a two step matrix splitting iteration for computing pagerank the pagerank algorithm plays an important role in determining the importance of web pages the core of this algorithm involves using the classical power method to compute the pagerank vector which is the principal eigenvector of the matrix representing the web link graph nevertheless it is well known that the power method may perform poorly when the second largest eigenvalue is close to the dominant one in this article we present a new approach that is based on the two step splitting iteration framework the description and convergence of the new algorithm are discussed in detail numerical examples are given to illustrate the performance of this algorithm

accelerating the arnoldi method via chebyshev polynomials for computing pagerank abstract in this paper by integrating the arnoldi method with the chebyshev acceleration technique we present the arnoldi chebyshev method for computing the pagerank vector of the google matrix convergence analysis of this method is exhibited in addition some numerical experiments are carried out to demonstrate the competitiveness of the new method with a few state of the art iterative methods

a reordering for the pagerank problem we describe a reordering particularly suited to the pagerank problem which reduces the computation of the pagerank vector to that of solving a much smaller system and then using forward substitution to get the full solution vector we compare the theoretical rates of convergence of the original pagerank algorithm to that of the new reordered pagerank algorithm showing that the new algorithm can do no worse than the original algorithm we present results of an experimental comparison on five datasets which demonstrate that the reordered pagerank algorithm can provide a speedup of as much as a factor of 6 we also note potential additional benefits that result from the proposed reordering

a note on the two step matrix splitting iteration for computing pagerank computing pagerank plays an important part in determining the importance of web pages based on the classical power method and the inner outer iteration proposed by gleich et al 2010 gu et al 2015 presented a two step splitting iteration i e the power inner outer pio iteration for the computation of pagerank in this paper we develop a variant of the pio iteration by applying multi step power method to combine with the inner outer iteration the new method is denoted as the mpio iteration its convergence is analyzed in detail numerical experiments on several pagerank problems are used to illustrate the effectiveness of the mpio iteration

parallel multisplitting iteration methods based on m splitting for the pagerank problem a class of iterative schemes based on the regular splitting is proposed for pagerank computation by some specific splitting these methods will reduce to some well known iterative schemes like jacobi method and successive overrelaxation method to suit computational requirements of the modern high speed multiprocessor environments we further study synchronous parallel counterparts for the new splitting iteration methods by using the m splitting of the coefficient matrix we establish the convergence theory of these synchronous multisplitting iteration methods finally some numerical experiments are presented to illustrate the feasibility stability and effectiveness of the proposed algorithms

adaptive methods for the computation of pagerank we observe that the convergence patterns of pages in the pagerank algorithm have a nonuniform distribution specifically many pages converge to their true pagerank quickly while relatively few pages take a much longer time to converge furthermore we observe that these slow converging pages are generally those pages with high pagerank we use this observation to devise a simple algorithm to speed up the computation of pagerank in which the pagerank of pages that have converged are not recomputed at each iteration after convergence this algorithm which we call adaptive pagerank speeds up the computation of pagerank by nearly 30

on lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres the n dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the n 1 dimensional complex projective space which is isometric to the real grassmann manifold of oriented 2 planes and is a compact hermitian symmetric space of rank 2 in this paper we study geometry of compact lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres from this viewpoint we provide a classification theorem of compact homogeneous lagrangian submanifolds in complex hyperquadrics by using the moment map technique moreover we determine the hamiltonian stability of compact minimal lagrangian submanifolds embedded in complex hyperquadrics which are obtained as gauss images of isoparametric hypersurfaces in spheres with g 1 2 3 distinct principal curvatures

a preconditioning approach to the pagerank computation problem abstract some spectral properties of the transition matrix of an oriented graph indicate the preconditioning of euler richardson er iterative scheme as a good way to compute efficiently the vertexrank vector associated with such graph we choose the preconditioner from an algebra u of matrices thereby obtaining an er u method and we observe that er u can outperform er in terms of rate of convergence the proposed preconditioner can be updated at a very low cost whenever the graph changes as is the case when it represents a generic set of information the particular u utilized requires a surplus of operations per step and memory allocations which make er u superior to er for not too wide graphs however the observed high improvement in convergence rate obtained by preconditioning and the general theory developed are a reason for investigating different choices of u more efficient for huge graphs

