for aq q symmetric matrix whereq is the matrix of factors the example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation

some contributions to efficient statistics in structural models specification and estimation of moment structures current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments e g means and covariances and to distribution theory based on multivariate normality in psychometrics the multinormality assumption is often incorrect so that statistical tests on parameters or model goodness of fit will frequently be incorrect as well it is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary structural representations are developed for generalizations of the bentler weeks joreskog keesling wiley and factor analytic models some asymptotically distribution free efficient estimators for such arbitrary structural models are developed limited information estimators are obtained as well the special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail the argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds many open research areas are described

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

polynomial preconditioning on vector computers abstract this paper is concerned with the computation of the minimal eigenpair of the generalized eigenproblem with the preconditioned conjugate gradient method the ssor polynomial preconditioner and the additive polynomial preconditioner have been analyzed the results of an extensive computer experimentation on cray y mp are reported

the sor method on parallel computers the jacobi overrelaxation jor method is usually cited as a perfect parallel algorithm whereas the successive overrelaxation sor method is considered as quite the opposite for linear systems with dense matrices an algorithm for the sor method is presented which is suited for parallelization nearly in the same way as jor for systems with band matrices an algorithm is described which using a pipeline principle yields a speedups p 1 p 1 p m 1 wherep denotes the number of processors andm the number of sor iterations thus form p there is no speedup whereas the speedup tends to its maximal valuep ifm tends to infinity

a new block parallel sor method and its analysis as the development of the psor method a point parallel sor method by mesh domain partitioning proposed in siam j sci comput 20 1999 pp 2261 2281 this paper introduces a new mesh domain partition and ordering the multitype partition and ordering and proposes a new block parallel sor bpsor method for numerically solving 2 dimensional 2d or three dimensional 3d elliptic boundary problems a general mathematical analysis shows that the bpsor method can have the same asymptotic convergence rate as the corresponding sequential block sor method if the coefficient matrix of the block linear system is consistently ordered it also shows that the original sequential ordering can be maintained in the parallel implementation of the bpsor method so that the bpsor method can be effectively applied to solve complex elliptic boundary problems furthermore three particular multitype orderings are proposed based on strip and block mesh partitions which lead to three effective bpsor methods for solving the five point like linear systems in two dimensions and the seven point like linear systems in three dimensions in addition it is shown that the psor method can be generated from bpsor if each block equation is solved approximately by only one point sor iteration thus the psor method is well defined without involving any interprocessor data communication operations and extended to solving three dimensional 3d problems finally numerical results are presented which confirm the theoretical results and show that bpsor has a good parallel performance on a parallel mimd computer

a community detection method based on local similarity and degree clustering information community detection is of great importance to understand the structures and functions of networks in this paper a novel algorithm is proposed based on local similarity and degree clustering information local similarity is employed to measure the similarity between nodes and their neighbors in order to form communities within which nodes are closely connected degree clustering information a hybrid criterion combining local neighborhood ratio with degree ratio make a large number of nodes with low degree to embrace a small amount of nodes with high degree furthermore each node in small scale communities has the duty to try to connect the nodes with high degree to expand communities and finally the optimal community structure can be obtained simulation results on real and artificial networks show that the proposed algorithm has the excellent performance in terms of accuracy

modularity and extreme edges of the internet we study the spectral properties of a diffusion process taking place on the internet network focusing on the slowest decaying modes these modes identify an underlying modular structure roughly corresponding to individual countries for instance in the slowest decaying mode the diffusion current flows from russia to u s military sites quantitatively the modular structure manifests itself in a 10 times larger participation ratio of its slow decaying modes compared to a random scale free network we propose to use the fraction of nodes participating in slow decaying modes as a general measure of the modularity of a network for the 100 slowest decaying modes of the internet this fraction is 30 finally we suggest that the degree of isolation of an individual module can be assessed by comparing its participation in different diffusion modes

a novel cosine distance for detecting communities in complex networks abstract detecting communities is significant to understand the potential structures and functions of complex systems in order to detect communities more accurately and reasonably a novel algorithm is proposed based on cosine distance and core node in this paper cosine distances between nodes are regarded as their similarity measure and network node vectors can be extracted directly from the similarity matrix without calculating eigenvectors core nodes as the initial communities are found by cosine distance threshold and degree threshold furthermore the initial communities are expanded by adding other nodes with the nearest cosine distance to core nodes through changing degree and cosine distance thresholds constantly the optimal community structure of complex networks can be obtained by optimizing modularity with high accuracy experimental results on both real world and synthetic networks demonstrate the feasibility and effectiveness of the proposed algorithm

a gauss newton algorithm for exploratory factor analysis it is shown that the scoring algorithm for maximum likelihood estimation in exploratory factor analysis can be developed in a way that is many times more efficient than a direct development based on information matrices and score vectors the algorithm offers a simple alternative to current algorithms and when used in one step mode provides the simplest and fastest method presently available for moving from consistent to efficient estimates perhaps of greater importance is its potential for extension to the confirmatory model the algorithm is developed as a gauss newton algorithm to facilitate its application to generalized least squares and to maximum likelihood estimation

detecting overlapping communities of weighted networks via a local algorithm abstract identification of communities is significant in understanding the structures and functions of networks since some nodes naturally belong to several communities the study of overlapping communities has attracted increasing attention recently and many algorithms have been designed to detect overlapping communities in this paper an overlapping communities detecting algorithm is proposed whose main strategies are finding an initial partial community from a node with maximal node strength and adding tight nodes to expand the partial community seven real world complex networks and one synthetic network are used to evaluate the algorithm experimental results demonstrate that the algorithm proposed is efficient for detecting overlapping communities in weighted networks