community detection algorithms a comparative analysis uncovering the community structure exhibited by real networks is a crucial step toward an understanding of complex systems that goes beyond the local organization of their constituents many algorithms have been proposed so far but none of them has been subjected to strict tests to evaluate their performance most of the sporadic tests performed so far involved small networks with known community structure and or artificial graphs with a simplified structure which is very uncommon in real systems here we test several methods against a recently introduced class of benchmark graphs with heterogeneous distributions of degree and community size the methods are also tested against the benchmark by girvan and newman proc natl acad sci u s a 99 7821 2002 and on random graphs as a result of our analysis three recent algorithms introduced by rosvall and bergstrom proc natl acad sci u s a 104 7327 2007 proc natl acad sci u s a 105 1118 2008 blondel et al j stat mech theory exp 2008 p10008 and ronhovde and nussinov phys rev e 80 016109 2009 have an excellent performance with the additional advantage of low computational complexity which enables one to analyze large systems

wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

a relative error bound for balanced stochastic truncation recently several methods have appeared for the approximation of power spectra notably balanced stochastic truncation bst it is shown that bst satisfies a relative error bound approximately twice the bound for the relative error method rem proper this offers a quantitative basis for the observation that bst and rem produce similar reduced order models balanced stochastic truncation can therefore be interpreted as providing a computationally simple algorithm for relative error approximation

benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities many complex networks display a mesoscopic structure with groups of nodes sharing many links with the other nodes in their group and comparatively few with nodes of different groups this feature is known as community structure and encodes precious information about the organization and the function of the nodes many algorithms have been proposed but it is not yet clear how they should be tested recently we have proposed a general class of undirected and unweighted benchmark graphs with heterogeneous distributions of node degree and community size an increasing attention has been recently devoted to develop algorithms able to consider the direction and the weight of the links which require suitable benchmark graphs for testing in this paper we extend the basic ideas behind our previous benchmark to generate directed and weighted networks with built in community structure we also consider the possibility that nodes belong to more communities a feature occurring in real systems such as social networks as a practical application we show how modularity optimization performs on our benchmark

passivity preserving model reduction by analytic interpolation abstract antoulas and sorensen have recently proposed a passivity preserving model reduction method of linear systems based on krylov projections the idea is to approximate a positive real rational transfer function with one of lower degree the method is based on an observation by antoulas in the single input single output case that if the approximant is preserving a subset of the spectral zeros and takes the same values as the original transfer function in the mirror points of the preserved spectral zeros then the approximant is also positive real however this turns out to be a special solution in the theory of analytic interpolation with degree constraint developed by byrnes georgiou and lindquist namely the maximum entropy central solution by tuning the interpolation points and the spectral zeros as prescribed by this theory one is able to obtain considerably better reduced order models we also show that in the multi input multi output case sorensen s algorithm actually amounts to tangential nevanlinna pick interpolation

stochastic balancing and approximation stability and minimality a new method of stochastic model reduction has recently been introduced by desai et al 1 2 the stability of the reduced order model has not previously been considered in this paper we show that the stability of the reduced order model follows directly from the results of pernebo and silverman 3 it is also shown that the reduced order model is minimal in the controllability observability sense the relevance of this notion of minimality to stochastic minimality is made clear

frequency weighted spl lscr sub spl infin norm and optimal hankel norm model reduction a new relative error model reduction method is proposed using frequency weighted balanced realization and explicit spl lscr sub spl infin norm error bounds are also derived for the relative error and multiplicative error the method only needs to solve two lyapunov equations it is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase this paper also gives a complete solution to the frequency weighted hankel norm approximation with antistable weighting these results are then applied to spl lscr sub spl infin norm model reduction and several numerically effective algorithms are proposed it is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions

multiresolution community detection for megascale networks by information based replica correlations we use a potts model community detection algorithm to accurately and quantitatively evaluate the hierarchical or multiresolution structure of a graph our multiresolution algorithm calculates correlations among multiple copies replicas of the same graph over a range of resolutions significant multiresolution structures are identified by strongly correlated replicas the average normalized mutual information the variation in information and other measures in principle give a quantitative estimate of the best resolutions and indicate the relative strength of the structures in the graph because the method is based on information comparisons it can in principle be used with any community detection model that can examine multiple resolutions our approach may be extended to other optimization problems as a local measure our potts model avoids the resolution limit that affects other popular models with this model our community detection algorithm has an accuracy that ranks among the best of currently available methods using it we can examine graphs over 40 ifmmode times else texttimes fi 10 6 nodes and more than 1 ifmmode times else texttimes fi 10 9 edges we further report that the multiresolution variant of our algorithm can solve systems of at least 200 text 000 nodes and 10 ifmmode times else texttimes fi 10 6 edges on a single processor with exceptionally high accuracy for typical cases we find a superlinear scaling o l 1 3 for community detection and o l 1 3 text text log text n for the multiresolution algorithm where l is the number of edges and n is the number of nodes in the system

line graphs link partitions and overlapping communities in this paper we use a partition of the links of a network in order to uncover its community structure this approach allows for communities to overlap at nodes so that nodes may be in more than one community we do this by making a node partition of the line graph of the original network in this way we show that any algorithm that produces a partition of nodes can be used to produce a partition of links we discuss the role of the degree heterogeneity and propose a weighted version of the line graph in order to account for this

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

statistical mechanics of community detection starting from a general ansatz we show how community detection can be interpreted as finding the ground state of an infinite range spin glass our approach applies to weighted and directed networks alike it contains the ad hoc introduced quality function from j reichardt and s bornholdt phys rev lett 93 218701 2004 and the modularity q as defined by newman and girvan phys rev e 69 026113 2004 as special cases the community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices we elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study further we show how hierarchies and overlap in the community structure can be detected computationally efficient local update rules for optimization procedures to find the ground state are given we show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension finally we give expectation values for the modularity of random graphs which can be used in the assessment of statistical significance of community structure

on stochastic model reduction in a recent paper by zhou and khargonekar 2 an example seemed to indicate that the stochastic model reduction procedure of jonckheere and helton 1 falls short of what could actually be achieved in this note we show that a variant of the algorithm of jonckheere and helton already proposed in the original manuscript 1 sect iv c yields results competitive with those of zhou and khargonekar 2

on inner outer factorization abstract the calculation of an inner outer factorization of a proper rational matrix by state space methods is considered particular attention is given to the extraction of the inner factor so that a minimal realization is obtained for this factor

pseudo paths towards minimum energy states in network dynamics abstract the dynamics of networks forming on heider balance theory moves towards lower tension states the condition derived from this theory enforces agents to reevaluate and modify their interactions to achieve equilibrium these possible changes in network s topology can be considered as various paths that guide systems to minimum energy states based on this theory the final destination of a system could reside on a local minimum energy jammed state or the global minimum energy balanced states the question we would like to address is whether jammed states just appear by chance or there exist some pseudo paths that bound a system towards a jammed state we introduce an indicator to suspect the location of a jammed state based on the inverse participation ratio method ipr we provide a margin before a local minimum where the number of possible paths dramatically drastically decreases this is a condition that proves adequate for ending up on a jammed states

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

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

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

bundle reduction and the alignment distance on spaces of state space lti systems this paper introduces a large class of differential geometric distances between finite dimensional linear dynamical systems collectively called the alignment distance contrary to the existing distances the alignment distance is based on the state space description of dynamical systems and is defined on the manifolds of systems of fixed order and fixed input output dimension under a matrix rank constraint e g minimality controllability or observability while the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory distances natural to this structure have not been studied the starting point for defining such a distance is to identify a linear system of order n with its equivalence class of state space realizations all related by the so called similarity action i e state space change of basis under gl n the lie group of nonsingular n times n matrices the main idea of the alignment distance is to first find the best state space change of basis that brings a realization of a system as close as possible to a realization of another system the alignment step and then compare the aligned realizations a direct implementation of this idea due to noncompactness of gl n is complicated however using the notion of reduction of the structure group of a principal bundle we show that the change of basis can be restricted to an orthogonal change of basis provided one uses realizations in a reduced subbundle this key observation brings about significant computational benefits as a technical contribution possibly of independent interest we show that several forms of realization balancing available in the control literature have differential geometric significance and are indeed examples of reducing the structure group from gl n to its subgroup of orthogonal matrices o n the alignment distance can be defined for stable and unstable systems discrete or continuous time and stochastic systems

relative error bound for discrete balanced stochastic truncation fundamental differences between discrete time balanced stochastic truncation bst and its continuous time counterpart are shown to preclude a straightforward application of bilinear mappings for directly generalizing existing continuous time error bounds nevertheless discrete time bst is shown to satisfy precisely the same lk error bound as continuous time bst although the discrete time derivations are somewhat more complex

a binary independent component analysis approach to tree topology inference using multicast probes to infer network topologies and internal link node characteristics is an attractive approach due to its bandwidth efficiency and suitability for large scale measurements in this paper we propose a new approach to tree topologies inference by exploiting dependence among end point receivers we first show that under the assumption of independent failure of intermediate nodes or links inferring tree topology is a special instance of the more general problem of binary independent component analysis bica and thus is amiable to existing analytical results and algorithms for bica then we propose the seqbica algorithm that is tailored for tree topology inference evaluation study shows that the proposed algorithm outperforms existing approaches in convergence speed and accuracy even when the number of measurements is small

eigenvectors of network complement reveal community structure more accurately abstract we propose a general spectral method to find communities of a network based on network complement and anti community concepts analytical and numerical results show that the eigenspace of matrices corresponding to a network complement reveals the community structure of a network more accurately than the eigenspace of matrices corresponding to the network itself it is shown that the laplacian eigenspace is the best candidate for spectral community detection especially in networks with a heterogeneous community structure the method is applied to some computer generated and real world networks with known community structures

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

non linear systems stabilization and coprimeness a theory of coprimeness is developed for non linear systems with the intention of constructing analytic tools for the solution of the problem of stabilizing a non linear system through the application of additive non linear feedback the qualitative features of the resulting theory are strikingly similar to the situation in the linear case

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

on constructing regular filter banks from domain bounded polynomials the design of regular two channel bi orthogonal filter banks is shown to be reducible to the design of pairs of real polynomials with domain bounded to the interval spl lsqb spl minus 1 1 spl rsqb techniques for designing polynomials satisfying various constraints are outlined transformation of polynomials to multidimensional bi orthogonal filter banks is presented

order reduction of siso nonminimum phase stochastic systems we consider the order reduction problem for single input single output nonminimum phase linear stochastic systems the problem is formulated as that of approximating the original system output process by another stochastic process generated via a reduced order model it is shown that in order to obtain a correct phase approximation it is necessary to consider some higher order statistics of the original process in addition to the usual second order statistics a computationally feasible approximation criterion is proposed and analyzed an example is also presented to illustrate the proposed method

state space formulae for the factorization of all pass matrix functions abstract we consider the factorization of an all pass matrix function e s using proper stable minimum phase factors stale space formulae for the stable minimum phase factors of e s are derived when they exist as well as state space formulae for the wiener hopf factors this is achieved using the state space characterization of all pass matrix functions given by glover 1984 in conjunction with the results of green and anderson 1987

system identification using balanced parametrizations some general issues in the black box identification of multivariable systems are first discussed it is then suggested that balanced parametrizations can be used to give identifiable forms a particular advantage is that balanced parametrizations are known for several useful classes of linear dynamic models including stable minimal models minimum phase models positive real models and normalized coprime factor models before optimizing the parameters of balanced parametrizations an initial model must be found we use realization based methods and so called subspace methods for this purpose these methods are very effective at finding accurate initial models without preliminary estimation of various structural indexes the paper ends with two simulation examples which compare the use of balanced parametrizations with more traditional ones and three real examples based on practical problems a distillation column an industrial dryer and the irrational spectrum or sea waves

the discrete wavelet transform wedding the a trous and mallat algorithms two separately motivated implementations of the wavelet transform are brought together it is observed that these algorithms are both special cases of a single filter bank structure the discrete wavelet transform the behavior of which is governed by the choice of filters in fact the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact moreover it is shown that the commonly used lagrange a trous filters are in one to one correspondence with the convolutional squares of the daubechies filters for orthonormal wavelets of compact support a systematic framework for the discrete wavelet transform is provided and conditions are derived under which it computes the continuous wavelet transform exactly suitable filter constraints for finite energy and boundedness of the discrete transform are also derived relevant signal processing parameters are examined and it is observed that orthonormality is balanced by restrictions on resolution

power spectrum reduction by optimal hankel norm approximation of the phase of the outer spectral factor a new approach to stochastic processes power spectrum reduction based on akaike s canonical correlation between the process past and future is developed as a first result the correlation between the canonical past and the canonical future is given its precise functional analytic interpretation as the hankel operator generated lay the all pass function characterizing the phase of the process outer spectral factor the canonical correlation coefficients turn out to he nothing else but the singular values of this hankel operator next the all pass function characterizing the phase of the outer spectral factor is reduced using the optimal hankel norm procedure of adamjan arov and krein the reduction of the all pass phase function is implemented with the safeguards necessary for the magnitude information to be carried all the way through this involves bode s relations between the gain and the phase of an outer function nehari s extension and the winding number of the all pass phase function which is itself shown to be related to the definition of the past and the future finally the net result is a reduced order outer spectral factor that matches the high order spectral factor fairly well

detection of community structures in networks via global optimization abstract we present an analysis of communality structure in networks based on the application of simulated annealing techniques in this case we use as cost function the already introduced modularity q 1 which is based on the relative number of links within a commune against the number of links that would correspond in case the links were distributed randomly we compare the results of our approach against other methodologies based on betweenness analysis and show that in all cases a better community structure can be attained

balanced stochastic realizations abstract the balanced stochastic realization is introduced as a balanced solution to the continuous time positive real equations and the dual positive real equations the structure of the associated balanced spectral factors is derived the main result being a product decomposition of the spectral factors the properties of the stochastic model reduction technique of balanced stochastic truncation are considered in particular it is shown that balanced stochastic truncation preserves the right half plane zeros of the spectral factors leading to a transformation approach to the approximation of non minimum phase stochastic systems

how many clusters an information theoretic perspective clustering provides a common means of identifying structure in complex data and there is renewed interest in clustering as a tool for the analysis of large data sets in many fields a natural question is how many clusters are appropriate for the description of a given system traditional approaches to this problem are based on either a framework in which clusters of a particular shape are assumed as a model of the system or on a two step procedure in which a clustering criterion determines the optimal assignments for a given number of clusters and a separate criterion measures the goodness of the classification to determine the number of clusters in a statistical mechanics approach clustering can be seen as a trade off between energy and entropy like terms with lower temperature driving the proliferation of clusters to provide a more detailed description of the data for finite data sets we expect that there is a limit to the meaningful structure that can be resolved and therefore a minimum temperature beyond which we will capture sampling noise this suggests that correcting the clustering criterion for the bias that arises due to sampling errors will allow us to find a clustering solution at a temperature that is optimal in the sense that we capture maximal meaningful structure without having to define an external criterion for the goodness or stability of the clustering we show that in a general information theoretic framework the finite size of a data set determines an optimal temperature and we introduce a method for finding the maximal number of clusters that can be resolved from the data in the hard clustering limit

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

modeling of normalized coprime factors with spl nu metric uncertainty modeling of uncertain systems with normalized coprime factor description is investigated where the experimental data is given by a finite set of frequency response measurement samples of the open loop plant that is linear time invariant and possibly infinite dimensional the objective is not only to identify the nominal model but also to quantify the modeling error with sup norm bounds in frequency domain the uncertainty to be identified and quantified is chosen as the spl nu metric proposed by vinnicombe 1993 because of its compatibility with h sub spl infin based robust control an algorithm is developed to model the normalized coprime factors of the given plant using techniques of discrete fourier analysis and balanced stochastic truncation and is shown to be robust in the presence of the worst case noise upper bounds are derived for the associated modeling error based on the minimum a priori information of the underlying model set and of the noise level in the measurement data a simulation example is used to illustrate the effectiveness of the proposed algorithm

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

community detection based on the clumpiness matrix in complex networks abstract the clumpiness matrix of a network is used to develop a method to identify its community structure a projection space is constructed from the eigenvectors of the clumpiness matrix and a border line is defined using some kind of angular distance in this space the community structure of the network is identified using this borderline and or hierarchical clustering methods the performance of our algorithm is tested on some computer generated and real world networks the accuracy of the results is checked using normalized mutual information the effect of community size heterogeneity on the accuracy of the method is also discussed

state space approach to approximation by phase matching a state space realization of the stable part of the phase function of a proper spectral factor is provided the phase matching problem jonckheere and helton 1985 harshavardhana and jonckheere 1985 can now be solved completely in the state space framework using existing numerically reliable procedures further a connection is made between the stochastic balancing approach and the phase approximation procedure for the stochastic realization approximation

paper relative multiplicative model reduction for unstable and non minimum phase systems this paper considers the relative multiplicative model reduction for unstable and non minimum phase systems using generalized frequency weighted normal representations the proposed method only requires solving two lyapunov equations with possibly indefinite solutions and preserves the number of unstable poles and the number of zeros in the square case furthermore a priori l norm error bounds are also derived for the relative approximation error and the multiplicative approximation error

inversion method for content based networks in this paper we generalize a recently introduced expectation maximization em method for graphs and apply it to content based networks the em method provides a classification of the nodes of a graph and allows one to infer relations between the different classes content based networks are ideal models for graphs displaying any kind of community and or multipartite structure we show both numerically and analytically that the generalized em method is able to recover the process that led to the generation of such networks we also investigate the conditions under which our generalized em method can recover the underlying contentbased structure in the presence of randomness in the connections two entropies sq and sc are defined to measure the quality of the node classification and to what extent the connectivity of a given network is content based sq and sc are also useful in determining the number of classes for which the classification is optimal

identification of overlapping community structure in complex networks using fuzzy c means clustering abstract identification of overlapping communities clusters in a complex network is a general problem in data mining of network data sets in this paper we devise a novel algorithm to identify overlapping communities in complex networks by the combination of a new modularity function based on generalizing ng s q function an approximation mapping of network nodes into euclidean space and fuzzy c means clustering experimental results indicate that the new algorithm is efficient at detecting both good clusterings and the appropriate number of clusters

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

factorization of all pass matrix functions we consider the factorization of an all pass matrix function e s into proper stable minimum phase factors this is achieved by applying the state space wiener hopf factorization theory of bart et al 1983 to the state space characterization of all pass matrix functions given by glover 1984 important connections are made between the properties of e s as a nehari extension its partial wiener hopf factorization indices and its hankel singular values

singular optimal control a geometric approach linear quadratic singular optimal control problem is solved for nonminimum phase and noninvertible systems a state space decomposition is obtained and a reduced order nonsingular subproblem is solved the optimal stabilizing input of the singular problem has been found when there are no transmission zeros on the imaginary axis

model reduction with relative multiplicative error bounds and relations to controller reduction balanced model reduction with a priori relative multiplicative error bounds in l sub spl infin norm is studied inverse weighted balanced truncation iwbt is proposed that is a variant of the enns algorithm it is shown that iwbt is equivalent to the balanced stochastic truncation bst this equivalence leads to the extension of an improved relative multiplicative error bound for bst obtained by wang and safonov to general nonsquare transfer matrices natural relations between iwbt and observer based controller reduction are investigated which motivate the simultaneous reduction of the plant and controller robust stability conditions for simultaneous reduction of the plant and the controller are established for the closed loop system consisting of full order plant and reduced order controller as a by product robust stability and stabilization problems are solved for general coprime factor plant descriptions with relative multiplicative h sub spl infin norm bounded uncertainties similar results are obtained for general coprime factors of the feedback controller involving relative multiplicative h sub spl infin norm bounded uncertainties

a transformation approach to stochastic model reduction a new and direct approach to stochastic model reduction is developed the order reduction algorithm is obtained by establishing an equivalence between canonical correlation analysis and solutions to algebraic riccati equations also the concept of balanced stochastic realization bsr plays a fundamental role asymptotic stability of the reduced order realization is established and spectral domain interpretations for the bsr are given

zeros of spectral factors the geometry of splitting subspaces and the algebraic riccati inequality in this paper it is shown how the zero dynamics of not necessarily square spectral factors relate to the splitting subspace geometry of stationary stochastic models and to the corresponding algebraic riccati inequality the notion of output induced subspace of a minimal markovian splitting subspace which is the stochastic analogue of the supremal output nulling subspace in geometric control theory is introduced through this concept the analysis can be made coordinate free and straightforward geometric methods can be applied it is shown how the zero structure of the family of spectral factors relates to the geometric structure of the family of minimal markovian splitting subspaces in the sense that the relationship between the zeros of different spectral factors is reflected in the partial ordering of minimal splitting subspaces finally the well known characterization of the solutions of the algebraic riccati equation is generalized in terms of lagrangian subspaces invariant under the corresponding hamiltonian to the larger solution set of the algebraic riccati inequality

acausal models and balanced realizations of stationary processes abstract we study acausal realizations of stationary or stationary increment processes in particular we characterize the family of models whose corresponding spectral factors have a fixed zero structure acausal models with a fixed zero structure are related to each other by a certain group of state feedback transformations which is naturally parametrized by the solution set of a homogeneous algebraic riccati equation each feedback transformation reflects some of the eigenvalues of the generator matrix a of the representation to a mirror image with respect to the imaginary axis dually acausal models with a fixed pole structure are parametrized by a dual riccati equation and by a corresponding family of output injection transformations from a general standpoint the results of this study clarify the role played by dual pairs of riccati equations in spectral factorization and may be relevant to other problem areas than stochastic modelling one natural application of the concepts discussed in the paper is to stochastic balancing balancing of models with an essentially arbitrary eigenvalue location can be accomodated very naturally in this framework a balancing algorithm involving the solution of a dual pair of riccati equations is discussed

multiplicative approximation of transfer functions with frequency weighting abstract an algorithm for transfer function order reduction is presented which generalizes the balanced stochastic truncation algorithm to allow for input weighting an example illustrates use of the algorithm to secure smaller db error in selected frequency bands through the introduction of the weighting

estimating the number of clusters in a data set via the gap statistic we propose a method the gap statistic for estimating the number of clusters groups in a set of data the technique uses the output of any clustering algorithm e g k means or hierarchical comparing the change in within cluster dispersion with that expected under an appropriate reference null distribution some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature

partitioning and modularity of graphs with arbitrary degree distribution we solve the graph bipartitioning problem in dense graphs with arbitrary degree distribution using the replica method we find the cut size to scale universally with k in contrast earlier results studying the problem in graphs with a poissonian degree distribution had found a scaling with kfu and anderson j phys a 19 1605 1986 our results also generalize to the problem of q partitioning they can be used to find the expected modularity q newman and girvan phys rev e 69 026113 2004 of random graphs and allow for the assessment of the statistical significance of the output of community detection algorithms given a graph or network gn m of n nodes and m edges the problem of graph partitioning is finding a partition of the nodes into q equal sized parts such that the number of edges connecting different parts the cut size is minimal the solution of this problem has many important practical applications in multiprocessor scheduling for parallel computing very large scale integrated chip design vlsi and data mining 1 due to the problem being np complete 2 i e no algorithm is known which is guaranteed to find an optimal solution in a number of steps that grows only polynomially with the size of the graph only heuristics exist to approximate a solution furthermore one cannot determine in polynomial time whether a solution found is indeed optimal the only way to assess the quality of a heuristic algorithm or a particular result is to compare with rigorous bounds on the cut size

flexible design of multidimensional perfect reconstruction fir 2 band filters using transformations of variables an approach to designing multidimensional linear phase fir diamond subband filters having the perfect reconstruction property is presented it is based on a transformation of variables technique and is equivalent to the generalized mcclellan transformation methods for designing a whole class of transformation are given the approach consists of two parts design of the transformation and design of the 1 d filters the use of lagrange halfband filters to design the 1 d filters is discussed the modification of a particular lagrange halfband filter which gives a pair of simple 1 d filters that are almost similar to each other in their frequency characteristics but still form a perfect reconstruction pair is presented the design technique is extended to other types of two channel sampling lattice and subband shapes in particular the parallelogram and the diagonally quadrant subband cases several numerical design examples are presented to illustrate the flexibility of the design method