some applications of covariance identities and inequalities to functions of multivariate normal variables abstract we apply general covariance identities and inequalities to some functions of multivariate normal variables we recover in particular a recent covariance identity due to siegel and provide simple estimates on the variance of order statistics we also present some computations

spectra and eigenvectors of scale free networks we study the spectra and eigenvectors of the adjacency matrices of scale free networks when bidirectional interaction is allowed so that the adjacency matrix is real and symmetric the spectral density shows an exponential decay around the center followed by power law long tails at both spectrum edges the largest eigenvalue ensuremath lambda 1 depends on system size n as ensuremath lambda 1 ensuremath sim n 1 4 for large n and the corresponding eigenfunction is strongly localized at the hub the vertex with largest degree the component of the normalized eigenfunction at the hub is of order unity we also find that the mass gap scales as n ensuremath 0 68

poincare type inequalities via stochastic integrals martingales and stochastic integrals are applied to prove poincare type inequalities involving probability distributions on the euclidean space these inequalities generalize and improve several results in the literature and are shown to yield weighted poincare inequalities on some special compact manifolds this leads to a new method of calculating all the eigenvalues and eigenfunctions of the laplacian on then sphere as a by product the eigenvalues are shown to be related to the moments of a probability distribution

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

second order poincare inequalities and clts on wiener space we prove infinite dimensional second order poincare inequalities on wiener space thus closing a circle of ideas linking limit theorems for functionals of gaussian fields stein s method and malliavin calculus we provide two applications i to a new second order characterization of clts on a fixed wiener chaos and ii to linear functionals of gaussian subordinated fields

on clusterings good bad and spectral we motivate and develop a natural bicriteria measure for assessing the quality of a clustering that avoids the drawbacks of existing measures a simple recursive heuristic is shown to have poly logarithmic worst case guarantees under the new measure the main result of the article is the analysis of a popular spectral algorithm one variant of spectral clustering turns out to have effective worst case guarantees another finds a good clustering if one exists

zero crossing rates of mixtures and products of gaussian processes formulas for the expected zero crossing rate of non gaussian mixtures and products of gaussian processes are obtained the approach we take is to first derive the expected zero crossing rate in discrete time and then obtain the rate in continuous time by an appropriate limiting argument the processes considered which are non gaussian but derived from gaussian processes serve to illustrate the variability of the zero crossing rate in terms of the normalized autocorrelation function p t of the process for gaussian processes rice s formula gives the expected zero crossing rate in continuous time as 1 spl pi spl radic spl rho 0 we show processes exist with expected zero crossing rates given by spl kappa spl pi spl radic spl rho 0 with either spl kappa spl gt 1 or spl kappa spl lt 1 consequently such processes can have an arbitrarily large or small zero crossing rate as compared to a gaussian process with the same autocorrelation function

the spectral distribution of random mixed graphs abstract let g be a mixed graph with n vertices h g the hermitian adjacency matrix of g and 1 g 2 g n g the eigenvalues of h g the hermitian energy of g is defined as e h g i 1 n i g in this paper we characterize the limiting spectral distribution of the hermitian adjacency matrices of random mixed graphs and as an application we give an estimation of the hermitian energy for almost all mixed graphs

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

entrance and escape dynamics for the typical set according to the asymptotic equipartition property sufficiently long sequences of random variables converge to a set that is typical while the size and probability of this set are central to information theory and statistical mechanics they can often only be estimated accurately in the asymptotic limit due to the exponential growth in possible sequences here we derive a time inhomogeneous dynamics that constructs the properties of the typical set for all finite length sequences of independent and identically distributed random variables these dynamics link the finite properties of the typical set to asymptotic results and allow the typical set to be applied to small and transient systems the main result is a geometric mapping the triangle map relating sequences of random variables of length n to those of length n 1 we show that the number of points in this map needed to quantify the properties of the typical set grows linearly with sequence length despite the exponential growth in the number of typical sequences we illustrate the framework for the bernoulli process and the schlogl model for autocatalytic chemical reactions and demonstrate both the convergence to asymptotic limits and the ability to reproduce exact calculations

laplacian spectra as a diagnostic tool for network structure and dynamics we examine numerically the three way relationships among structure laplacian spectra and frequency synchronization dynamics on complex networks we study the effects of clustering degree distribution and a particular type of coupling asymmetry input normalization all of which are known to have effects on the synchronizability of oscillator networks we find that these topological factors produce marked signatures in the laplacian eigenvalue distribution and in the localization properties of individual eigenvectors using a set of coordinates based on the laplacian eigenvectors as a diagnostic tool for synchronization dynamics we find that the process of frequency synchronization can be visualized as a series of quasi independent transitions involving different normal modes particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes for example there are important partially synchronized states in which a set of low lying modes remain unlocked while those in the main spectral peak are locked we find therefore that spectra are correlated with dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue

fluctuations of eigenvalues and second order poincare inequalities linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems the proofs of such results are usually rather difficult involving hard computations specific to the model in question in this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments in the process we introduce a notion of second order poincare inequalities just as ordinary poincare inequalities give variance bounds second order poincare inequalities give central limit theorems the proof of the main result employs stein s method of normal approximation a number of examples are worked out some of which are new one of the new results is a clt for the spectrum of gaussian toeplitz matrices

