new block quadrature rules for the approximation of matrix functions abstract golub and meurant have shown how to use the symmetric block lanczos algorithm to compute block gauss quadrature rules for the approximation of certain matrix functions we describe new block quadrature rules that can be computed by the symmetric or nonsymmetric block lanczos algorithms and yield higher accuracy than standard block gauss rules after the same number of steps of the symmetric or nonsymmetric block lanczos algorithms the new rules are block generalizations of the generalized averaged gauss rules introduced by spalevic applications to network analysis are presented

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

dynamical scaling in branching models for seismicity we propose a branching process based on a dynamical scaling hypothesis relating time and mass in the context of earthquake occurrence we show that experimental power laws in size and time distribution naturally originate solely from this scaling hypothesis we present a numerical protocol able to generate a synthetic catalog with an arbitrary large number of events the numerical data reproduce the hierarchical organization in time and magnitude of experimental interevent time distribution

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

augmented implicitly restarted lanczos bidiagonalization methods new restarted lanczos bidiagonalization methods for the computation of a few of the largest or smallest singular values of a large matrix are presented restarting is carried out by augmentation of krylov subspaces that arise naturally in the standard lanczos bidiagonalization method the augmenting vectors are associated with certain ritz or harmonic ritz vectors computed examples show the new methods to be competitive with available schemes

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

analysis of self organized criticality in ehrenfest s dog flea model the self organized criticality in ehrenfest s historical dog flea model is analyzed by simulating the underlying stochastic process the fluctuations around the thermal equilibrium in the model are treated as avalanches we show that the distributions for the fluctuation length differences at subsequent time steps are in the shape of a q gaussian the distribution which is obtained naturally in the context of nonextensive statistical mechanics if one avoids the finite size effects by increasing the system size we provide clear numerical evidence that the relation between the exponent ensuremath tau of avalanche size distribution obtained by maximum likelihood estimation and the q value of appropriate q gaussian obeys the analytical result recently introduced by caruso et al phys rev e 75 055101 r 2007 this allows us to determine the value of q parameter a priori from one of the well known exponents of such dynamical systems

network analysis via partial spectral factorization and gauss quadrature large scale networks arise in many applications it is often of interest to be able to identify the most important nodes of a network or to ascertain the ease of traveling between nodes these and related quantities can be determined by evaluating expressions of the form mathbf u tf a mathbf w where a is the adjacency matrix that represents the graph of the network f is a nonlinear function such as the exponential function and mathbf u and mathbf w are vectors for instance axis vectors this paper describes a novel technique for determining upper and lower bounds for expressions mathbf u tf a mathbf w when a is symmetric and bounds for many vectors mathbf u and mathbf w are desired the bounds are computed by first evaluating a low rank approximation of a which is used to determine rough bounds for the desired quantities for all nodes these rough bounds indicate for which vectors mathbf u and mathbf w more accurate bounds should be computed with the aid

dynamic scaling approach to study time series fluctuations we propose an approach for properly analyzing stochastic time series by mapping the dynamics of time series fluctuations onto a suitable nonequilibrium surface growth problem in this framework the fluctuation sampling time interval plays the role of time variable whereas the physical time is treated as the analog of spatial variable in this way we found that the fluctuations of many real world time series satisfy the analog of the family viscek dynamic scaling ansatz this finding permits us to use the powerful tools of kinetic roughening theory to classify model and forecast the fluctuations of real world time series

comparison of detrending methods for fluctuation analysis abstract we examine several recently suggested methods for the detection of long range correlations in data series based on similar ideas as the well established detrended fluctuation analysis dfa in particular we present a detailed comparison between the regular dfa and two recently suggested methods the centered moving average cma method and a modified detrended fluctuation analysis mdfa we find that cma performs the same as dfa in long data with weak trends and is slightly superior to dfa in short data with weak trends when comparing standard dfa to mdfa we observe that dfa performs slightly better in almost all examples we studied we also discuss how several types of trends affect different types of dfa for weak trends in the data the new methods are comparable with dfa in these respects however if the functional form of the trend in data is not a priori known dfa remains the method of choice only a comparison of dfa results using different detrending polynomials yields full recognition of the trends a comparison with independent methods is recommended for proving long range correlations

long range correlations in the electric signals that precede rupture the smoluchowski chapman kolmogorov functional equation is applied to the electric signals that precede rupture the results suggest a non markovian character of the analyzed data the rescaled range hurst and detrended fluctuation analyses as well as that related with the mean distance a walker spanned lead to power law exponents which are consistent with the existence of long range correlations a universality in the power spectrum characteristics of these signals emerges if an analysis is made not in the conventional time frame but in the natural time domain within this frame it seems that certain power spectrum characteristics of ion current fluctuations in membrane channels distinguish them from the electric signals preceding rupture the latter exhibit a behavior compatible with that expected from a model based on the random field ising hamiltonian at the critical point