an efficient elimination strategy for solving pagerank problems in web link structures similar link distributions often occur especially for pages from same hosts for pagerank problems similar in link distributions of pages result in similar row patterns of the transition matrix we demonstrate that common row patterns in the transition matrix determine identical sub rows thus the identical sub rows with a large proportion of nonzeros can be eliminated to decrease the density of pagerank problems we propose an elimination strategy that exploits such identical sub rows and generates an elimination operator for transferring the problem to an equivalent but more sparse problem numerical experiments are reported to illustrate the effectiveness of this strategy for decreasing the computational cost of solving pagerank problems

the 25 000 000 000 eigenvector the linear algebra behind google google s success derives in large part from its pagerank algorithm which ranks the importance of web pages according to an eigenvector of a weighted link matrix analysis of the pagerank formula provides a wonderful applied topic for a linear algebra course instructors may assign this article as a project to more advanced students or spend one or two lectures presenting the material with assigned homework from the exercises this material also complements the discussion of markov chains in matrix algebra maple and mathematica files supporting this material can be found at www rose hulman edu bryan

isoparametric functions on exotic spheres abstract this paper extends widely the work in 11 existence and non existence results of isoparametric functions on exotic spheres and eells kuiper projective planes are established in particular every homotopy n sphere n 4 carries an isoparametric function with certain metric with 2 points as the focal set in strong contrast to the classification of cohomogeneity one actions on homotopy spheres 26 only exotic kervaire spheres admit cohomogeneity one actions besides the standard spheres as an application we improve a beautiful result of berard bergery 2 see also pp 234 235 of 3

updating kernel methods in spectral decomposition by affinity perturbations abstract many machine learning based algorithms contain a training step that is done once the training step is usually computational expensive since it involves processing of huge matrices if the training profile is extracted from an evolving dynamic dataset it has to be updated as some features of the training dataset are changed this paper proposes a solution how to update this profile efficiently therefore we investigate how to update the training profile when the data is constantly evolving we assume that the data is modeled by a kernel method and processed by a spectral decomposition in many algorithms for clustering and classification a low dimensional representation of the affinity kernel graph of the embedded training dataset is computed then it is used for classifying newly arrived data points we present methods for updating such embeddings of the training datasets in an incremental way without the need to perform the entire computation upon the occurrences of changes in a small number of the training samples efficient computation of such an algorithm is critical in many web based applications

subgame perfection in quitting games with perfect information and differential equations we introduce a new approach to studying subgame perfect equilibrium payoffs in stochastic games the differential equations approach we apply our approach to quitting games with perfect information those are sequential games in which at every stage one ofn players is chosen each player is chosen with probability 1 n the chosen playeri decides whether to quit in which case the game terminates and the terminal payoff is some vectora i rn or whether to continue in which case the game continues to the next stage if no player ever quits the payoff is some vectora rn we define a certain differential inclusion prove that it has at least one solution and prove that every vector on a solution of this differential inclusion is a subgame perfect equilibrium payoff

rational extrapolation for the pagerank vector an important problem in web search is to determine the importance of each page from the mathematical point of view this problem consists in finding the nonnegative left eigenvector of a matrix corresponding to its dominant eigenvalue 1 since this matrix is neither stochastic nor irreducible the power method has convergence problems so the matrix is replaced by a convex combination depending on a parameter c with a rank one matrix its left principal eigenvector now depends on c and it is the pagerank vector we are looking for however when c is close to 1 the problem is ill conditioned and the power method converges slowly so the idea developed in this paper consists in computing the pagerank vector for several values of c and then to extrapolate them by a conveniently chosen rational function at a point near 1 the choice of this extrapolating function is based on the mathematical expression of the pagerank vector as a function of c numerical experiments end the paper

