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

embedded tori with prescribed mean curvature abstract we construct a sequence of compact oriented embedded two dimensional surfaces of genus one into euclidean 3 space with prescribed almost constant mean curvature of the form h x 1 a x for x large when a 0 and 0 2 such surfaces are close to sections of unduloids with small neck size folded along circumferences centered at the origin and with larger and larger radii the construction involves a deep study of the corresponding jacobi operators an application of the lyapunov schmidt reduction method and some variational argument

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

estimating normal means with a dirichlet process prior abstract in this article the dirichlet process prior is used to provide a nonparametric bayesian estimate of a vector of normal means in the past there have been computational difficulties with this model this article solves the computational difficulties by developing a gibbs sampler algorithm the estimator developed in this article is then compared to parametric empirical bayes estimators peb and nonparametric empirical bayes estimators npeb in a monte carlo study the monte carlo study demonstrates that in some conditions the peb is better than the npeb and in other conditions the npeb is better than the peb the monte carlo study also shows that the estimator developed in this article produces estimates that are about as good as the peb when the peb is better and produces estimates that are as good as the npeb estimator when that method is better

gibbs sampling methods for stick breaking priors a rich and flexible class of random probability measures which we call stick breaking priors can be constructed using a sequence of independent beta random variables examples of random measures that have this characterization include the dirichlet process its two parameter extension the two parameter poisson dirichlet process finite dimensional dirichlet priors and beta two parameter processes the rich nature of stick breaking priors offers bayesians a useful class of priors for nonparametric problems while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them in this article we present two general types of gibbs samplers that can be used to fit posteriors of bayesian hierarchical models based on stick breaking priors the first type of gibbs sampler referred to as a polya urn gibbs sampler is a generalized version of a widely used gibbs sampling method currently employed for dirichlet process computing this method applies t

weak limit and blowup of approximate solutions to h systems abstract let h r 3 r be a continuous function such that h p h 0 r as p fixing a domain in r 2 we study the behaviour of a sequence u n of approximate solutions to the h system u 2 h u u x u y in assuming that sup p r 3 h p h 0 p 1 we show that the weak limit of the sequence u n solves the h system and u n u strongly in h 1 apart from a countable set s made by isolated points moreover if in addition h p h 0 o 1 p as p then in correspondence of each point of s we prove that the sequence u n blows either an h bubble or an h 0 sphere

bubbles with prescribed mean curvature the variational approach abstract let h r 3 r be a c 1 mapping such that h p h 0 as p we show that when h satisfies some global conditions then there exists an h bubble namely a sphere s in r 3 such that the mean curvature of s at any regular point p s equals h p

point process modelling for directed interaction networks abstract network data often take the form of repeated interactions between senders and receivers tabulated over time a primary question to ask of such data is which traits and behaviors are predictive of interaction to answer this question a model is introduced for treating directed interactions as a multivariate point process a cox multiplicative intensity model using covariates that depend on the history of the process consistency and asymptotic normality are proved for the resulting partial likelihood based estimators under suitable regularity conditions and an efficient fitting procedure is described multicast interactions those involving a single sender but multiple receivers are treated explicitly the resulting inferential framework is then employed to model message sending behavior in a corporate e mail network the analysis gives a precise quantification of which static shared traits and dynamic network effects are predictive of message recipient selection

blow up analysis for the prescribed mean curvature equation on r2 abstract we consider the h system u 2 h u u x u y on r 2 where h c 0 r 3 r satisfies h q h o 1 q as q and sup q r 3 h q h q 1 for some h r 0 we show that a sequence of approximate solutions of the h system on r 2 admits a limit configuration made by h bubbles namely nonconstant solutions of h systems on r 2 and h can be the mapping h or the constant h

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

a bayesian semiparametric multivariate joint model for multiple longitudinal outcomes and a time to event motivated by a real data example on renal graft failure we propose a new semiparametric multivariate joint model that relates multiple longitudinal outcomes to a time to event to allow for greater flexibility key components of the model are modelled nonparametrically in particular for the subject specific longitudinal evolutions we use a spline based approach the baseline risk function is assumed piecewise constant and the distribution of the latent terms is modelled using a dirichlet process prior formulation additionally we discuss the choice of a suitable parameterization from a practitioner s point of view to relate the longitudinal process to the survival outcome specifically we present three main families of parameterizations discuss their features and present tools to choose between them

positive solutions of a nonlinear three point boundary value problem in this paper by using krasnoselskii s fixed point theorem in a cone we study the existence of single and multiple positive solutions to the three point boundary value problem bvp y t a t f y t 0 0

multiple solutions of h systems and rellich s conjecture soit x y r 2 x 2 y 2 1 on considere une fonction u r 3 satisfaisant le systeme h u 2hu x u y sur avec l une des conditions suivantes soit dirichlet u sur soit plateau u x 2 u y 2 u x u y 0 sur u et u non decroissante sur