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks

hypocenter interval statistics between successive earthquakes in the two dimensional burridge knopoff model abstract we study statistical properties of spatial distances between successive earthquakes the so called hypocenter intervals produced by a two dimensional 2d burridge knopoff model involving stick slip behavior it is found that cumulative distributions of hypocenter intervals can be described by the q exponential distributions with q 1 which is also observed in nature the statistics depend on a friction and stiffness parameters characterizing the model and a threshold of magnitude the conjecture which states that q t q r 2 where q t and q r are an entropy index of time intervals and spatial intervals respectively can be reproduced semi quantitatively it is concluded that we provide a new perspective on the burridge knopoff model which addresses that the model can be recognized as a realistic one in view of the reproduction of the spatio temporal interval statistics of earthquakes on the basis of nonextensive statistical mechanics

analysis of the spatial and temporal distributions between successive earthquakes nonextensive statistical mechanics viewpoint abstract the spatial and temporal distributions between successive earthquakes are treated in the framework of nonextensive statistical mechanics we find temporal distributions exhibit the power law behavior q exponential with q 1 it means the earthquakes are strongly correlated in time the spatial distributions obey the q exponential form with q 1 we also examine the dependence of the q exponent on magnitude range covering period time interval and size of the region where data are gathered the conjecture of abe et al s abe n suzuki physica a 350 2005 588 has been examined for different categories of data the results show a strange relation between q values of the spatial and temporal distributions

quadrature rule based bounds for functions of adjacency matrices bounds for entries of matrix functions based on gauss type quadrature rules are applied to adjacency matrices associated with graphs this technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities estrada index subgraph centrality communicability that describe properties of networks

memory in the occurrence of earthquakes we study the statistics of the recurrence times ensuremath tau between earthquakes above a certain magnitude m in six one global and five regional earthquake catalogs we find that the distribution of the recurrence times strongly depends on the previous recurrence time ensuremath tau 0 such that small and large recurrence times tend to cluster in time this dependence on the past is reflected in both the conditional mean recurrence time and the conditional mean residual time until the next earthquake which increase monotonically with ensuremath tau 0 as a consequence the risk of encountering the next event within a certain time span after the last event depends significantly on the past an effect that has to be taken into account in any effective earthquake prognosis

volatility of linear and nonlinear time series previous studies indicated that nonlinear properties of gaussian distributed time series with long range correlations u i can be detected and quantified by studying the correlations in the magnitude series ensuremath mid u i ensuremath mid the volatility however the origin for this empirical observation still remains unclear and the exact relation between the correlations in u i and the correlations in ensuremath mid u i ensuremath mid is still unknown here we develop analytical relations between the scaling exponent of linear series u i and its magnitude series ensuremath mid u i ensuremath mid moreover we find that nonlinear time series exhibit stronger or the same correlations in the magnitude time series compared with linear time series with the same two point correlations based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series the nonlinear multifractal time series is generated by multiplying a long range correlated time series that represents the magnitude series with uncorrelated time series that represents the sign series mathrm sgn u i we apply our techniques on daily deep ocean temperature records from the equatorial pacific the region of the el nin o phenomenon and find i long range correlations from several days to several years with 1 f power spectrum ii significant nonlinear behavior as expressed by long range correlations of the volatility series and iii broad multifractal spectrum

matrices moments and rational quadrature abstract many problems in science and engineering require the evaluation of functionals of the form f u a u t f a u where a is a large symmetric matrix u a vector and f a nonlinear function a popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the lanczos procedure applied to a with initial vector u and then evaluating pairs of gauss and gauss radau quadrature rules associated with the tridiagonal matrix determined by the lanczos procedure the present paper extends this approach to allow the use of rational gauss quadrature rules