predictable dynamics of opinion forming for networks with antagonistic interactions for communities of agents which are not necessarily cooperating distributed processes of opinion forming are naturally represented by signed graphs with positive edges representing friendly and cooperative interactions and negative edges the corresponding antagonistic counterpart unlike for nonnegative graphs the outcome of a dynamical system evolving on a signed graph is not obvious and it is in general difficult to characterize even when the dynamics are linear in this paper we identify a significant class of signed graphs for which the linear dynamics are however predictable and show many analogies with positive dynamical systems these cases correspond to adjacency matrices that are eventually positive for which the perron frobenius property still holds and implies the existence of an invariant cone contained inside the positive orthant as examples of applications we determine cases in which it is possible to anticipate or impose unanimity of opinion in decision voting processes even in presence of stubborn agents and show how it is possible to extend the pagerank algorithm to include negative links

off diagonal low rank preconditioner for difficult pagerank problems abstract pagerank problem is the cornerstone of google search engine and is usually stated as solving a huge linear system moreover when the damping factor approaches 1 the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve in this paper we demonstrate that the coefficient matrix of this system can be transferred into a block form by partitioning its rows into special sets in particular the off diagonal part of the block coefficient matrix can be compressed by a simple low rank factorization which can be beneficial for solving the pagerank problem hence a matrix partition method is proposed to discover the special sets of rows for supporting the low rank factorization then a preconditioner based on the low rank factorization is proposed for solving difficult pagerank problems numerical experiments are presented to support the discussions and to illustrate the effectiveness of the proposed methods

recursive repeated games with absorbing states we show the existence of stationary limiting average e equilibria e 0 for two person recursive repeated games with absorbing states these are stochastic games where all states but one are absorbing and in the nonabsorbing state all payoffs are equal to zero a state is called absorbing if the probability of a transition to any other state is zero for all available pairs of actions for the purpose of our proof we introduce properness for stationary strategy pairs our result is sharp since it extends neither to the case with more nonabsorbing states nor to the n person case with n 2 moreover it is well known that the result cannot be strengthened to the existence of 0 equilibria and that repeated games with absorbing states generally do not admit stationary e equilibria

a gmres power algorithm for computing pagerank problems abstract the pagerank algorithm for determining the importance of web pages has become a central technique in web search we propose a new method to speed up the convergence performance for computing pagerank when the damping factor is close to one called as gmres power which is based on a periodic combination of the power method with the gmres algorithm the description and convergence analysis of the new algorithm are discussed in detail numerical results are reported to confirm the efficiency of the new algorithm

regenerative analysis and steady state distributions for markov chains we apply regenerative theory to derive certain relations between steady state probabilities of a markov chain these relations are then used to develop a numerical algorithm to find these probabilities the algorithm is a modification of the gauss jordan method in which all elements used in numerical computations are nonnegative as a consequence the algorithm is numerically stable

isoparametric foliations and critical sets of eigenfunctions jakobson and nadirashvili j differ geom 53 1 177 182 1999 constructed a sequence of eigenfunctions on t 2 with a bounded number of critical points answering in the negative the question raised by yau problem section seminar on differential geometry princeton university press princeton 1982 which asks that whether the number of the critical points of eigenfunctions for the laplacian increases with the corresponding eigenvalues the present paper finds three interesting eigenfunctions on the minimal isoparametric hypersurface m n in s n 1 1 the corresponding eigenvalues are n 2n and 3n while their critical sets consist of 8 points a submanifold infinite many points and 8 points respectively on one of its focal submanifolds a similar phenomenon occurs

two player nonzero sum stopping games in discrete time we prove that every two player non zero sum stopping game in discrete time admits an equilibrium in randomized strategies for every 0 we use a stochastic variation of ramsey theorem which enables us to reduce the problem to that of studying properties of equilibria in a simple class of stochastic games with finite state space

monte carlo methods in pagerank computation when one iteration is sufficient pagerank is one of the principle criteria according to which google ranks web pages pagerank can be interpreted as a frequency of visiting a web page by a random surfer and thus it reflects the popularity of a web page google computes the pagerank using the power iteration method which requires about one week of intensive computations in the present work we propose and analyze monte carlo type methods for the pagerank computation there are several advantages of the probabilistic monte carlo methods over the deterministic power iteration method monte carlo methods already provide good estimation of the pagerank for relatively important pages after one iteration monte carlo methods have natural parallel implementation and finally monte carlo methods allow one to perform continuous update of the pagerank as the structure of the web changes