nonparametric bayes estimators based on beta processes in models for life history data the article studies the problem of finding bayes estimators for cumulative hazard rates and related quantities w r t prior distributions that correspond to cumulative hazard rate processes with nonnegative independent increments a particular class of prior processes termed beta processes is introduced and is shown to constitute a conjugate class an important bonus of the approach based on cumulative hazards is that more complicated models for life history data than the simple life table situation can be treated for example time inhomogeneous markov chains we find posterior distributions and derive bayes estimators in such models and present a semiparametric bayesian analysis of the cox regression model

twin solutions of boundary value problems for ordinary differential equations and finite difference equations a new twin fixed point theorem is applied first to obtain the existence of at least two positive solutions for the right focal boundary value problem y f ity 0 0 t 1 y 0 y 1 0 it is applied later to obtain the existence of at least two positive solutions for the analogous discrete right focal boundary value problem 2u k g u k 0 k 0 n u 0 u n 1 0

semiparametric bayesian inference for stochastic frontier models in this paper we propose a semiparametric bayesian framework for the analysis of stochastic frontiers and efficiency measurement the distribution of inefficiencies is modelled nonparametrically through a dirichlet process prior we suggest prior distributions and implement a bayesian analysis through an efficient markov chain monte carlo sampler which allows us to deal with practically relevant sample sizes we also consider the case where the efficiency distribution varies with firm characteristics the methodology is applied to a cost frontier estimated from a panel data set on 382 u s hospitals

the dirichlet problem for h systems with small boundary data blowup phenomena and nonexistence results given h 3 of class c1 and bounded we consider a sequence un of solutions of the h system open image in new window in the unit open disc open image in new window satisfying the boundary condition un n on open image in new window in the first part of this paper assuming that un is bounded in h1 open image in new window 3 we study the behavior of un when the boundary data n shrink to zero we show that either un 0 strongly in h1 open image in new window 3 or un blows up at least one h bubble namely a nonconstant conformal solution of the h system on 2 under additional assumptions on h we can obtain more precise information on the blow up in the second part of this paper we investigate the multiplicity of solutions for the dirichlet problem on the disc with small boundary datum we detect a family of nonconstant functions h even close to a nonzero constant in any reasonable topology for which the dirichlet problem cannot admit a large solution at a mountain pass level when the boundary datum is small

a semiparametric bayesian model for randomised block designs summary a model is proposed for a bayesian semiparametric analysis of randomised block experiments the model is a hierarchical model in which a dirichlet process is inserted at the middle stage for the distribution of the block effects this model allows an arbitrary distribution of block effects and it results in effective estimates of treatment contrasts block effects and the distribution of block effects an effective computational strategy is presented for describing the posterior distribution

size biased sampling of poisson point processes and excursions some general formulae are obtained for size biased sampling from a poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function these formulae explain why in certain cases gamma and stable the size biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking residual allocation scheme defined by independent beta random variables an application is made to length biased sampling of excursions of a markov process away from a recurrent point of its statespace with emphasis on the brownian and bessel cases when the associated inverse local time is a stable subordinator results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by brownian motion

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

h bubbles in a perturbative setting the finite dimensional reduction s method given a regular function h colon mathbb r 3 to mathbb r we look for h bubbles that is regular surfaces in mathbb r 3 parametrized on the sphere mathbb s 2 with mean curvature h at every point here we study the case of h u h 0 varepsilon h 1 u h varepsilon u where h 0 is a nonzero constant varepsilon is the smallness parameter and h 1 is any c 2 function we prove that if bar p in mathbb r 3 is a good stationary point for the melnikov type function gamma p int q p h 0 1 h 1 q dq then for varepsilon small there exists an h varepsilon bubble omega varepsilon that converges to a sphere of radius h 0 1 centered at bar p as varepsilon to 0

bayesian nonparametric estimation for incomplete data via successive substitution sampling in the problem of estimating an unknown distribution function f in the presence of censoring one can use a nonparametric estimator such as the kaplan meier estimator or one can consider parametric modeling there are many situations where physical reasons indicate that a certain parametric model holds approximately in these cases a nonparametric estimator may be very inefficient relative to a parametric estimator on the other hand if the parametric model is only a crude approximation to the actual model then the parametric estimator may perform poorly relative to the nonparametric estimator and may even be inconsistent the bayesian paradigm provides a reasonable framework for this problem in a bayesian approach one would try to put a prior distribution on f that gives most of its mass to small neighborhoods of the entire parametric family we show that certain priors based on the dirichlet process prior can be used to accomplish this for these priors the posterior distribution of f given the censored data appears to be analytically intractable we provide a method for approximating this posterior distribution through the use of a successive substitution sampling algorithm we also show convergence of the algorithm

sparse exchangeable graphs and their limits via graphon processes in a recent paper caron and fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in r here we show that by generalizing the classical definition of graphons as functions over probability spaces to functions over finite measure spaces we can model a large family of exchangeable graphs including the caron fox graphs and the traditional exchangeable dense graphs as special cases explicitly modelling the underlying space of features by a finite measure space s s and the connection probabilities by an integrable function w s s 0 1 we construct a random family gt t 0 of growing graphs such that the vertices of gt are given by a poisson point process on s with intensity t with two points x y of the point process connected with probability w x y we call such a random family a graphon process we prove that a graphon process has convergent subgraph frequencies with possibly infinite limits and that in the natural extension of the cut metric to our setting the sequence converges to the generating graphon we also show that the underlying graphon is identifiable only as an equivalence class over graphons with cut distance zero more generally we study metric convergence for arbitrary not necessarily random sequences of graphs and show that a sequence of graphs has a convergent subsequence if and only if it has a subsequence satisfying a property we call uniform regularity of tails finally we prove that every graphon is equivalent to a graphon on r equipped with lebesgue measure

positive solutions of singular three point boundary value problems for the one dimensional p laplacian abstract we use the fixed point index theory to establish the existence of at least one or two positive solutions for the singular three point boundary value problems f p y a t f y t 0 0

markov chain monte carlo in approximate dirichlet and beta two parameter process hierarchical models summary we present some easy to construct random probability measures which approximate the dirichlet process and an extension which we will call the beta two parameter process the nature of these constructions makes it simple to implement markov chain monte carlo algorithms for fitting nonparametric hierarchical models and mixtures of nonparametric hierarchical models for the dirichlet process we consider a truncation approximation as well as a weak limit approximation based on a mixture of dirichlet processes the same type of truncation approximation can also be applied to the beta two parameter process both methods lead to posteriors which can be fitted using markov chain monte carlo algorithms that take advantage of blocked coordinate updates these algorithms promote rapid mixing of the markov chain and can be readily applied to normal mean mixture models and to density estimation problems we prefer the truncation approximations since a simple device for monitoring the adequacy of the approximation can be easily computed from the output of the gibbs sampler furthermore for the dirichlet process the truncation approximation offers an exponentially higher degree of accuracy over the weak limit approximation for the same computational effort we also find that a certain beta two parameter process may be suitable for finite mixture modelling because the distinct number of sampled values from this process tends to match closely the number of components of the underlying mixture distribution

bayesian density regression summary the paper considers bayesian methods for density regression allowing a random probability distribution to change flexibly with multiple predictors the conditional response distribution is expressed as a non parametric mixture of regression models with the mixture distribution changing with predictors a class of weighted mixture of dirichlet process priors is proposed for the uncountable collection of mixture distributions it is shown that this specification results in a generalized polya urn scheme which incorporates weights that are dependent on the distance between subjects predictor values to allow local dependence in the mixture distributions we propose a kernel based weighting scheme a gibbs sampling algorithm is developed for posterior computation the methods are illustrated by using simulated data examples and an epidemiologic application

network cross validation by edge sampling summary while many statistical models and methods are now available for network analysis resampling of network data remains a challenging problem cross validation is a useful general tool for model selection and parameter tuning but it is not directly applicable to networks since splitting network nodes into groups requires deleting edges and destroys some of the network structure in this paper we propose a new network resampling strategy based on splitting node pairs rather than nodes that is applicable to cross validation for a wide range of network model selection tasks we provide theoretical justification for our method in a general setting and examples of how the method can be used in specific network model selection and parameter tuning tasks numerical results on simulated networks and on a statisticians citation network show that the proposed cross validation approach works well for model selection

three positive fixed points of nonlinear operators on ordered banach spaces we generalize the fixed point theorem of leggett williams which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a banach space we then show how to apply our theorem to prove the existence of three positive solutions to a second order discrete boundary value problem 2001 elsevier science ltd all rights reserved

a multiplicity result for the yamabe problem on sn abstract we prove a multiplicity result for the yamabe problem on the manifold s g where g is a perturbation of the standard metric g 0 of s n solutions are found by variational methods via an abstract perturbation result

the two parameter poisson dirichlet distribution derived from a stable subordinator the two parameter poisson dirichlet distribution denoted mathsf pd alpha theta is a probability distribution on the set of decreasing positive sequences with sum 1 the usual poisson dirichlet distribution with a single parameter theta introduced by kingman is mathsf pd 0 theta known properties of mathsf pd 0 theta including the markov chain description due to vershik shmidt and ignatov are generalized to the two parameter case the size biased random permutation of mathsf pd alpha theta is a simple residual allocation model proposed by engen in the context of species diversity and rediscovered by perman and the authors in the study of excursions of brownian motion and bessel processes for 0 alpha 1 mathsf pd alpha 0 is the asymptotic distribution of ranked lengths of excursions of a markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index alpha formulae in this case trace back to work of darling lamperti and wendel in the 1950s and 1960s the distribution of ranked lengths of excursions of a one dimensional brownian motion is mathsf pd 1 2 0 and the corresponding distribution for a brownian bredge is mathsf pd 1 2 1 2 the mathsf pd alpha 0 and mathsf pd alpha alpha distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable markov process whose zero set is the range of a stable subordinator of index alpha

variational perturbative methods and bifurcation of bound states from the essential spectrum this paper consists of two main parts the first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results the second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum

bayesian density estimation and inference using mixtures abstract we describe and illustrate bayesian inference in models for density estimation using mixtures of dirichlet processes these models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions efficient simulation methods are used to approximate various prior posterior and predictive distributions this allows for direct inference on a variety of practical issues including problems of local versus global smoothing uncertainty about density estimates assessment of modality and the inference on the numbers of components also convergence results are established for a general class of normal mixture models

multiple solutions to the plateau problem for nonconstant mean curvature we prove the existence of at least two solutions to the plateau problem for any curvature function h close enough in l norm to a nonzero constant h 0 such that h 0 1 where denotes the l norm of the boundary condition

positive solutions for multipoint boundary value problems with one dimensional p laplacian operator in this paper we study the existence of positive solutions for the following nonlinear m point boundary value problem with p laplacian p u a t f u t 0 0 t 1 u 0 i 1m 2aiu i u 1 i 1kbiu i i k 1sbiu i i s 1m 2biu i where p s is p laplacian operator i e p s s p 2s p 1 q p 1 1p 1q 1 1 k s m 2 ai bi 0 with 0 i 1kbi i k 1sbi 1 0 i 1m 2ai 1 0 1 2 m 2 1 a t c 0 1 0 f c 0 0 we show that there exists one or two positive solutions by using fixed point theorem for operator on a cone the conclusions in this paper essentially extend and improve the known results

oracle inequalities for network models and sparse graphon estimation inhomogeneous random graph models encompass many network models such as stochastic block models and latent position models we consider the problem of statistical estimation of the matrix of connection probabilities based on the observations of the adjacency matrix of the network taking the stochastic block model as an approximation we construct estimators of network connection probabilities the ordinary block constant least squares estimator and its restricted version we show that they satisfy oracle inequalities with respect to the block constant oracle as a consequence we derive optimal rates of estimation of the probability matrix our results cover the important setting of sparse networks another consequence consists in establishing upper bounds on the minimax risks for graphon estimation in the l 2 norm when the probability matrix is sampled according to a graphon model these bounds include an additional term accounting for the agnostic error induced by the variability of the latent unobserved variables of the graphon model in this setting the optimal rates are influenced not only by the bias and variance components as in usual nonparametric problems but also include the third component which is the agnostic error the results shed light on the differences between estimation under the empirical loss the probability matrix estimation and under the integrated loss the graphon estimation

multiple positive solutions of four point nonlinear boundary value problems for a higher order p laplacian operator with all derivatives abstract in this paper we study the existence of multiple positive solutions for nonlinear four point boundary value problems for a higher order p laplacian operator with all derivatives some new results are obtained for the multiplicity of positive solutions by applying the avery henderson fixed point theorem and the leggett williams fixed point theorem in a cone which is different from the previous approaches

exchangeable and partially exchangeable random partitions call a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn1 nk the probability that the partition breaks the firstn1 nk integers intok particular classes of sizesn1 nk in order of their first elements has the same valuep n1 nk for every possible choice of classes subject to the sizes constraint a random partition is exchangeable iff it is partially exchangeable for a symmetric functionp n1 nk a representation is given for partially exchangeable random partitions which provides a useful variation of kingman s representation in the exchangeable case results are illustrated by the two parameter generalization of ewens partition structure

nonparametric hierarchical bayes via sequential imputations we consider the empirical bayes estimation of a distribution using binary data via the dirichlet process let d denote a dirichlet process with being a finite measure on 0 1 instead of having direct samples from an unknown random distribution f from d we assume that only indirect binomial data are observable this paper presents a new interpretation of lo s formula and thereby relates the predictive density of the observations based on a dirichlet process model to likelihoods of much simpler models as a consequence the log likelihood surface as well as the maximum likelihood estimate of c 0 1 is found when the shape of is assumed known together with a formula for the fisher information evaluated at the estimate the sequential imputation method of kong liu and wong is recommended for overcoming computational difficulties commonly encountered in this area the related approximation formulas are provided an analysis of the tack data of beckett and diaconis which motivated this study is supplemented to illustrate our methods