network properties revealed through matrix functions the emerging field of network science deals with the tasks of modeling comparing and summarizing large data sets that describe complex interactions because pairwise affinity data can be stored in a two dimensional array graph theory and applied linear algebra provide extremely useful tools here we focus on the general concepts of centrality communicability and betweenness each of which quantifies important features in a network some recent work in the mathematical physics literature has shown that the exponential of a network s adjacency matrix can be used as the basis for defining and computing specific versions of these measures we introduce here a general class of measures based on matrix functions and show that a particular case involving a matrix resolvent arises naturally from graph theoretic arguments we also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering we finish with computational examples showing the new matrix resolvent version applied to real networks

effect of extreme data loss on long range correlated and anticorrelated signals quantified by detrended fluctuation analysis detrended fluctuation analysis dfa is an improved method of classical fluctuation analysis for nonstationary signals where embedded polynomial trends mask the intrinsic correlation properties of the fluctuations to better identify the intrinsic correlation properties of real world signals where a large amount of data is missing or removed due to artifacts we investigate how extreme data loss affects the scaling behavior of long range power law correlated and anticorrelated signals we introduce a segmentation approach to generate surrogate signals by randomly removing data segments from stationary signals with different types of long range correlations the surrogate signals we generate are characterized by four parameters i the dfa scaling exponent alpha of the original correlated signal u i ii the percentage p of the data removed from u i iii the average length mu of the removed or remaining data segments and iv the functional form p l of the distribution of the length l of the removed or remaining data segments we find that the global scaling exponent of positively correlated signals remains practically unchanged even for extreme data loss of up to 90 in contrast the global scaling of anticorrelated signals changes to uncorrelated behavior even when a very small fraction of the data is lost these observations are confirmed on two examples of real world signals human gait and commodity price fluctuations we further systematically study the local scaling behavior of surrogate signals with missing data to reveal subtle deviations across scales we find that for anticorrelated signals even 10 of data loss leads to significant monotonic deviations in the local scaling at large scales from the original anticorrelated to uncorrelated behavior in contrast positively correlated signals show no observable changes in the local scaling for up to 65 of data loss while for larger percentage of data loss the local scaling shows overestimated regions with higher local exponent at small scales followed by underestimated regions with lower local exponent at large scales finally we investigate how the scaling is affected by the average length probability distribution and percentage of the remaining data segments in comparison to the removed segments we find that the average length mu r of the remaining segments is the key parameter which determines the scales at which the local scaling exponent has a maximum deviation from its original value interestingly the scales where the maximum deviation occurs follow a power law relationship with mu r whereas the percentage of data loss determines the extent of the deviation the results presented in this paper are useful to correctly interpret the scaling properties obtained from signals with extreme data loss

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

statistics of return intervals in long term correlated records we consider long term correlated data with several distribution densities gaussian exponential power law and log normal and various correlation exponents ensuremath gamma 0l ensuremath gamma l1 and study the statistics of the return intervals r j between events above some threshold q we show that irrespective of the distribution the return intervals are long term correlated in the same way as the original record but with additional uncorrelated noise due to this noise the correlations are difficult to observe by the detrended fluctuation analysis which exhibits a crossover behavior but show up very clearly in the autocorrelation function the distribution p q r of the return intervals is characterized at large scales by a stretched exponential with exponent ensuremath gamma and at short scales by a power law with exponent ensuremath gamma ensuremath 1 we discuss in detail the occurrence of finite size effects for large threshold values for all considered distributions we show that finite size effects are most pronounced in exponentially distributed data sets where they can even mask the stretched exponential behavior in records of up to 10 6 data points finally in order to quantify the clustering of extreme events due to the long term correlations in the return intervals we study the conditional distribution function and the related moments we find that they show pronounced memory effects irrespective of the distribution of the original data

low rank matrix approximation using the lanczos bidiagonalization process with applications low rank approximation of large and or sparse matrices is important in many applications and the singular value decomposition svd gives the best low rank approximations with respect to unitarily invariant norms in this paper we show that good low rank approximations can be directly obtained from the lanczos bidiagonalization process applied to the given matrix without computing any svd we also demonstrate that a so called one sided reorthogonalization process can be used to maintain an adequate level of orthogonality among the lanczos vectors and produce accurate low rank approximations this technique reduces the computational cost of the lanczos bidiagonalization process we illustrate the efficiency and applicability of our algorithm using numerical examples from several applications areas

