hypothesis testing for automated community detection in networks summary community detection in networks is a key exploratory tool with applications in a diverse set of areas ranging from finding communities in social and biological networks to identifying link farms in the world wide web the problem of finding communities or clusters in a network has received much attention from statistics physics and computer science however most clustering algorithms assume knowledge of the number of clusters k we propose to determine k automatically in a graph generated from a stochastic block model by using a hypothesis test of independent interest our main contribution is twofold first we theoretically establish the limiting distribution of the principal eigenvalue of the suitably centred and scaled adjacency matrix and use that distribution for our test of the hypothesis that a random graph is of erd s renyi noise type secondly we use this test to design a recursive bipartitioning algorithm which naturally uncovers nested community structure using simulations and quantifiable classification tasks on real world networks with ground truth we show that our algorithm outperforms state of the art methods

partially observed semi markov zero sum games with average payoff we study a zero sum partially observed semi markov game with average payoff on a countable state space under certain ergodicity conditions we show that a saddle point equilibrium exists we achieve this by solving the corresponding average cost optimality equation using a span contraction method the average value is shown to be the unique zero of a lipschitz continuous function a value iteration scheme is developed to compute the value

estimation of monotone treatment effects in network experiments abstractrandomized experiments on social networks pose statistical challenges due to the possibility of interference between units we propose new methods for finding confidence intervals on the attributable treatment effect in such settings the methods do not require partial interference but instead require an identifying assumption that is similar to requiring nonnegative treatment effects network or spatial information can be used to customize the test statistic in principle this can increase power without making assumptions on the data generating process supplementary materials for this article are available online

structural and functional discovery in dynamic networks with non negative matrix factorization time series of graphs are increasingly prevalent in modern data and pose unique challenges to visual exploration and pattern extraction this paper describes the development and application of matrix factorizations for exploration and time varying community detection in time evolving graph sequences the matrix factorization model allows the user to home in on and display interesting underlying structure and its evolution over time the methods are scalable to weighted networks with a large number of time points or nodes and can accommodate sudden changes to graph topology our techniques are demonstrated with several dynamic graph series from both synthetic and real world data including citation and trade networks these examples illustrate how users can steer the techniques and combine them with existing methods to discover and display meaningful patterns in sizable graphs over many time points

network cross validation for determining the number of communities in network data abstractthe stochastic block model sbm and its variants have been a popular tool for analyzing large network data with community structures in this article we develop an efficient network cross validation ncv approach to determine the number of communities as well as to choose between the regular stochastic block model and the degree corrected block model dcbm the proposed ncv method is based on a block wise node pair splitting technique combined with an integrated step of community recovery using sub blocks of the adjacency matrix we prove that the probability of under selection vanishes as the number of nodes increases under mild conditions satisfied by a wide range of popular community recovery algorithms the solid performance of our method is also demonstrated in extensive simulations and two data examples supplementary materials for this article are available online

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

community extraction for social networks analysis of networks and in particular discovering communities within networks has been a focus of recent work in several fields and has diverse applications most community detection methods focus on partitioning the entire network into communities with the expectation of many ties within communities and few ties between however many networks contain nodes that do not fit in with any of the communities and forcing every node into a community can distort results here we propose a new framework that extracts one community at a time allowing for arbitrary structure in the remainder of the network which can include weakly connected nodes the main idea is that the strength of a community should depend on ties between its members and ties to the outside world but not on ties between nonmembers the proposed extraction criterion has a natural probabilistic interpretation in a wide class of models and performs well on simulated and real networks for the case of the block model we establish asymptotic consistency of estimated node labels and propose a hypothesis test for determining the number of communities

asymptotic normality of maximum likelihood and its variational approximation for stochastic blockmodels variational methods for parameter estimation are an active research area potentially offering computationally tractable heuristics with theoretical performance bounds we build on recent work that applies such methods to network data and establish asymptotic normality rates for parameter estimates of stochastic blockmodel data by either maximum likelihood or variational estimation the result also applies to various sub models of the stochastic blockmodel found in the literature

statistical clustering of temporal networks through a dynamic stochastic block model statistical node clustering in discrete time dynamic networks is an emerging field that raises many challenges here we explore statistical properties and frequentist inference in a model that combines a stochastic block model sbm for its static part with independent markov chains for the evolution of the nodes groups through time we model binary data as well as weighted dynamic random graphs with discrete or continuous edges values our approach motivated by the importance of controlling for label switching issues across the different time steps focuses on detecting groups characterized by a stable within group connectivity behavior we study identifiability of the model parameters propose an inference procedure based on a variational expectation maximization algorithm as well as a model selection criterion to select for the number of groups we carefully discuss our initialization strategy which plays an important role in the method and compare our procedure with existing ones on synthetic datasets we also illustrate our approach on dynamic contact networks one of encounters among high school students and two others on animal interactions an implementation of the method is available as a r package called dynsbm