the general inner outer iteration method based on regular splittings for the pagerank problem abstract in this paper combined the regular splittings of the coefficient matrix i p with the inner outer iteration framework 9 a general inner outer gio iteration method is presented for solving the pagerank problem firstly the aor and modified aor maor methods for solving the pagerank problem are constructed and several comparison results are also given next the gio iteration scheme is developed and its overall convergence is analyzed in detail furthermore the preconditioner derived from the gio iteration can be used to accelerate the krylov subspace methods such as gmres method finally some numerical experiments on several pagerank problems are provided to illustrate the efficiency of the proposed algorithm

ricci curvature of double manifolds via isoparametric foliations abstract given a closed manifold m and a vector bundle of rank n over m by gluing two copies of the disc bundle of we can obtain a closed manifold d m the so called double manifold in this paper we firstly prove that each sphere bundle s r of radius r 0 is an isoparametric hypersurface in the total space of equipped with a connection metric and for r 0 small enough the induced metric of s r has positive ricci curvature under the additional assumptions that m has a metric with positive ricci curvature and n 3 as an application if m admits a metric with positive ricci curvature and n 2 then we construct a metric with positive ricci curvature on d m moreover under the same metric d m admits a natural isoparametric foliation for a compact minimal isoparametric hypersurface y n in s n 1 1 which separates s n 1 1 into s n 1 and s n 1 one can get double manifolds d s n 1 and d s n 1 inspired by tang xie and yan s work on scalar curvature of such manifolds with isoparametric foliations cf 25 we study ricci curvature of them with isoparametric foliations in the last part

relative error bounds for the lu decomposition via the gth algorithm recently the author showed that the grassmann taksar heyman gth algorithm computes the steady state distribution of a finite state markov chain with low relative error here it is shown that the lu decomposition computed in the course of the gth algorithm also has low relative error the proof requires a refinement of the methods used in the earlier paper

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

new look ahead lanczos type algorithms for linear systems a breakdown due to a division by zero can arise in the algorithms for implementing lanczos method because of the non existence of some formal orthogonal polynomials or because the recurrence relationship used is not appropriate such a breakdown can be avoided by jumping over the polynomials involved this strategy was already used in some algorithms such as the mrz and its variants in this paper we propose new implementations of the recurrence relations of these algorithms which only need the storage of a fixed number of vectors independent of the length of the jump these new algorithms are based on horner s rule and on a different way for computing the coefficients of the recurrence relationships moreover these new algorithms seem to be more stable than the old ones and they provide better numerical results numerical examples and comparisons with other algorithms will be given

on computing pagerank via lumping the google matrix computing google s pagerank via lumping the google matrix was recently analyzed in i c f ipsen t m selee pagerank computation with special attention to dangling nodes siam j matrix anal appl 29 2007 1281 1296 it was shown that all of the dangling nodes can be lumped into a single node and the pagerank could be obtained by applying the power method to the reduced matrix furthermore the stochastic reduced matrix had the same nonzero eigenvalues as the full google matrix and the power method applied to the reduced matrix had the same convergence rate as that of the power method applied to the full matrix therefore a large amount of operations could be saved for computing the full pagerank vector in this note we show that the reduced matrix obtained by lumping the dangling nodes can be further reduced by lumping a class of nondangling nodes called weakly nondangling nodes to another single node and the further reduced matrix is also stochastic with the same nonzero eigenvalues as the google matrix

isoparametric hypersurfaces in complex hyperbolic spaces we review some results of an ongoing research on isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in the complex hyperbolic space in order to motivate this topic we recall first the main steps of its long history

a note on the pagerank algorithm abstract in this paper we present some notes of the pagerank algorithm including its l1 condition number and some observation of the numerical tests of two variant algorithms which are based on the extrapolation method

on perturbations of principal eigenvectors of substochastic matrices in this report we consider substochastic matrices with some zero rows and study the sensitivity of their eigenvectors to the modifications of zero entries the analysis is prompted by and directly applied to the google method of ranking the web sites which substitutes the pages without outlinks dangling nodes in the original web link matrix by non zero rows dangling vectors we present an analysis of the influence of artificial links attributed to the dangling nodes on the principal eigenvectors of the web matrix we clarify when the choice of the dangling vector does not change the original eigenvectors and give an evaluation for perturbations of the principal eigenvectors when they are subject to modification

isoparametric hypersurfaces with g m 6 2 we prove that isoparametric hypersurfaces with g m 6 2 are homogeneous which answers dorfmeister neher s conjecture armatively and solves yau s problem in the case g 6

isoparametric foliations a problem of eells lemaire and conjectures of leung in this paper two sequences of minimal isoparametric hypersurfaces are constructed via representations of clifford algebras based on these we give estimates on eigenvalues of the laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres this improves results of z z tang and w j yan isoparametric foliation and yau conjecture on the first eigenvalue j differential geom 94 2013 521 540 z z tang y q xie and w j yan isoparametric foliation and yau conjecture on the first eigenvalue ii j funct anal 266 2014 6174 6199 eells and lemaire selected topics in harmonic maps c b m s regional conference series in mathematics 50 american mathematical society providence ri 1983 posed a problem to characterize the compact riemannian manifold m for which there is an eigenmap from m to sn as another application of our constructions the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres most importantly by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres we provide infinitely many counterexamples to two conjectures of leung minimal submanifolds in a sphere ii bull london math soc 23 1991 387 390 posed in 1991 on minimal submanifolds in unit spheres note that these conjectures of leung have been proved in the case that the normal connection is flat t hasanis and t vlachos ricci curvatures and minimal submanifolds pacific j math 197 2001 13 24

comparison of partitioning techniques for two level iterative solvers on large sparse markov chains experimental results for large sparse markov chains especially the ill conditioned nearly completely decomposable ncd ones are few we believe there is need for further research in this area specifically to aid in the understanding of the effects of the degree of coupling of ncd markov chains and their nonzero structure on the convergence characteristics and space requirements of iterative solvers the work of several researchers has raised the following questions that led to research in a related direction how must one go about partitioning the global coefficient matrix into blocks when the system is ncd and a two level iterative solver such as block sor is to be employed are block partitionings dictated by the ncd form of the stochastic one step transition probability matrix necessarily superior to others is it worth investigating alternative partitionings better yet for a fixed labeling and partitioning of the states how does the performance of block sor or even that of point sor compare to the performance of the iterative aggregation disaggregation iad algorithm finally is there any merit in using two level iterative solvers when preconditioned krylov subspace methods are available we seek answers to these questions on a test suite of 13 markov chains arising in 7 applications

isoparametric hypersurfaces with four principal curvatures let m be an isoparametric hypersurface in the sphere s n with four distinct principal curvatures munzner showed that the four principal curvatures can have at most two distinct multiplicities m1 m 2 and stolz showed that the pair m1 m 2 must either be 2 2 4 5 or be equal to the multiplicities of an isoparametric hypersurface of fkm type constructed by ferus karcher and munzner from orthogonal representations of clifford algebras in this paper we prove that if the multiplicities satisfy m2 2m1 1 then the isoparametric hypersurface m must be of fkm type together with known results of takagi for the case m1 1 and ozeki and takeuchi for m1 2 this handles all possible pairs of multiplicities except for four cases for which the classification problem remains open

a new extrapolation method for pagerank computations abstract the pagerank algorithm is widely considered these years because of its great significance in search engine technology and other scientific domains though the power method is the initial measure to settle the pagerank problem it gives poor convergence when the damping factor is sufficiently close to 1 this difficulty encourages researchers to present improved iterative methods for accelerating pagerank computations in this paper a cheap and practical extrapolation approach which is determined by the trace of the google matrix is proposed and it is more attractive when combined with the well known arnoldi type algorithm convergence analysis of our algorithms is given numerical examples illustrate the efficiency of these accelerated schemes

isoparametric foliation and a problem of besse on generalizations of einstein condition abstract the focal sets of isoparametric hypersurfaces in spheres with g 4 are all willmore submanifolds being minimal but mostly non einstein 25 22 inspired by a gray s view the present paper shows that these focal sets are all a manifolds but rarely ricci parallel except possibly for the only unclassified case as a byproduct it gives infinitely many simply connected examples to the problem 16 56 i of besse concerning generalizations of the einstein condition

vector extrapolation methods with applications to solution of large systems of equations and to pagerank computations an important problem that arises in different areas of science and engineering is that of computing the limits of sequences of vectors x n where x n c n with n very large such sequences arise for example in the solution of systems of linear or nonlinear equations by fixed point iterative methods and lim n x n are simply the required solutions in most cases of interest however these sequences converge to their limits extremely slowly one practical way to make the sequences x n converge more quickly is to apply to them vector extrapolation methods in this work we review two polynomial type vector extrapolation methods that have proved to be very efficient convergence accelerators namely the minimal polynomial extrapolation mpe and the reduced rank extrapolation rre we discuss the derivation of these methods describe the most accurate and stable algorithms for their implementation along with the effective modes of usage in solving systems of equations nonlinear as well as linear and present their convergence and stability theory we also discuss their close connection with the method of arnoldi and with gmres two well known krylov subspace methods for linear systems we show that they can be used very effectively to obtain the dominant eigenvectors of large sparse matrices when the corresponding eigenvalues are known and provide the relevant theory as well one such problem is that of computing the pagerank of the google matrix which we discuss in detail in addition we show that a recent extrapolation method of kamvar et al that was proposed for computing the pagerank is very closely related to mpe we present a generalization of the method of kamvar et al along with a very economical algorithm for this generalization we also provide the missing convergence theory for it

power law strength degree correlation from resource allocation dynamics on weighted networks many weighted scale free networks are known to have a power law correlation between strength and degree of nodes which however has not been well explained we investigate the dynamic behavior of resource traffic flow on scale free networks the dynamical system will evolve into a kinetic equilibrium state where the strength defined by the amount of resource or traffic load is correlated with the degree in a power law form with tunable exponent the analytical results agree well with simulations

isoparametric foliation and yau conjecture on the first eigenvalue ii abstract this is a continuation of 23 which investigated the first eigenvalues of minimal isoparametric hypersurfaces with g 4 distinct principal curvatures and focal submanifolds in unit spheres for the focal submanifolds with g 6 the present paper obtains estimates on all the eigenvalues among others giving an affirmative answer in one case to the problem posed in 23 which may be regarded as a generalization of yau s conjecture in two of the four unsettled cases in 23 for focal submanifolds m 1 of ot fkm type we prove the first eigenvalues to be their respective dimensions

comparison of krylov subspace methods on the pagerank problem pagerank algorithm plays a very important role in search engine technology and consists in the computation of the eigenvector corresponding to the eigenvalue one of a matrix whose size is now in the billions the problem incorporates a parameter a that determines the difficulty of the problem in this paper the effectiveness of stationary and nonstationary methods are compared on some portion of real web matrices for different choices of a we see that stationary methods are very reliable and more competitive when the problem is well conditioned that is for small values of a however for large values of the parameter a the problem becomes more difficult and methods such as preconditioned bicgstab or restarted preconditioned gmres become competitive with stationary methods in terms of mflops count as well as in number of iterations necessary to reach convergence

practical acceleration for computing the hits expertrank vectors a meaningful rank as well as efficient methods for computing such a rank are necessary in many areas of applications major methodologies for ranking often exploit principal eigenvectors kleinberg s hits model is one of such methodologies the standard approach for computing the hits rank is the power method unlike the pagerank calculations where many acceleration schemes have been proposed relatively few works on accelerating hits rank calculation exist this is mainly because the power method often works quite well in the hits setting however there are cases where the power method is ineffective moreover a systematic acceleration over the power method is desirable even when the power method works well we propose a practical acceleration scheme for hits rank calculations based on the filtered power method by adaptive chebyshev polynomials for cases where the gap ratio is below 0 85 for which the power method works well our scheme is about twice faster than the power method for cases where gap ratio is unfavorable for the power method our scheme can provide significant speedup when the ranking problems are of very large scale even a single matrix vector product can be expensive for which accelerations are highly necessary the scheme we propose is desirable in that it provides consistent reduction in number of matrix vector products as well as cpu time over the power method with little memory overhead

an arnoldi extrapolation algorithm for computing pagerank the arnoldi type algorithm proposed by golub and greif g golub c greif an arnoldi type algorithm for computing pagerank bit 46 2006 759 771 is a restarted krylov subspace method for computing pagerank however this algorithm may not be efficient when the damping factor is high and the dimension of the search subspace is small in this paper we first develop an extrapolation method based on ritz values we then consider how to periodically knit this extrapolation method together with the arnoldi type algorithm the resulting algorithm is the arnoldi extrapolation algorithm the convergence of the new algorithm is analyzed numerical experiments demonstrate the numerical behavior of this algorithm

stochastic games with additive transitions abstract we deal with n player at stochastic games where at stands for additive transitions these are stochastic games in which the transition probability vector ps as for action combination a s a s 1 a s n in state s can be decomposed into player dependent components as p s a s i 1 n s i p s i a s i where s i 0 1 for all players i and i 1 n s i 1 and where p s i a s i is a probability distribution on the finite set of states s here s i reflects the influence of player i on the transitions in state s as such the class of at stochastic games covers several other well known classes such as perfect information stochastic games stochastic games with switching control and so called arat stochastic games with respect to the average reward it is not clear whether e equilibria always exist in general n player stochastic games for the class of n player at games we establish the existence of 0 equilibria although the strategies involved may be history dependent in addition we have the following results for the two player case 1 for zero sum at games stationary 0 optimal strategies always exist 2 for two player general sum at absorbing games there always exist stationary e equilibria for all e 0 several examples are provided to clarify the issues and to demonstrate the sharpness of the results

a multi class approach for ranking graph nodes after the phenomenal success of the pagerank algorithm many researchers have extended the pagerank approach to ranking graphs with richer structures in addition to the simple linkage structure indeed in some scenarios we have to deal with networks modeling multi parameters data where each node has additional features and there are important relationships between such features this paper addresses the need of a systematic approach to deal with multi parameter data we propose models and ranking algorithms that can be applied to a large variety of networks bibliographic data patent data twitter and social data healthcare data we focus on several aspects not previously addressed in the literature 1 we propose different models for ranking multi parameters data and a class of numerical algorithms for efficiently computing the ranking score of such models 2 we analyze stability and convergence of the proposed numerical schemes and we derive a fast and stable ranking algorithm 3 we analyze the robustness of our models when data are incomplete the comparison of the rank on the incomplete data with the rank on the full structure shows that our models compute consistent rankings whose correlation is up to 60 when just 10 of the links of the attributes are maintained

algebraic schwarz methods for the numerical solution of markov chains the convergence of additive and multiplicative schwarz methods for computing certain characteristics of markov chains such as stationary probability vectors and mean first passage matrices is studied the main result is a convergence theorem for multiplicative schwarz iterations when applied to singular systems as a byproduct a convergence result for alternating iterations is also obtained it is also shown that when the markov chain is ergodic additive and multiplicative schwarz methods can be applied to the nonsingular systems that result from reducing the equations the so called coarse grid corrections are also studied

isoparametric foliations diffeomorphism groups and exotic smooth structures abstract in this paper we are concerned with interactions between isoparametric theory and differential topology two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves using differential topology we obtain several results towards the classification problem of isoparametric foliations up to equivalence in particular we show that each homotopy n sphere has the same isoparametric foliations as the standard sphere s n has except for n 4 reducing the classification problem on homotopy spheres to that on the standard sphere moreover we prove the uniqueness up to equivalence of isoparametric foliations with two points as the focal submanifolds on each sphere s n except for n 5 besides we show that the uniqueness holds on s 5 if and only if 0 diff s 4 z 2 i e pseudo isotopy implies isotopy for diffeomorphisms on s 4 at last some ideas behind the proofs enable us to discover new exotic smooth structures on certain manifolds

on the effects of using the grassmann taksar heyman method in interative aggregation disaggregation iterative aggregation disaggregation iad is an effective method for solving finite nearly completely decomposable ncd markov chains small perturbations in the transition probabilities of these chains may lead to considerable changes in the stationary probabilities ncd markov chains are known to be ill conditioned during an iad step this undesirable condition is inherited by the coupling matrix and one confronts the problem of finding the stationary probabilities of a stochastic matrix whose diagonal elements are close to 1 in this paper the effects of using the grassmann taksar heyman gth method to solve the coupling matrix formed in the aggregation step are investigated then the idea is extended in such a way that the same direct method can be incorporated into the disaggregation step finally the effects of using the gth method in the iad algorithm on various examples are demonstrated and the conditions under which it should be employed are explained

an inner outer iteration for computing pagerank we present a new iterative scheme for pagerank computation the algorithm is applied to the linear system formulation of the problem using inner outer stationary iterations it is simple can be easily implemented and parallelized and requires minimal storage overhead our convergence analysis shows that the algorithm is effective for a crude inner tolerance and is not sensitive to the choice of the parameters involved the same idea can be used as a preconditioning technique for nonstationary schemes numerical examples featuring matrices of dimensions exceeding 100 000 000 in sequential and parallel environments demonstrate the merits of our technique our code is available online for viewing and testing along with several large scale examples

conditioning of the entries in the stationary vector of a google type matrix abstract using combinatorial and analytic techniques we give conditioning bounds for the stationary vector t of a stochastic matrix of the form ca 1 c b where c 0 1 is a scalar and a and b are stochastic matrices the latter being rank one such matrices and their stationary vectors arise as a key component in google s pagerank algorithm the conditioning bounds considered include normwise absolute componentwise and relative componentwise and the bounds depend on c and on quantities such as the number of dangling nodes which correspond to rows of a having all entries equal or the lengths of certain cycles in the directed graph associated with a it is shown that if vertex j is on only long cycles in that directed graph then the corresponding entry in t exhibits better conditioning properties and that for dangling nodes the sensitivity of the corresponding entries in t decreases as the number of dangling nodes increases conditions are given that are sufficient to ensure that an iterate of the power method accurately reflects the relative ordering of two entries in t

multiplicities of dupin hypersurfaces all known examples of isoparametric hypersurfaces of spheres are of the following two types 1 the homogeneous examples which are principal orbits of isotropy representations of symmetric spaces of rank 2 2 the clifford examples of ferus karcher munzner which are constructed from modules over clifford algebras our main result which extends work of munzner thorbergsson and grove halperin is that any isoparametric hypersurface has the same number of principal curvatures and the same multiplicities as some hypersurface of type 1 or 2 more generally this holds for any dupin hypersurface the proof is homotopy theoretic the main tools are the hopf invariant and the ehp sequence

aggregation algorithms for perturbed markov chains with applications to networks modeling a powerly perturbed markov chain mc is a family of finite mcs given by transition probability matrices p varepsilon p ij varepsilon or generators for continuous time mcs such that p ij varepsilon delta ij varepsilon d ij o varepsilon d ij varepsilon rightarrow0 in the simplest case we have popular nearly completely decomposable ncd chains it has been shown by directed forest expansions that solutions of linear systems related to p varepsilon have a form x i varepsilon alpha i varepsilon a i o varepsilon a i for some vectors boldsymbol alpha alpha i and mathbf a a i many characteristics of mcs such as stationary distributions mean passage times or limiting matrices can be defined in this way we present efficient combinatorial aggregation algorithms for computing mathbf a and boldsymbol alpha the algorithms rely on grouping the states of mcs in such a way that the probability of changing the state inside the group is of a greater order of magnitude than the interactions between groups in contrast to existing methods our approach is based on a combinatorial framework and can be seen as an algorithmization of the famous matrix tree theorem we establish some preliminary results on the complexity of our algorithms numerical experiments on several network models such as queueing systems or the google problem show the potential applicability of the method in real life problems