networks of recurrent events a theory of records and an application to finding causal signatures in seismicity we propose a method to search for signs of causal structure in spatiotemporal data making minimal a priori assumptions about the underlying dynamics to this end we generalize the elementary concept of recurrence for a point process in time to recurrent events in space and time an event is defined to be a recurrence of any previous event if it is closer to it in space than all the intervening events as such each sequence of recurrences for a given event is a record breaking process this definition provides a strictly data driven technique to search for structure defining events to be nodes and linking each event to its recurrences generates a network of recurrent events significant deviations in statistical properties of that network compared to networks arising from acausal random processes allows one to infer attributes of the causal dynamics that generate observable correlations in the patterns we derive analytically a number of properties for the network of recurrent events composed by a random process in space and time we extend the theory of records to treat not only the variable where records happen but also time as continuous in this way we construct a fully symmetric theory of records leading to a number of results those analytic results are compared in detail to the properties of a network synthesized from time series of epicenter locations for earthquakes in southern california significant disparities from the ensemble of acausal networks that can be plausibly attributed to the causal structure of seismicity are as follows 1 invariance of network statistics with the time span of the events considered 2 the appearance of a fundamental length scale for recurrences independent of the time span of the catalog which is consistent with observations of the rupture length 3 hierarchy in the distances and times of subsequent recurrences as expected almost all of the statistical properties of a network constructed from a surrogate in which the original magnitudes and locations of earthquake epicenters are randomly shuffled are completely consistent with predictions from the acausal null model

a nonextensive modification of the gutenberg richter law q stretched exponential form abstract we study the cumulative distribution for the magnitude of earthquakes in the context of nonextensive statistical mechanics a new modification of the gutenberg richter gr law is introduced we use seismic data sets which were recorded in two different regions iran and california to compute the cumulative distribution of the magnitudes the empirical data are fitted extremely well by our suggested expression for the modified gr law over the whole range of magnitudes

discussion one step sparse estimates in nonconcave penalized likelihood models who cares if it is a white cat or a black cat 1 an insider s minor comments section 2 3 seems to be the reason that i am a discussant there it was first stated that the proposed lla algorithm is an instance of the mm algorithm as termed by lange hunter and yang 8 then it was shown under certain conditions that it is also an em algorithm my initial re action was hmmm the authors reading of lange hunter and yang 8 must have ceased before reaching its discussions because a more general mm em re sult using the same laplace transform technique was the base for a key inquiry of meng 10 a discussion of lange hunter and yang 8 upon a more careful reading i realized that the authors construction though mathematically equiva lent to mine gives a different interpretation to the constructed missing data latent variable this is rather interesting especially if my initial reaction was correct for the current paper this mm em result appears to be of minor interest especially because its potential benefit is not explored in the paper instead the only punch line seems to be a logically unsubstantiated one thus theorem 3 also indicates that mm algorithms are more flexible than em algorithms to clar ify these issues which seems to be what i have been asked to do i ll focus my discussion first on this algorithmic connection and qualify the statements that i just made i will then pose some questions going beyond those algorithmic and computational considerations the two parts are connected via the cat title for those who are patient enough to read both to do so most effectively let me invoke an author s privilege to reproduce sec tion 2 of meng 10 in its entirety to map out its mathematical connections with section 2 3 of the current paper the acronym sm below was my suggestion because the general recipe lange hunter and yang 8 put forward consists of a surrogate step and a maximization minimization step in their rejoinder lange hunter and yang 8 coined the term mm partially in fear of the as sociation of sm with sm in deed for readers who enjoy collecting acronyms the gaecm below is another treasure to hunt