latent space approaches to social network analysis network models are widely used to represent relational information among interacting units in studies of social networks recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors we develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved social space we make inference for the social space within maximum likelihood and bayesian frameworks and propose markov chain monte carlo procedures for making inference on latent positions and the effects of observed covariates we present analyses of three standard datasets from the social networks literature and compare the method to an alternative stochastic blockmodeling approach in addition to improving on model fit for these datasets our method provides a visual and interpretable model based spatial representation of social relationships and improv

hypergraph based anomaly detection of high dimensional co occurrences this paper addresses the problem of detecting anomalous multivariate co occurrences using a limited number of unlabeled training observations a novel method based on using a hypergraph representation of the data is proposed to deal with this very high dimensional problem hypergraphs constitute an important extension of graphs which allow edges to connect more than two vertices simultaneously a variational expectation maximization algorithm for detecting anomalies directly on the hypergraph domain without any feature selection or dimensionality reduction is presented the resulting estimate can be used to calculate a measure of anomalousness based on the false discovery rate the algorithm has o np computational complexity where n is the number of training observations and p is the number of potential participants in each co occurrence event this efficiency makes the method ideally suited for very high dimensional settings and requires no tuning bandwidth or regularization parameters the proposed approach is validated on both high dimensional synthetic data and the enron email database where p 75 000 and it is shown that it can outperform other state of the art methods

modeling non stationary processes through dimension expansion in this article we propose a novel approach to modeling nonstationary spatial fields the proposed method works by expanding the geographic plane over which these processes evolve into higher dimensional spaces transforming and clarifying complex patterns in the physical plane by combining aspects of multidimensional scaling group lasso and latent variable models a dimensionally sparse projection is found in which the originally nonstationary field exhibits stationarity following a comparison with existing methods in a simulated environment dimension expansion is studied on a classic test bed dataset historically used to study nonstationary models following this we explore the use of dimension expansion in modeling air pollution in the united kingdom a process known to be strongly influenced by rural urban effects amongst others which gives rise to a nonstationary field

risk sensitive control of markov processes in countable state space in this paper we consider infinite horizon risk sensitive control of markov processes with discrete time and denumerable state space this problem is solved by proving under suitable conditions that there exists a bounded solution to the dynamic programming equation the dynamic programming equation is transformed into an isaacs equation for a stochastic game and the vanishing discount method is used to study its solution in addition we prove that the existence conditions are also necessary

stochastic blockmodels for directed graphs abstract holland and leinhardt 1981 proposed the p 1 model for the analysis of binary directed graph data in network studies such a model provides information about the attractiveness and expansiveness of the individual nodes in the network as well as the tendency of a pair of nodes to reciprocate relational ties when the nodes are a priori partitioned into subgroups based on attributes such as race and sex the density of ties from one subgroup to another can differ considerably from that relating another pair of subgroups thus creating a situation called blocking in social networks the p 1 model completely ignores this extra piece of information and is therefore unable to explain the block structure blockmodels that are simple extensions of the p 1 model are proposed specifically for such data an iterative scaling algorithm is presented for fitting the model parameters by maximum likelihood the methodology is illustrated in detail on two empirical examples

attribute fusion in a latent process model for time series of graphs hypothesis testing on time series of attributed graphs has applications in diverse areas e g social network analysis wherein vertices represent individual actors or organizations connectome inference wherein vertices are neurons or brain regions and text processing wherein vertices represent authors or documents we consider the problem of anomaly change point detection given the latent process model for time series of graphs with categorical attributes on the edges presented in n h lee and c e priebe a latent process model for time series of attributed random graphs statist inference stoch process vol 14 pp 231 253 2011 various attributed graph invariants are considered and their power for detection as a function of a linear fusion parameter is presented our main result is that inferential performance in mathematically tractable first order and second order approximation models does provide guidance for methodological choices applicable to the exact realistic but intractable model furthermore to the extent that the exact model is realistic we may tentatively conclude that approximation model investigations have some bearing on real data applications