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

a generalization of chernoff inequality via stochastic analysis let be a probability measure on rdwith density c exp 2u x where u c2 rd left nabla u x right 2 delta u x to infty and u x as x by using stochastic analysis and theorems in schrodinger operators we have the following result there exists a constant c 0 such that var mu f underline leqslant c e mu left nabla f right 2 1 for any f l1 with a well defined distributional gradient f under our condition the operator frac 1 2 delta nabla u cdot nabla in l2 has discrete spectrum 0 1 2 m m 1 with corresponding eigenfunctions n which form a c o n s complete orthonormal system if the r h s of 1 is finite then equality holds iff f sum limits i 1 m b i phi i for some b1 bm r moreover the constant c can be taken as 2 2 1

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

multivariate inequalities of chernoff type for classical orthogonal polynomials abstract in this paper we present two sided chernoff type inequalities for the error of the best approximation of a smooth d variate function by polynomials of total degree less than k in the l w 2 norm it is supposed that the d variate weight function w has one dimensional classical component weights which satisfy pearson differential equations similarly as in the univariate case the leading coefficients of the multivariate classical polynomials orthonormal with respect to the weight function w play an important role in the presented estimates

asymptotics for high dimensional regression m estimates fixed design results we investigate the asymptotic distributions of coordinates of regression m estimates in the moderate p n regime where the number of covariates p grows proportionally with the sample size n under appropriate regularity conditions we establish the coordinate wise asymptotic normality of regression m estimates assuming a fixed design matrix our proof is based on the second order poincare inequality and leave one out analysis some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices we also show a counterexample namely an anova type design to emphasize that the technical assumptions are not just artifacts of the proof finally numerical experiments confirm and complement our theoretical results

zero safe nets comparing the collective and individual token approaches abstract the main feature of zero safe nets is a primitive notion of transition synchronization to this aim besides ordinary places called stable places zero safe nets are equipped with zero places which in an observable marking cannot contain any token this yields the notion of transaction a basic atomic computation which may use zero tokens as triggers but defines an evolution between observable markings only the abstract counterpart of a generic zero safe net b consists of an ordinary p t net whose places are the stable places of b and whose transitions represent the transactions of b the two nets offer the refined and the abstract models of the same system respectively where the former can be much smaller than the latter because of the transition synchronization mechanism depending on the chosen approach collective vs individual token philosophy two notions of transaction may be defined each leading to different operational and abstract models their comparison is fully discussed on the basis of a multicasting system example in the second part of the paper we make use of category theory to analyze and motivate our framework more precisely the two operational semantics of zero safe nets are characterized as adjunctions and the derivation of abstract p t nets as coreflections

structural information and dynamical complexity of networks in 1953 shannon proposed the question of quantification of structural information to analyze communication systems the question has become one of the longest great challenges in information science and computer science here we propose the first metric for structural information given a graph g we define the k dimensional structural information of g or structure entropy of g denoted by mathcal h k g to be the minimum overall number of bits required to determine the k dimensional code of the node that is accessible from random walk in g the k dimensional structural information provides the principle for completely detecting the natural or true structure which consists of the rules regulations and orders of the graphs for fully distinguishing the order from disorder in structured noisy data and for analyzing communication systems solving the shannon s problem and opening up new directions the k dimensional structural information is also the first metric of dynamical complexity of networks measuring the complexity of interactions communications operations and even evolution of networks the metric satisfies a number of fundamental properties including additivity locality robustness local and incremental computability and so on we establish the fundamental theorems of the one and two dimensional structural information of networks including both lower and upper bounds of the metrics of classic data structures general graphs the networks of models and the networks of natural evolution we propose algorithms to approximate the k dimensional structural information of graphs by finding the k dimensional structure of the graphs that minimizes the k dimensional structure entropy we find that the k dimensional structure entropy minimization is the principle for detecting the natural or true structures in real world networks consequently our structural information provides the foundation for knowledge discovering from noisy data we establish a black hole principle by using the two dimensional structure information of graphs we propose the natural rank of locally listing algorithms by the structure entropy minimization principle providing the basis for a next generation search engine

pattern formation by boundary forcing in convectively unstable oscillatory media with and without differential transport convectively unstable open reactive flows of oscillatory media whose phase is fixed or periodically modulated at the inflow boundary are known to result in stationary and traveling waves respectively the latter are implicated in biological segmentation the boundary controlled pattern selection by this flow distributed oscillator fdo mechanism has been generalized to include differential flow difi and differential diffusion turing modes our present goal is to clarify the relationships among these mechanisms in the general case where there is differential flow as well as differential diffusion to do so we analyze the dispersion relation for linear perturbations in the presence of periodic boundary forcing and show how the solutions are affected by differential transport we find that the difi and fdo modes are closely related and lie in the same frequency range while the turing mechanism gives rise to a distinct set of unstable modes in a separate frequency range finally we substantiate the linear analysis by nonlinear simulations and touch upon the issue of competition of spatial modes

hardy s inequality for dirichlet forms abstract we prove an abstract form of hardy s l2 inequality in which the dirichlet integral is replaced by the dirichlet form of a general symmetric markov process a number of examples are provided

detecting communities in large networks we develop an algorithm to detect community structure in complex networks the algorithm is based on spectral methods and takes into account weights and link orientation since the method detects efficiently clustered nodes in large networks even when these are not sharply partitioned it turns to be specially suitable for the analysis of social and information networks we test the algorithm on a large scale data set from a psychological experiment of word association in this case it proves to be successful both in clustering words and in uncovering mental association patterns

a note on an inequality involving the normal distribution the following inequality is useful in studying a variation of the classical isoperimetric problem let x be normally distributed with mean 0 and variance 1 if g is absolutely continuous and g x has finite variance then e lbrack g x rbrack 2 geq operatorname var lbrack g x rbrack with equality if and only if g x is linear in x the proof involves expanding g x in hermite polynomials

fisher information inequalities and the central limit theorem we give conditions for an o 1 n rate of convergence of fisher information and relative entropy in the central limit theorem we use the theory of projections in l 2 spaces and poincare inequalities to provide a better understanding of the decrease in fisher information implied by results of barron and brown we show that if the standardized fisher information ever becomes finite then it converges to zero

discrete isoperimetric and poincare type inequalities we study some discrete isoperimetric and poincare type inequalities for product probability measures n on the discrete cube 0 1 n and on the lattice z n in particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube more generally we characterize those probability distributions on z which satisfy these inequalities on z n the class of these distributions can be described by a certain class of monotone transforms of the two sided exponential measure a similar characterization of distributions on r which satisfy poincare inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes

analysis of nonlinear synchronization dynamics of oscillator networks by laplacian spectral methods we analyze the synchronization dynamics of phase oscillators far from the synchronization manifold including the onset of synchronization on scale free networks with low and high clustering coefficients we use normal coordinates and corresponding time averaged velocities derived from the laplacian matrix which reflects the network s topology in terms of these coordinates synchronization manifests itself as a contraction of the dynamics onto progressively lower dimensional submanifolds of phase space spanned by laplacian eigenvectors with lower eigenvalues differences between high and low clustering networks can be correlated with features of the laplacian spectrum for example the inhibition of full synchoronization at high clustering is associated with a group of low lying modes that fail to lock even at strong coupling while the advanced partial synchronization at low coupling noted elsewhere is associated with high eigenvalue modes

some results on upper bounds for the variance of functions of the residual life random variables as a measure of maximum dispersion from the mean upper bounds on variance have applications in all areas of theoretical and applied mathematical sciences in this paper we obtain an upper bound for the variance of a function of the residual life random variable xt since one of the most important types of system structures is the parallel structure we give an upper bound for the variance of a function of this system consisting of n identical and independent components under the condition that at time t nr 1 r 1 n of its components are still working here we characterize the pareto distribution through cauchys functional equation for mean residual life it is shown that the underlying distribution function f can be recovered from the proposed mean and variance residual life function of the system for r 1 moreover we see that the variance residual lifetime of the components of the system is not necessarily a decreasing function of r and increasing of n for r 1 unlike their mean residual lifetime as an application the variance of xf1 p0 for all p0 0 1 is investigated and also a real data analysis is presented

on upper and lower bounds for the variance of a function of a random variable chernoff 1981 obtained an upper bound for the variance of a function of a standard normal random variable using hermite polynomials chen 1980 gave a different proof using the cauchy schwarz inequality and extended the inequality to the case of a multivariate normal here it is shown how similar upper bounds can be obtained for other distributions including discrete ones moreover by using a variation of the cramer rao inequality analogous lower bounds are given for the variance of a function of a random variable which satisfies the usual regularity conditions matrix inequalities are also obtained all these bounds involve the first two moments of derivatives or differences of the function

normal approximation on poisson spaces mehler s formula second order poincare inequalities and stabilization we prove a new class of inequalities yielding bounds for the normal approximation in the wasserstein and the kolmogorov distance of functionals of a general poisson process poisson random measure our approach is based on an iteration of the classical poincare inequality as well as on the use of malliavin operators of stein s method and of an integrated mehler s formula providing a representation of the ornstein uhlenbeck semigroup in terms of thinned poisson processes our estimates only involve first and second order difference operators and have consequently a clear geometric interpretation in particular we will show that our results are perfectly tailored to deal with the normal approximation of geometric functionals displaying a weak form of stabilization and with non linear functionals of poisson shot noise processes we discuss two examples of stabilizing functionals in great detail i the edge length of the k nearest neighbour graph ii intrinsic volumes of k faces of voronoi tessellations in all these examples we obtain rates of convergence in the kolmogorov and the wasserstein distance that one can reasonably conjecture to be optimal thus significantly improving previous findings in the literature as a necessary step in our analysis we also derive new lower bounds for variances of poisson functionals