vere jones self similar branching model motivated by its potential application to earthquake statistics as well as for its intrinsic interest in the theory of branching processes we study the exactly self similar branching process introduced recently by vere jones this model extends the etas class of conditional self excited branching point processes of triggered seismicity by removing the problematic need for a minimum as well as maximum earthquake size to make the theory convergent without the need for the usual ultraviolet and infrared cutoffs the distribution of magnitudes m sup of daughters of first generation of a mother of magnitude m has two branches m sup m with exponent beta d where beta and d are two positive parameters we investigate the condition and nature of the subcritical critical and supercritical regime in this and in an extended version interpolating smoothly between several models we predict that the distribution of magnitudes of events triggered by a mother of magnitude m over all generations has also two branches m sup m with exponent beta h with h d radical 1 s where s is the fraction of triggered events this corresponds to a renormalization of the exponent d into h by the hierarchy of successive generations ofmore triggered events for a significant part of the parameter space the distribution of magnitudes over a full catalog summed over an average steady flow of spontaneous sources immigrants reproduces the distribution of the spontaneous sources with a single branch and is blind to the exponents beta d of the distribution of triggered events since the distribution of earthquake magnitudes is usually obtained with catalogs including many sequences we conclude that the two branches of the distribution of aftershocks are not directly observable and the model is compatible with real seismic catalogs in summary the exactly self similar vere jones model provides an attractive new approach to model triggered seismicity which alleviates delicate questions on the role of magnitude cutoffs in other non self similar models the new prediction concerning two branches in the distribution of magnitudes of aftershocks could be tested with recently introduced stochastic reconstruction methods tailored to disentangle the different triggered sequences less

ranking hubs and authorities using matrix functions the notions of subgraph centrality and communicability based on the exponential of the adjacency matrix of the underlying graph have been effectively used in the analysis of undirected networks in this paper we propose an extension of these measures to directed networks and we apply them to the problem of ranking hubs and authorities the extension is achieved by bipartization i e the directed network is mapped onto a bipartite undirected network with twice as many nodes in order to obtain a network with a symmetric adjacency matrix we explicitly determine the exponential of this adjacency matrix in terms of the adjacency matrix of the original directed network and we give an interpretation of centrality and communicability in this new context leading to a technique for ranking hubs and authorities the matrix exponential method for computing hubs and authorities is compared to the well known hits algorithm both on small artificial examples and on more realistic real world networks a few other ranking algorithms are also discussed and compared with our technique the use of gaussian quadrature rules for calculating hub and authority scores is discussed

itineration of the internet over nonequilibrium stationary states in tsallis statistics the cumulative probability distribution of sparseness time interval in the internet is studied by the method of data analysis round trip time between a local host and a destination host through ten odd routers is measured using the ping command i e doing an echo experiment the data are found to be well described by q exponential distributions which maximize the tsallis entropy indexed by q less or larger than unity showing a scale invariant feature of the system the network is observed to itinerate over a series of the nonequilibrium stationary states characterized by tsallis statistics

properties of cross entropy minimization the principle of minimum cross entropy minimum directed divergence minimum discrimination information is a general method of inference about an unknown probability density when there exists a prior estimate of the density and new information in the form of constraints on expected values various fundamental properties of cross entropy minimization are proven and collected in one place cross entropy s well known properties as an information measure are extended and strengthened when one of the densities involved is the result of cross entropy minimization the interplay between properties of cross entropy minimization as an inference procedure and properties of cross entropy as an information measure is pointed out examples are included and general analytic and computational methods of finding minimum cross entropy probability densities are discussed

generalized block anti gauss quadrature rules golub and meurant describe how pairs of gauss and gauss radau quadrature rules can be applied to determine inexpensively computable upper and lower bounds for certain real valued matrix functionals defined by a symmetric matrix however there are many matrix functionals for which their technique is not guaranteed to furnish upper and lower bounds in this situation it may be possible to determine upper and lower bounds by evaluating pairs of gauss and anti gauss rules unfortunately it is difficult to ascertain whether the values determined by gauss and anti gauss rules bracket the value of the given real valued matrix functional therefore generalizations of anti gauss rules have recently been described such that pairs of gauss and generalized anti gauss rules may determine upper and lower bounds for real valued matrix functionals also when pairs of gauss and standard anti gauss rules do not the available generalization requires the matrix that defines the functional to be real and symmetric the present paper reviews available anti gauss and generalized anti gauss rules and extends them in several ways that allow applications in new situations in particular the genarlized anti gauss rules for a real valued non negative measure described in pranic and reichel j comput appl math 284 235 243 2015 are extended to allow the estimation of the error in matrix functionals defined by a non symmetric matrix as well as to matrix valued matrix functions modifications that give simpler formulas and thereby make the application of the rules both easier and applicable to a larger class of problems also are described