on the existence of stationary optimal policies for partially observed mdps under the long run average cost criterion abstract this paper studies the problem of the existence of stationary optimal policies for finite state controlled markov chains with compact action space and imperfect observations under the long run average cost criterion it presents sufficient conditions for existence of solutions to the associated dynamic programming equation that strengthen past results there is a detailed discussion comparing the different assumptions commonly found in the literature

estimation and prediction for stochastic blockstructures a statistical approach to a posteriori blockmodeling for digraphs and valued digraphs is proposed the probability model assumes that the vertices of the digraph are partitioned into several unobserved latent classes and that the probability distribution of the relation between two vertices depends only on the classes to which they belong a bayesian estimator based on gibbs sampling is proposed the basic model is not identified because class labels are arbitrary the resulting identifiability problems are solved by restricting inference to the posterior distributions of invariant functions of the parameters and the vertex class membership in addition models are considered where class labels are identified by prior distributions for the class membership of some of the vertices the model is illustrated by an example from the social networks literature kapferer s tailor shop

phase transitions for detecting latent geometry in random graphs random graphs with latent geometric structure are popular models of social and biological networks with applications ranging from network user profiling to circuit design these graphs are also of purely theoretical interest within computer science probability and statistics a fundamental initial question regarding these models is when are these random graphs affected by their latent geometry and when are they indistinguishable from simpler models without latent structure such as the erd s renyi graph mathcal g n p we address this question for two of the most well studied models of random graphs with latent geometry the random intersection and random geometric graph our results are as follows a the random intersection graph is defined by sampling n random sets s 1 s 2 ldots s n by including each element of a set of size d in each s i independently with probability delta and including the edge i j if s i cap s j ne emptyset we prove that the random intersection graph converges in total variation to an erd s renyi graph if d tilde omega n 3 and does not if d o n 3 for both dense and sparse edge densities p this resolves an open problem in fill et al random struct algorithms 16 2 156 176 2000 rybarczyk random struct algorithms 38 1 2 205 234 2011 and kim et al random struct algorithms 52 4 662 679 2018 the same result was obtained simultaneously and independently by bubeck et al when random intersection graphs lose geometry manuscript 2019 b we strengthen the preceding argument to show that the matrix of random intersection sizes s i cap s j converges in total variation to a symmetric matrix with independent poisson entries this yields the first total variation convergence result for tau random intersection graphs where the edge i j is included if s i cap s j ge tau more precisely our results imply that if p is bounded away from 1 then the tau random intersection graph with edge density p converges to mathcal g n p if d omega tau 3 n 3 c the random geometric graph on mathbb s d 1 is defined by sampling x 1 x 2 dots x n uniformly at random from mathbb s d 1 and including the edge i j if vert x i x j vert 2 le tau a result of bubeck et al random struct algorithms 49 503 532 2016 showed that this model converges to mathcal g n p in total variation where p is chosen so that the models have matching edge densities as long as d omega n 3 it was conjectured in bubeck et al 2016 that this threshold decreases drastically for p small we make the first progress towards this conjecture by showing convergence if d tilde omega left min pn 3 p 2 n 7 2 right our proofs are a hybrid of combinatorial arguments direct couplings and applications of information inequalities previous upper bounds on the total variation distance between random graphs with latent geometry and mathcal g n p have typically not been both combinatorial and information theoretic while this interplay is essential to the sharpness of our bounds

a spatial model for social networks abstract we study spatial embeddings of random graphs in which nodes are randomly distributed in geographical space we let the edge probability between any two nodes to be dependent on the spatial distance between them and demonstrate that this model captures many generic properties of social networks including the small world properties skewed degree distribution and most distinctively the existence of community structures

real representations of the finite orthogonal and symplectic groups of odd characteristic let q be a power of an odd prime p let sp 2n q denote the symplectic group of degree 2n over gf q let o n q denote the split orthogonal group of degree n over gf q corresponding to a symmetric form of maximal witt index and 0 n q denote the non split orthogonal group unless we need to be more specific we will simply write o n q for either of these two orthogonal groups in addition we will often abbreviate these symbols further to o n and sp 2n whenever q is understood to be fixed it is known from conjugacy class considerations that all complex irreducible characters of the finite orthogonal groups o n are real valued the same is true of the symplectic groups sp 2n q provided that q z 1 mod 4 when q 3 mod 4 not all characters of sp 2n q are realvalued however as the total number of irreducible characters is a manic polynomial in q of degree n and this is true of the number of real valued irreducible characters we can assert that the majority of characters is realvalued in this case it is of interest to know whether or not the real valued characters of these families of groups are the characters of representations that are defined over the real numbers this paper gives a solution to this problem by proving the following theorem

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

an interproduct competition model incorporating branding hierarchy and product similarities using store level data we develop and implement a bayesian semiparametric model of demand under interproduct competition that enables us to assess the respective contributions of brand sku stock keeping unit hierarchy and interproduct similarity to explaining and predicting demand to incorporate brand sku hierarchy effects we use bayesian hierarchical clustering inherent in a nested dirichlet process to simultaneously partition brands and skus conditional on brands into groups of similarity clusters we examine cluster memberships and postprocess the markov chain monte carlo output to infer cluster properties by accounting for parameter uncertainty our proposed approach lends to a spatial competition interpretation in latent attribute space and helps uncover the extent to which competition across skus in the latent attribute space is local or global in a related vein we discuss the implications of well defined groups of similar skus as subcategory or submarket boundaries in latent attribute space we empirically test our model using aggregate beer category sales data from a midsize u s retail chain we find that branding hierarchy effects dominate those from product similarity we find that the model partitions the 15 brands in the data into 4 brand clusters and the 96 skus into 25 sku clusters conditional on brand cluster membership in estimating a set of models of spatial interproduct competition we find that sku competition is more local than global in that only subsets of products compete within groups of comparable products finally we discuss the substantive implications of our results this paper was accepted by pradeep chintagunta marketing

optimal graphon estimation in cut distance consider the twin problems of estimating the connection probability matrix of an inhomogeneous random graph and the graphon of a w random graph we establish the minimax estimation rates with respect to the cut metric for classes of block constant matrices and step function graphons surprisingly our results imply that from the minimax point of view the raw data that is the adjacency matrix of the observed graph is already optimal and more involved procedures cannot improve the convergence rates for this metric this phenomenon contrasts with optimal rates of convergence with respect to other classical distances for graphons such as the l 1 or l 2 metrics

covariate assisted spectral clustering biological and social systems consist of myriad interacting units the interactions can be represented in the form of a graph or network measurements of these graphs can reveal the underlying structure of these interactions which provides insight into the systems that generated the graphs moreover in applications such as connectomics social networks and genomics graph data are accompanied by contextualizing measures on each node we utilize these node covariates to help uncover latent communities in a graph using a modification of spectral clustering statistical guarantees are provided under a joint mixture model that we call the node contextualized stochastic blockmodel including a bound on the misclustering rate the bound is used to derive conditions for achieving perfect clustering for most simulated cases covariate assisted spectral clustering yields results superior both to regularized spectral clustering without node covariates and to an adaptation of canonical correlation analysis we apply our clustering method to large brain graphs derived from diffusion mri data using the node locations or neurological region membership as covariates in both cases covariate assisted spectral clustering yields clusters that are easier to interpret neurologically

co clustering of nonsmooth graphons performance bounds are given for exploratory co clustering blockmodeling of bipartite graph data where we assume the rows and columns of the data matrix are samples from an arbitrary population this is equivalent to assuming that the data is generated from a nonsmooth graphon it is shown that co clusters found by any method can be extended to the row and column populations or equivalently that the estimated blockmodel approximates a blocked version of the generative graphon with estimation error bounded by o p n 1 2 analogous performance bounds are also given for degree corrected blockmodels and random dot product graphs with error rates depending on the dimensionality of the latent variable space

average cost dynamic programming equations for controlled markov chains with partial observations the value function for the average cost control of a class of partially observed markov chains is derived as the vanishing discount limit in a suitable sense of the value functions for the corresponding discounted cost problems the limiting procedure is justified by bounds derived using a simple coupling argument

average cost optimal stationary policies in infinite state markov decision processes with unbounded costs we deal with infinite state markov decision processes with unbounded costs three simple conditions based on the optimal discounted value function guarantee the existence of an expected average cost optimal stationary policy sufficient conditions are the existence of a distinguished state of smallest discounted value and a single stationary policy inducing an irreducible ergodic markov chain for which the average cost of a first passage from any state to the distinguished state is finite a result to verify this is also given two examples illustrate the ease of applying the criteria

a bayesian multidimensional scaling procedure for the spatial analysis of revealed choice data we present a new bayesian formulation of a vector multidimensional scaling procedure for the spatial analysis of binary choice data the gibbs sampler is gainfully employed to estimate the posterior distribution of the specified scalar products bilinear model parameters the computational procedure allows for the explicit estimation of a covariance matrix which can accommodate violations of iia due to context effects in addition posterior standard errors can be estimated which reflect differential degrees of consumer choice uncertainty and or brand position instability a marketing application concerning the analysis of consumers consideration sets for luxury automobiles is provided to illustrate the use of the proposed methodology

