edge exchangeable models for interaction networks abstractmany modern network datasets arise from processes of interactions in a population such as phone calls email exchanges co authorships and professional collaborations in such interaction networks the edges comprise the fundamental statistical units making a framework for edge labeled networks more appropriate for statistical analysis in this context we initiate the study of edge exchangeable network models and explore its basic statistical properties several theoretical and practical features make edge exchangeable models better suited to many applications in network analysis than more common vertex centric approaches in particular edge exchangeable models allow for sparse structure and power law degree distributions both of which are widely observed empirical properties that cannot be handled naturally by more conventional approaches our discussion culminates in the hollywood model which we identify here as the canonical family of edge exchangeable distributions the hollywood model i

a pieri type formula for isotropic flag manifolds we give the formula for multiplying a schubert class on an odd orthogonal or symplectic flag manifold by a special schubert class pulled back from the grassmannian of maximal isotropic subspaces this is also the formula for multiplying a type b respectively type c schubert polynomial by the schur p polynomial p m respectively the schur q polynomial q m geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold these intermediate results are concerned with the bruhat order of the infinite coxeter group b identities of the structure constants for the schubert basis of cohomology and intersections of schubert varieties we show that most of these identities follow from the pieri type formula and our analysis leads to a new partial order on the coxeter group b and formulas for many of these structure constants

asymptotics in directed exponential random graph models with an increasing bi degree sequence although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature it remains an open problem to study the statistical properties of directed network models in this paper we provide for the first time a rigorous analysis of directed exponential random graph models using the in degrees and out degrees as sufficient statistics with binary and non binary weighted edges we establish the uniform consistency and the asymptotic normality of the maximum likelihood estimator when the number of parameters grows and only one realized observation of the graph is available one key technique in the proofs is to approximate the inverse of the fisher information matrix using a simple matrix with high accuracy along the way we also establish a geometrically fast rate of convergence for the newton iterative algorithm which is used to obtain the maximum likelihood estimate numerical studies confirm our theoretical findings

estimating and understanding exponential random graph models we introduce a method for the theoretical analysis of exponential random graph models the method is based on a large deviations approximation to the normalizing constant shown to be consistent using theory developed by chatterjee and varadhan european j combin 32 2011 1000 1017 the theory explains a host of difficulties encountered by applied workers many distinct models have essentially the same mle rendering the problems practically ill posed we give the first rigorous proofs of degeneracy observed in these models here almost all graphs have essentially no edges or are essentially complete we supplement recent work of bhamidi bresler and sly 2008 ieee 49th annual ieee symposium on foundations of computer science focs 2008 803 812 ieee showing that for many models the extra sufficient statistics are useless most realizations look like the results of a simple erd s renyi model we also find classes of models where the limiting graphs differ from erd s renyi graphs a limitation of our approach inherited from the limitation of graph limit theory is that it works only for dense graphs

random networks graphical models and exchangeability we study conditional independence relationships for random networks and their interplay with exchangeability we show that for finitely exchangeable network models the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts we then characterize all possible markov structures for finitely exchangeable random graphs thereby identifying a new class of markov network models corresponding to bidirected kneser graphs in particular we demonstrate that the fundamental property of dissociatedness corresponds to a markov property for exchangeable networks described by bidirected line graphs finally we study those exchangeable models that are also summarized in the sense that the probability of a network depends only on the degree distribution and we identify a class of models that is dual to the markov graphs of frank and strauss particular emphasis is placed on studying consistency properties of network models under the process of forming subnetworks and we show that the only consistent systems of markov properties correspond to the empty graph the bidirected line graph of the complete graph and the complete graph

an exponential family of probability distributions for directed graphs abstract directed graph or digraph data arise in many fields especially in contemporary research on structures of social relationships we describe an exponential family of distributions that can be used for analyzing such data a substantive rationale for the general model is presented and several special cases are discussed along with some possible substantive interpretations a computational algorithm based on iterative scaling procedures for use in fitting data is described as are the results of a pilot simulation study an example using previously reported empirical data is worked out in detail an extension to multiple relationship data is discussed briefly

stochastic blockmodels and community structure in networks stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks most blockmodels however ignore variation in vertex degree making them unsuitable for applications to real world networks which typically display broad degree distributions that can significantly affect the results here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks we also propose a heuristic algorithm for community detection using this objective function or its non degree corrected counterpart and show that the degree corrected version dramatically outperforms the uncorrected one in both real world and synthetic networks

an econometric model of network formation with degree heterogeneity i introduce a model of undirected dyadic link formation which allows for assortative matching on observed agent characteristics homophily as well as unrestricted agent level heterogeneity in link surplus degree heterogeneity like in fixed effects panel data analyses the joint distribution of observed and unobserved agent level characteristics is left unrestricted two estimators for the common homophily parameter 0 are developed and their properties studied under an asymptotic sequence involving a single network growing large the first tetrad logit tl estimator conditions on a sufficient statistic for the degree heterogeneity the second joint maximum likelihood jml estimator treats the degree heterogeneity a i0 i 1 super n as additional incidental parameters to be estimated the tl estimate is consistent under both sparse and dense graph sequences whereas consistency of the jml estimate is shown only under dense graph sequences

exponential random simplicial complexes exponential random graph models have attracted significant research attention over the past decades these models are maximum entropy ensembles subject to the constraints that the expected values of a set of graph observables are equal to given values here we extend these maximum entropy ensembles to random simplicial complexes which are more adequate and versatile constructions to model complex systems in many applications we show that many random simplicial complex models considered in the literature can be casted as maximum entropy ensembles under certain constraints we introduce and analyze the most general random simplicial complex ensemble with statistically independent simplices our analysis is simplified by the observation that any distribution on any collection of objects including graphs and simplicial complexes is maximum entropy subject to the constraint that the expected value of is equal to the entropy of the distribution with the help of this observation we prove that ensemble is maximum entropy subject to the two types of constraints which fix the expected numbers of simplices and their boundaries

exponential random graph models for networks with community structure although the community structure organization is an important characteristic of real world networks most of the traditional network models fail to reproduce the feature therefore the models are useless as benchmark graphs for testing community detection algorithms they are also inadequate to predict various properties of real networks with this paper we intend to fill the gap we develop an exponential random graph approach to networks with community structure to this end we mainly built upon the idea of blockmodels we consider both the classical blockmodel and its degree corrected counterpart and study many of their properties analytically we show that in the degree corrected blockmodel node degrees display an interesting scaling property which is reminiscent of what is observed in real world fractal networks a short description of monte carlo simulations of the models is also given in the hope of being useful to others working in the field

several remarks on acceleration of convergence using generalized linear methods of summability some acceleration theorems for generalized linear summability methods a ank where ank are linear bounded operators from banach space b space x into b space y are derived these results are proved by the method of summability with given rapidity for this purpose some problems of summability with given rapidity are discussed

acceleration methods based on convergence tests in this paper we propose an acceleration method based on a general convergence test and depending on an auxiliary sequence x n for different choices of x n we obtain some known and some new transformations and for each one sufficient conditions for acceleration are given

nonemptiness of symmetric degeneracy loci let v be a rank n vector bundle on a d dimensional complex projective scheme x assume that v is equipped with a quadratic form with values in a line bundle l and that s 2 v l is ample suppose that the maximum rank of the quadratic form at any point of x is r 0 the main result of this paper is that if d n r the locus of points where the rank of the quadratic form is at most r 1 is nonempty we give some applications to subschemes of matrices and to degeneracy loci associated to embeddings in projective space the paper concludes with an appendix on gysin maps the main result of the appendix which may be of independent interest identifies a gysin map with the natural map from ordinary to relative cohomology

permanental partition models and markovian gibbs structures we study both time invariant and time varying gibbs distributions for configurations of particles into disjoint clusters specifically we introduce and give some fundamental properties for a class of partition models called permanental partition models whose distributions are expressed in terms of the permanent of a similarity matrix parameter we show that in the time invariant case the permanental partition model is a refinement of the celebrated pitman ewens distribution whereas in the time varying case the permanental model refines the ewens cut and paste markov chains j appl probab 43 3 778 791 2011 by a special property of the permanent the partition function can be computed exactly allowing us to make several precise statements about this general model including a characterization of exchangeable and consistent permanental models

the chow ring of the classifying space bso 2n c we compute the chow ring of the classifying space bso 2n c in the sense of totaro using the fibration gl 2n so 2n bso 2n bgl 2n and a computation of the chow ring of gl 2n so 2n in a previous paper we find this chow ring is generated by chern classes and a characteristic class defined by edidin and graham which maps to 2n 1 times the euler class under the usual class map from the chow ring to ordinary cohomology moreover we show this class represents 1 2n 1 n 1 times the nth chern class of the representation of so 2n whose highest weight vector is twice that of the half spin representation

applications of stein s method for concentration inequalities stein s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and gibbs measures in this paper we provide some extensions of the theory and three applications 1 we obtain a concentration inequality for the magnetization in the curie weiss model at critical temperature where it obeys a nonstandard normalization and super gaussian concentration 2 we derive exact large deviation asymptotics for the number of triangles in the erd s renyi random graph g n p when p 0 31 similar results are derived also for general subgraph counts 3 we obtain some interesting concentration inequalities for the ising model on lattices that hold at all temperatures

double schubert polynomials for the classical groups abstract for each infinite series of the classical lie groups of type b c or d we construct a family of polynomials parametrized by the elements of the corresponding weyl group of infinite rank these polynomials represent the schubert classes in the equivariant cohomology of the appropriate flag variety they satisfy a stability property and are a natural extension of the single schubert polynomials of billey and haiman which represent non equivariant schubert classes they are also positive in a certain sense and when indexed by maximal grassmannian elements or by the longest element in a finite weyl group these polynomials can be expressed in terms of the factorial analogues of schur s q or p functions defined earlier by ivanov

logit models and logistic regressions for social networks i an introduction to markov graphs and p spanning nearly sixty years of research statistical network analysis has passed through at least two generations of researchers and models beginning in the late 1930 s the first generation of research dealt with the distribution of various network statistics under a variety of null models the second generation beginning in the 1970 s and continuing into the 1980 s concerned models usually for probabilities of relational ties among very small subsets of actors in which various simple substantive tendencies were parameterized much of this research most of which utilized log linear models first appeared in applied statistics publications

some new convergence acceleration methods abstract the algorithm is a general extrapolation method which includes most of thesequence transformations actually known some new convergence acceleration methods arederived from the f algorithm by applying the so called procedure the algorithms thusobtained are studied some theoretical results are proved and numerical examples are given 1 introduction the general idea for accelerating the convergence of a slowlyconvergent sequence is to transform it into another sequence which will undercertain assumptions converge faster to the same limit among the most famoussequence transformations are linear summation processes richardson polynomialextrapolation aitken s a2 process 1 and shanks transformation 15 wynn s algorithm 17 an extensive study of these methods can be found in 3 whilenumerical examples applications and fortran subroutines are given in 4 each of these methods usually works in some particular cases since a universalmethod accelerating the convergence of wide classes of sequences cannot exist 10 however numerical examples 11 16 and also some theoretical results 6 8 revealthat the best sequence transformations that is those working in most of theexamples seem to be levin s u and v transforms 14 and the algorithm 2 the algorithm is a method derived from the rule of the e algorithm by a procedurewhich will be recalled in the next section recently a general sequence transformation as well as the corresponding recursivealgorithm was obtained by havie 13 and brezinski 5 this transformation calledthe transformation includes most of the convergence acceleration methods actu ally known thus it seemed interesting to apply to the algorithm the sameprocedure which was used to derive the algorithm from the e algorithm and tostudy the new convergence acceleration methods obtained in that way it is the aimof this paper 2 the procedure 8 let us first begin by some reminiscences about shanks transformation and the e algorithm let sn be a sequence of real or complex numbers shanks transformationconsists in transforming sn into a set of sequences ek sn given by

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

characteristic classes and quadric bundles in this paper we construct stiefel whitney and euler classes in chow cohomology for algebraic vector bundles with nondegenerate quadratic form these classes are not in the algebra generated by the chern classes of such bundles and are new characteristic classes in algebraic geometry on complex varieties they correspond to classes with the same name pulled back from the cohomology of the classifying space bso n c the classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups t2 eg we begin by using the geometry of quadric bundles to study chern classes of maximal isotropic subbundles if v x is a vector bundle with quadratic form and if e and f are maximal isotropic subbundles of v then we prove theorem 1 that ci e and ci f are equal mod 2 moreover if the rank of v is 2n then cn e cn f proving a conjecture of fulton we define stiefel whitney and euler classes as chow cohomology classes which pull back to chern classes of maximal isotropic subbundles of the pullback bundle using the above theorem we show theorem 2 that these classes exist and are unique even though v need not have a maximal isotropic subbundle these constructions also make it possible to give schubert presentations

predictive likelihood inference with applications summary conditional and marginal predictive likelihood are defined and used in predicting un observed or missing data a distinction is made between predictive and estimative goals of a data analysis such methodology and distinctions are illustrated in the context of outlier theory and mixed model ano va the prediction of unobserved or missing data is considered from a likelihood perspective a general predictive likelihood is proposed for such purposes and contrasted with parametric likelihood the resulting maximum likelihood predictor is also compared with the limiting imputation from the em algorithm the new methodology is illustrated and contrasted with estimative likelihood analysis in the context of outlier theory and mixed model ano va predictions are proposed that are based on conditional and marginal predictive likelihoods for the unobserved data which are predictive adaptations of the methods developed by kalbfleisch and sprott 1970 1973 these authors were concerned with the elimination of nuisance parameters in order to make inference about another set of structural parameters prediction in the frame work of parametric distributions involves the elimination of all parameters in such a way that inference about the unobserved data is possible therefore in the predictive setting all para meters are nuisance parameters and the unobserved data now play a role similar to that of the structural parameters various predictive approaches to statistical inference have been advocated principally by geisser 1971 1975 stone 1974 aitchison and dunsmore 1975 and guttman 1970 a survey is provided by geisser 1980 and a brief overview in a parametric setting is presented in the next section following this a conditional likelihood approach is developed and shown to be closely related to the lauritzen 1974 hinkley 1979 likelihood method but specified in a manner allowing for a wider range of applicability a marginal predictive likelihood procedure is also specified which when applicable essentially agrees with the conditional likelihood procedure for the examples considered for a general account of likelihood based inference see fraser

consistency under sampling of exponential random graph models the growing availability of network data and of scientific interest in distributed systems has led to the rapid development of statistical models of network structure typically however these are models for the entire network while the data consists only of a sampled sub network parameters for the whole network which is what is of interest are estimated by applying the model to the sub network this assumes that the model is consistent under sampling or in terms of the theory of stochastic processes that it defines a projective family focusing on the popular class of exponential random graph models ergms we show that this apparently trivial condition is in fact violated by many popular and scientifically appealing models and that satisfying it drastically limits ergm s expressive power these results are actually special cases of more general results about exponential families of dependent random variables which we also prove using such results we offer easily checked conditions for the consistency of maximum likelihood estimation in ergms and discuss some possible constructive responses

solution of the two star model of a network the p star model or exponential random graph is among the oldest and best known of network models here we give an analytic solution for the particular case of the two star model which is one of the most fundamental of exponential random graphs we derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry broken phase separated from the normal phase of the model by a conventional continuous phase transition

k theoretic chern class formulas for vexillary degeneracy loci abstract using raising operators and geometric arguments we establish formulas for the k theory classes of degeneracy loci in classical types we also find new determinantal and pfaffian expressions for classical cases considered by giambelli the loci where a generic matrix drops rank and where a generic symmetric or skew symmetric matrix drops rank in an appendix we construct a k theoretic euler class for even rank vector bundles with quadratic form refining the chow theoretic class introduced by edidin and graham we also establish a relation between top chern classes of maximal isotropic subbundles which is used in proving the type d degeneracy locus formulas

steenrod operations on the chow ring of a classifying space we use the steenrod algebra to study the chow ring ch bg of the classifying space of an algebraic group g we describe a localization property which relates a given g to its elementary abelian subgroups and we study a number of particular cases namely symmetric groups and chevalley groups it turns out that the chow rings of these groups are completely determined by the abelian subgroups and their fusion

critique of q entropy for thermal statistics during the past 12 years there have been numerous papers on a relation between entropy and probability which is nonadditive and has a parameter q that depends on the nature of the thermodynamic system under consideration for q 1 this relation corresponds to the boltzmann gibbs entropy but for other values of q it is claimed that it leads to a formalism that is consistent with the laws of thermodynamics however it is shown here that the joint entropy for systems having different values of q is not defined in this formalism and consequently fundamental thermodynamic concepts such as temperature and heat exchange cannot be considered for such systems moreover for q ensuremath ne 1 the probability distribution for weakly interacting systems does not factor into the product of the probability distribution for the separate systems leading to spurious correlations and other unphysical consequences e g nonextensive energy that have been ignored in various applications given in the literature

on the acceleration of two classes of sequences by the q d algorithm the algorithm is a general extrapolation procedure for accelerating the convergence of sequences acceleration theorems for two classes of sequences of real numbers are proved theoretical results are illustrated by numerical examples

solution for the properties of a clustered network we study strauss s model of a network with clustering and present an analytic mean field solution which is exact in the limit of large network size previous computer simulations have revealed a degenerate region in the model s parameter space in which triangles of adjacent edges clump together to form unrealistically dense subgraphs and perturbation calculations have been found to break down in this region at all orders our solution shows that this region corresponds to a classic symmetry broken phase and that the onset of the degeneracy corresponds to a first order phase transition in the density of the network

sparse maximum entropy random graphs with a given power law degree distribution even though power law or close to power law degree distributions are ubiquitously observed in a great variety of large real networks the mathematically satisfactory treatment of random power law graphs satisfying basic statistical requirements of realism is still lacking these requirements are sparsity exchangeability projectivity and unbiasedness the last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints here we prove that the hypersoft configuration model belonging to the class of random graphs with latent hyperparameters also known as inhomogeneous random graphs or w random graphs is an ensemble of random power law graphs that are sparse unbiased and either exchangeable or projective the proof of their unbiasedness relies on generalized graphons and on mapping the problem of maximization of the normalized gibbs entropy of a random graph ensemble to the graphon entropy maximization problem showing that the two entropies converge to each other in the large graph limit

zero dimensional donaldson thomas invariants of calabi yau 4 folds abstract we study hilbert schemes of points on a smooth projective calabi yau 4 fold x we define dt 4 invariants by integrating the euler class of a tautological vector bundle l n against the virtual class we conjecture a formula for their generating series which we prove in certain cases when l corresponds to a smooth divisor on x a parallel equivariant conjecture for toric calabi yau 4 folds is proposed this conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples combining the equivariant conjecture with a vertex calculation we find explicit positive rational weights which can be assigned to solid partitions the weighted generating function of solid partitions is given by exp m q 1 where m q denotes the macmahon function

local dependence in random graph models characterization properties and statistical inference dependent phenomena such as relational spatial and temporal phenomena tend to be characterized by local dependence in the sense that units which are close in a well defined sense are dependent in contrast with spatial and temporal phenomena though relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent owing to the challenge of characterizing local dependence and constructing random graph models with local dependence many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference we take first steps to characterize local dependence in random graph models inspired by the notion of finite neighbourhoods in spatial statistics and m dependence in time series and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference we show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy but conventional exponential family random graph models do not satisfy in addition we establish a central limit theorem for random graph models with local dependence which suggests that random graph models with local dependence are amenable to statistical inference we discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure in the absence of observed neighbourhood structure we take a bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures we present simulation results and applications to two real world networks with ground truth