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

a measure of similarity between graph vertices applications to synonym extraction and web searching we introduce a concept of similarity between vertices of directed graphs let ga and gb be two directed graphs with respectively na and nb vertices we define an nb times na similarity matrix s whose real entry sij expresses how similar vertex j in ga is to vertex i in gb we say that sij is their similarity score the similarity matrix can be obtained as the limit of the normalized even iterates of sk 1 bskat btska where a and b are adjacency matrices of the graphs and s0 is a matrix whose entries are all equal to 1 in the special case where ga gb g the matrix s is square and the score sij is the similarity score between the vertices i and j of g we point out that kleinberg s hub and authority method to identify web pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them in analogy to kleinberg we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix potential applications of our similarity concept are numerous we illustrate an application for the automatic extraction of synonyms in a monolingual dictionary

the e mom approach for approximating matrix functionals abstract the approximation of matrix functionals appears in many applications arising from the fields of mathematics statistics mechanics networks machine learning and physics in this paper we estimate matrix functionals of the form x t f a y where a r p p is a given diagonalizable matrix x y r p k are skinny block vectors with k p columns and f is a smooth function defined on the spectrum of the matrix a we apply a direct approach based on the extrapolation of the moments of the given matrix for estimating this kind of matrix functionals this approach avoids the application of the polarization identity is fairly inexpensive and leads to a stable procedure moreover we develop a detailed backward error analysis for the derived estimates several numerical results illustrating the effectiveness of the direct method are presented and concrete classes of matrices suitable for the extrapolation estimates are proposed

improved surrogate data for nonlinearity tests current tests for nonlinearity compare a time series to the null hypothesis of a gaussian linear stochastic process for this restricted null assumption random surrogates can be constructed which are constrained by the linear properties of the data we propose a more general null hypothesis allowing for nonlinear rescalings of a gaussian linear process we show that such rescalings cannot be accounted for by a simple amplitude adjustment of the surrogates which leads to spurious detection of nonlinearity an iterative algorithm is proposed to make appropriate surrogates which have the same autocorrelations as the data and the same probability distribution

analysis of directed networks via the matrix exponential abstract the matrix exponential has been identified as a useful tool for the analysis of undirected networks with sound theoretical justifications for its ability to model important aspects of a given network its use for directed networks however is less developed and has been less successful so far in this article we discuss some methods to identify important nodes in a directed network using the matrix exponential taking into account that the notion of importance changes whether we consider the influence of a given node along the edge directions downstream influence or how it is influenced by directed paths that point to it upstream influence in addition we introduce a family of importance measures based on counting walks that are allowed to reverse their direction a limited number of times thus capturing relationships arising from influencing the same nodes or being influenced by the same nodes without sacrificing information about edge direction these measures provide information about branch points

missing data in aftershock sequences explaining the deviations from scaling laws in this paper we extend the branching aftershock sequence model to study the role of missing data at short times and small amplitudes after a mainshock we apply this model which contains three parameters characterizing the missing data to the magnitude and temporal statistics of four aftershock sequences in california we find that the observed time dependent deviations of the frequency magnitude scaling from the gutenberg richter power law dependency can be described quantitatively by the model we also show that for the same set of parameters the model is able to explain quantitatively the observed magnitude dependent deviations of the temporal decay of aftershocks from omori s law in addition we show that the same sets of data can also reproduce quite well the various functional forms of the probability density functions of the return times between consecutive events with magnitudes above a prescribed threshold as well as the violation of scaling at short and intermediate time scales

nonextensivity and natural time the case of seismicity nonextensive statistical mechanics pioneered by tsallis has recently achieved a generalization of the gutenberg richter law for earthquakes this remarkable generalization is combined here with natural time analysis which enables the distinction of two origins of self similarity i e the process memory and the process increments infinite variance by using also detrended fluctuation analysis for the detection of long range temporal correlations we demonstrate the existence of both temporal and magnitude correlations in real seismic data of california and japan natural time analysis reveals that the nonextensivity parameter q in contrast to some published claims cannot be considered as a measure of temporal organization but the tsallis formulation does achieve a satisfactory description of real seismic data for japan for q 1 66 when supplemented by long range temporal correlations

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators