a quasi invariance theorem for measures on banach spaces we show that for a measure on a banach space directional differentiability implies quasi translation invariance this result is shown to imply the cameron martin theorem a second application is given in which ty is the image of a gaussian measure under a suitably regular map

equations differentielles sur l espace de wiener et formules de cameron martin non lineaires proprietes algebriques des approximations sur r n une formule de perturbation des flots et convergence des approximations formule de changement de variable sur x et equation differentielle

reciprocity of networks with degree correlations and arbitrary degree sequences although most of the real networks contain a mixture of directed and bidirectional reciprocal connections the reciprocity r has received little attention as a subject of theoretical understanding we study the expected reciprocity of networks with arbitrary input and output degree sequences and given 2 node degree correlations by means of statistical ensemble approach we demonstrate that degree correlations are crucial to understand the reciprocity in real networks and a hierarchy of correlation contributions to r is revealed numerical experiments using network randomization methods show very good agreement to our analytical estimations

mechanism for linear preferential attachment in growing networks abstract the network properties of a graph ensemble subject to the constraints imposed by the expected degree sequence are studied it is found that the linear preferential attachment is a fundamental rule as it keeps the maximal entropy in sparse growing networks this provides theoretical evidence in support of the linear preferential attachment widely exists in real networks and adopted as a crucial assumption in growing network models besides in the sparse limit we develop a method to calculate the degree correlation and clustering coefficient in our ensemble model which is suitable for all kinds of sparse networks including the ba model proposed by barabasi and albert

a poincare inequality on loop spaces we show that the laplacian on the loop space over a class of riemannian manifolds has a spectral gap the laplacian is defined using the levi civita connection the brownian bridge measure and the standard bismut tangent spaces

orthogonal polynomial eigenfunctions of second order partial differerential equations in this paper we show that for several second order partial differential equations l u a x y uxx 2b x y uxy c x y uyy d x y ux e x y uy atu which have orthogonal polynomial eigenfunctions these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable this is important since otherwise it is very difficult to explicitly find formulas for these polynomial solutions from this observation and characterization we are able to produce additional examples of such orthogonal polynomials together with their orthogonality that widens the class found by h l krall and sheffer in their seminal work in 1967 moreover from our approach we can answer some open questions raised by krall and sheffer

on bivariate classical orthogonal polynomials abstract we deduce new characterizations of bivariate classical orthogonal polynomials associated with a quasi definite moment functional and we revise old properties for these polynomials more precisely new characterizations of classical bivariate orthogonal polynomials satisfying a diagonal pearson type equation are proved they are solutions of two separate partial differential equations one for every partial derivative their partial derivatives are again orthogonal and every vector polynomial can be expressed in terms of its partial derivatives by means of a linear relation involving only three terms of consecutive total degree moreover we study general solutions of the matrix second order partial differential equation satisfied by classical orthogonal polynomials and we deduce the explicit expressions for the matrix coefficients of the structure relation finally some illustrative examples are given

smoothness of heat kernel measures on infinite dimensional heisenberg like groups abstract we study measures associated to brownian motions on infinite dimensional heisenberg like groups in particular we prove that the associated path space measure and heat kernel measure satisfy a strong definition of smoothness

partial differential equations having orthogonal polynomial solutions abstract we show that if a second order partial differential equation l u auxx 2buxy cuyy dux euy nu has orthogonal polynomial solutions then the differential operator l must be symmetrizable and can not be parabolic in any nonempty open subset of the plane we also find rodrigues type formula for orthogonal polynomial solutions of such differential equations

random graph model with power law distributed triangle subgraphs clustering is well known to play a prominent role in the description and understanding of complex networks and a large spectrum of tools and ideas have been introduced to this end in particular it has been recognized that the abundance of small subgraphs is important here we study the arrangement of triangles in a model for scale free random graphs and determine the asymptotic behavior of the clustering coefficient the average number of triangles as well as the number of triangles attached to the vertex of maximum degree we prove that triangles are power law distributed among vertices and characterized by both vertex and edge coagulation when the degree exponent satisfies 2l ensuremath beta l2 5 furthermore a finite density of triangles appears as ensuremath beta 2 1 3

anticipating flows on the wiener space generated by vector fields of low regularity abstract sobolev spaces for banach space valued wiener functionals are constructed using the operator norm for the h derivatives the corresponding sobolev norms are shown to be strictly weaker than the usual malliavin sobolev norms existing results on flows on an abstract wiener space generated by a vector field are generalized to the case where the vector field is a sum of a not necessarily bounded skew adjoint operator and a non linear part whose regularity is measured by the new sobolev norms this is equivalent to a non linear perturbation of a semi group of rotations

clustering of correlated networks we obtain the clustering coefficient the degree dependent local clustering and the mean clustering of networks with arbitrary correlations between the degrees of the nearest neighbor vertices the resulting formulas allow one to determine the nature of the clustering of a network

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

percolation on correlated networks we reconsider the problem of percolation on an equilibrium random network with degree degree correlations between nearest neighboring vertices focusing on critical singularities at a percolation threshold we obtain criteria for degree degree correlations to be irrelevant for critical singularities we present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents

malliavin calculus for the stochastic 2d navier stokes equation we consider the incompressible two dimensional navier stokes equation with periodic boundary conditions under the effect of an additive white in time stochastic forcing under mild restrictions on the geometry of the scales forced we show that any finite dimensional projection of the solution possesses a smooth strictly positive density with respect to lebesgue measure in particular our conditions are viscosity independent we are mainly interested in forcing that excites a very small number of modes all of the results rely on proving the nondegeneracy of the infinite dimensional malliavin matrix c 2006 wiley periodicals inc

origin of degree correlations in the internet and other networks it has been argued that the observed anticorrelation between the degrees of adjacent vertices in the network representation of the internet has its origin in the restriction that no two vertices have more than one edge connecting them here we propose a formalism for modeling ensembles of graphs with single edges only and derive values for the exponents and correlation coefficients characterizing them our results confirm that the conjectured mechanism does indeed give rise to correlations of the kind seen in the internet although only a part of the measured correlation can be accounted for in this way

the taylor map on complex path groups let w g denote the path group of an arbitrary complex connected lie group the existence of a heat kernel measure t on w g has been shown in m cecil b k driver heat kernel measure on loop and path groups preprint http www math uconn edu cecil papers p2 pdf infin dimens anal quantum probab relat top submitted for publication the present work establishes an isometric map the taylor map from the space of l2 t holomorphic functions on w g to a subspace of the dual of the universal enveloping algebra of lie h g where h g is the lie subgroup of finite energy paths this map is shown to be surjective in the case where g is a simply connected graded lie group

growing network with local rules preferential attachment clustering hierarchy and degree correlations the linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power law degree distributions it is then quite important to determine if this mechanism is the consequence of a general principle based on local rules in this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules it is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks these conclusions are based on both analytical and numerical results for different local rules including some models already proposed in the literature

mean field theory for clustering coefficients in barabasi albert networks we applied a mean field approach to study clustering coefficients in barabasi albert ba networks we found that the local clustering in ba networks depends on the node degree analytic results have been compared to extensive numerical simulations finding a very good agreement for nodes with low degrees clustering coefficient of a whole network calculated from our approach perfectly fits numerical data

quasi invariance of the wiener measure on the path space over a complete riemannian manifold we prove a generalization of the cameron martin theorem for a geometrically and stochastically complete riemannian manifold namely the wiener measure on the path space over such a manifold is quasiinvariant under the flow generated by a cameron martin vector field

diversity of graphs with highly variable connectivity a popular approach for describing the structure of many complex networks focuses on graph theoretic properties that characterize their large scale connectivity while it is generally recognized that such descriptions based on aggregate statistics do not uniquely characterize a particular graph and also that many such statistical features are interdependent the relationship between competing descriptions is not entirely understood this paper lends perspective on this problem by showing how the degree sequence and other constraints e g connectedness no self loops or parallel edges on a particular graph play a primary role in dictating many features including its correlation structure building on recent work we show how a simple structural metric characterizes key differences between graphs having the same degree sequence more broadly we show how the often implicit choice of a background set against which to measure graph features has serious implications for the interpretation and comparability of graph theoretic descriptions

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

a cameron martin type quasi invariance theorem for brownian motion on a compact riemannian manifold abstract let m be a compact manifold without boundary o be a fixed base point in m g be a riemannian metric on m and be a g compatible covariant derivative on tm the tangent space to m assume the torsion t of satisfies the skew symmetry condition g t x y y 0 for all vector fields x and y on m for example take to be the levi civita covariant derivative on m g also let denote the wiener measure on w0 m c 0 1 m 0 o and let h s denote stochastic parallel translation relative to along the path w0 m up to time s 0 1 given a c1 function h 0 1 t0m it is shown that the differential equation t h t h with initial condition 0 id w m w m has a solution r maps w m w m the measurable maps from w m to w m this function is a flow on w m i e for all t r t t furthermore t has the quasi invariance property the law t of t with respect to the wiener measure is equivalent to for all t r this result is used to prove an integration by parts formula for the h derivative hf defined by h f d dt 0 f t where f is a c2 cylinder function on w m

degree correlations in random geometric graphs spatially embedded networks are important in several disciplines the prototypical spatial network we assume is the random geometric graph of which many properties are known here we present new results for the two point degree correlation function in terms of the clustering coefficient of the graphs for two dimensional space in particular with extensions to arbitrary finite dimensions

logarithmic sobolev inequalities for pinned loop groups abstract letgbe a connected compact type lie group equipped with anadg invariant inner product on the lie algebra g ofg given this data there is a well known left invariant h1 riemannian structure on l l g the infinite dimensional group of continuous based loops ing using this riemannian structure we define and construct a heat kernel t g0 associated to the laplace beltrami operator on l g heret 0 g0 l g and t g0 is a certain probability measure on l g for fixedg0 l g andt 0 we use the measure t g0 and the riemannian structure on l g to construct a classical pre dirichlet form the main theorem of this paper asserts that this pre dirichlet form admits a logarithmic sobolev inequality

orthogonal polynomials in two variables and second order partial differential equations abstract we study the second order partial differential equations l u au xx 22 bu xy cu yy du x eu y n u which have orthogonal polynomials in two variables as solutions by using formal functional calculus on moment functionals we first give new simpler proofs and improvements of the results by krall and sheffer and littlejohn we then give a two variable version of al salam and chihara s characterization of classical orthogonal polynomials in one variable we also study in detail the case when l belongs to the basic class that is a y c x 0 in particular we characterize all such differential equations which have a product of two classical orthogonal polynomials in one variable as solutions

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

precise asymptotics of certain wiener functionals abstract in this article we present a method for developing certain wiener integrals in an asymptotic expansion as the variance of the wiener goes to 0 the basic ideas are a combination of methods borrowed from the theory of large deviations together with techniques from the calculus of wiener space

weak classical orthogonal polynomials in two variables orthogonal polynomials in two variables constitute a very old subject in approximation theory usually they are studied as solutions of second order partial differential equations in this work we study two variable orthogonal polynomials associated with a moment functional satisfying the two variable analogue of the pearson differential equation from this approach we derive the extension of some of the usual characterizations of the classical orthogonal polynomials in one variable

structural transitions in scale free networks real growing networks such as the world wide web or personal connection based networks are characterized by a high degree of clustering in addition to the small world property and the absence of a characteristic scale appropriate modifications of the barabasi albert preferential attachment network growth capture all these aspects we present a scaling theory to describe the behavior of the generalized models and the mean field rate equation for clustering this is solved for a specific case with the result c k approximately 1 k for the clustering of a node of degree k this mean field exponent agrees with simulations and reproduces the clustering of many real networks

integration by parts and quasi invariance for heat kernel measures on loop groups abstract integration by parts formulas are established both for wiener measure on the path space of a loop group and for the heat kernel measures on the loop group the wiener measure is defined to be the law of a certain loop group valued brownian motion and the heat kernel measures are time t t 0 distributions of this brownian motion a corollary of either of these integrations by parts formulas is the closability of the pre dirichlet form considered by b k driver and t lohrenz 1996 j functional anal 140 381 448 we also show that the heat kernel measures are quasi invariant under right under right and left translations by finite energy loops

homogeneous complex networks abstract we discuss various ensembles of homogeneous complex networks and a monte carlo method of generating graphs from these ensembles the method is quite general and can be applied to simulate micro canonical canonical or grand canonical ensembles for systems with various statistical weights it can be used to construct homogeneous networks with desired properties or to construct a non trivial scoring function for problems of advanced motif searching

second order partial differential equations for gradients of orthogonal polynomials in two variables in this work we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients the usual gradient operator and the divergence operator here we show that the successive gradients of these polynomials also satisfy a matrix second order partial differential equation closely related to the first one

a semiclassical perspective on multivariate orthogonal polynomials differential properties for orthogonal polynomials in several variables are studied we consider multivariate orthogonal polynomials whose gradients satisfy some quasi orthogonality conditions we obtain several characterizations for these polynomials including the analogues of the semiclassical pearson differential equation the structure relation and a differential difference equation

quasi invariance of brownian measures on the group of circle homeomorphisms and infinite dimensional riemannian geometry abstract a cameron martin type theorem is proved for the canonical brownian motion on the group of homeomorphisms of the circle

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

topology and correlations in structured scale free networks we study a recently introduced class of scale free networks showing a high clustering coefficient and nontrivial connectivity correlations we find that the connectivity probability distribution strongly depends on the fine details of the model we solve exactly the case of low average connectivity providing also exact expressions for the clustering and degree correlation functions the model also exhibits a lack of small world properties in the whole parameter range we discuss the physical properties of these networks in the light of the present detailed analysis

correlated random networks we develop a statistical theory of networks a network is a set of vertices and links given by its adjacency matrix c and the relevant statistical ensembles are defined in terms of a partition function z summation operator exp betah c the simplest cases are uncorrelated random networks such as the well known erdos renyi graphs here we study more general interactions h c which lead to correlations for example between the connectivities of adjacent vertices in particular such correlations occur in optimized networks described by partition functions in the limit beta infinity they are argued to be a crucial signature of evolutionary design in biological networks

heat kernel analysis on infinite dimensional heisenberg groups abstract we introduce a class of non commutative heisenberg like infinite dimensional lie groups based on an abstract wiener space the ricci curvature tensor for these groups is computed and shown to be bounded brownian motion and the corresponding heat kernel measures t t 0 are also studied we show that these heat kernel measures admit 1 gaussian like upper bounds 2 cameron martin type quasi invariance results 3 good l p bounds on the corresponding radon nikodym derivatives 4 integration by parts formulas and 5 logarithmic sobolev inequalities the last three results heavily rely on the boundedness of the ricci tensor

class of correlated random networks with hidden variables we study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices we find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices the expressions obtained are checked by means of numerical simulations in a particular example the general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure we also present an extension of the class to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system

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

two dimensional krall sheffer polynomials and integrable systems two dimensional krall sheffer polynomials are analogues of the classical orthogonal polynomial they are eigenfunctions of second order linear partial differential operators and moreover satisfy orthogonality conditions we show that all krall sheffer polynomials are connected with two dimensional superintegrable systems on spaces with constant curvature

semiclassical orthogonal polynomials in two variables in this work semiclassical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated with a quasi definite linear functional satisfying a matrix pearson type differential equation semiclassical functionals are characterized by means of the analogue of the structure relation in one variable moreover non trivial examples of semiclassical orthogonal polynomials in two variables are given

the geometry of loop groups l espace g des boucles basees sur un groupe de lie compact admet une metrique de kahler on exprime sa courbure en termes d operateurs de toeplitz et on definit des classes de chern par analogie avec la theorie de chern weil aux dimensions finies il y a une structure de fredhohn naturelle sur g induite a partir de la famille des operateurs de toeplitz on obtient des formules explicites pour su n

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

new steps on sobolev orthogonality in two variables sobolev orthogonal polynomials in two variables are defined via inner products involving gradients such a kind of inner product appears in connection with several physical and technical problems matrix second order partial differential equations satisfied by sobolev orthogonal polynomials are studied in particular we explore the connection between the coefficients of the second order partial differential operator and the moment functionals defining the sobolev inner product finally some old and new examples are given

symmetrizability of differential equations having orthogonal polynomial solutions abstract we show that if a linear differential equation of spectral type with polynomial coefficients l n y x i 0 n l i x n y x has an orthogonal polynomial system of solutions then the differential operator ln must be symmetrizable we also give a few applications of this result

an estimate of the gap of spectrum of schrodinger operators which generate hyperbounded semigroups abstract logarithmic sobolev inequalities with potential functions on loop spaces were proved by l gross 1991 j funct anal 102 265 313 s aida 1996 in proceedings of the fifth gregynog symposium stochastic analysis and applications pp 1 19 world scientific singapore and f z gong and z m ma 1998 j funct anal 599 623 the generators of the dirichlet forms are schrodinger operators on loop spaces and they generate hyperbounded semigroups recently f z gong m rockner and l m wu 2000 poincare inequality for weighted first order sobolev spaces on loop spaces submitted proved the existence of the gap of spectrum at the lowest eigenvalue of the schrodinger operators by using m hino s exponential decay estimates of the semigroup 2000 osaka j math 37 603 624 and the author s strong ergodicity property of the semi group s aida 1998 j funct anal 158 152 185 in this paper we will give a lower bound on the gap by using a weak poincare inequality which was introduced by m rockner and f y wang 2000 weak poincare inequalities and l 2 convergence rates of markov semigroups preprint also we will give estimates on the distribution function of ground states using the weak poincare inequality

higher order distributions and nongrowing complex networks without multiple connections we study stochastic processes that generate nongrowing complex networks without self loops and multiple edges simple graphs the work concentrates on understanding and formulation of constraints which keep the rewiring stochastic processes within the class of simple graphs to formulate these constraints a different concept of wedge distribution paths of length 2 is introduced and its relation to degree degree correlation is studied the analysis shows that the constraints together with edge selection rules do not even allow the formulation of a closed master equation in the general case we also introduce a particular stochastic process which does not contain edge selection rules but which we believe can provide some insight into the complexities of simple graphs