remarks on the existence of solutions to the average cost optimality equation in markov decision processes necessary conditions are given for the existence of a bounded solution to the optimality equation arising in markov decision processes under a long run expected average cost criterion the relationships of some of our results to known sufficient conditions are also shown

self dual representations with vectors fixed under an iwahori subgroup abstract let g be the group of f points of a split connected reductive f group over a non archimedean local field f of characteristic 0 let be an irreducible smooth self dual representation of g the space w of carries a non degenerate g invariant bilinear form which is unique up to scaling the form is easily seen to be symmetric or skew symmetric and we set e 1 accordingly in this article we show that e 1 when is a generic representation of g with non zero vectors fixed under an iwahori subgroup i

necessary and sufficient conditions for a bounded solution to the optimality equation in average reward markov decision chains abstract we consider average reward markov decision processes with discrete time parameter and denumerable state space we are concerned with the following problem find necessary and sufficient conditions so that for arbitrary bounded reward function the corresponding average reward optimality equation has a bounded solution this problem is solved for a class of systems including the case in which under the action of any stationary policy the state space is an irreducible positive recurrent class

optimization via low rank approximation for community detection in networks author s le can m levina elizaveta vershynin roman abstract community detection is one of the fundamental problems of network analysis for which a number of methods have been proposed most model based or criteria based methods have to solve an optimization problem over a discrete set of labels to find communities which is computationally infeasible some fast spectral algorithms have been proposed for specific methods or models but only on a case by case basis here we propose a general approach for maximizing a function of a network adjacency matrix over discrete labels by projecting the set of labels onto a subspace approximating the leading eigenvectors of the expected adjacency matrix this projection onto a low dimensional space makes the feasible set of labels much smaller and the optimization problem much easier we prove a general result about this method and show how to apply it to several previously proposed community detection criteria establishing its consistency for label estimation in each case and demonstrating the fundamental connection between spectral properties of the network and various model based approaches to community detection simulations and applications to real world data are included to demonstrate our method performs well for multiple problems over a wide range of parameters

bayesian multidimensional scaling and choice of dimension multidimensional scaling is widely used to handle data that consist of similarity or dissimilarity measures between pairs of objects we deal with two major problems in metric multidimensional scaling configuration of objects and determination of the dimension of object configuration within a bayesian framework a markov chain monte carlo algorithm is proposed for object configuration along with a simple bayesian criterion called mdsic for choosing their dimension simulation results are presented as are real data our method provides better results than does classical multidimensional scaling and alscal for object configuration and mdsic seems to work well for dimension choice in the examples considered

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

average cost markov decision processes optimality conditions we are concerned in this paper with discrete time markov decision processes mdps with borel state and action spaces x and a respectively and the long run expected average cost criterion when x is a denumerable set man y necessary and or sufficient conditions for the existence of optimal control policies are known however when x is a borel space i e a borel subset of a complete separable metric space most of the available results impose on the mdp very restrictive topological conditions e g compactness and or strong recurrence assumptions such as doeblin s condition see e g 4 9 12 and their references another related work is 7 where we have studied mdps from the viewpoint of the recurrence or ergodicity properties of the state process ln the present paper however we are concerned with the existence of average optimal policies by looking at static optimization problems see condition cs in section 4 related in sorne cases equivalent to the existence of a bounded solution to the so called optimality equation see c4 in section 4 these optimization problems are dual in the sense that under appropriate conditions the existence of an optimal solution to one of the problems implies existence of an optimal

a consistent adjacency spectral embedding for stochastic blockmodel graphs we present a method to estimate block membership of nodes in a random graph generated by a stochastic blockmodel we use an embedding procedure motivated by the random dot product graph model a particular example of the latent position model the embedding associates each node with a vector these vectors are clustered via minimization of a square error criterion we prove that this method is consistent for assigning nodes to blocks as only a negligible number of nodes will be misassigned we prove consistency of the method for directed and undirected graphs the consistent block assignment makes possible consistent parameter estimation for a stochastic blockmodel we extend the result in the setting where the number of blocks grows slowly with the number of nodes our method is also computationally feasible even for very large graphs we compare our method with laplacian spectral clustering through analysis of simulated data and a graph derived from wikipedia documents

a block model for node popularity in networks with community structure summary the community structure that is observed in empirical networks has been of particular interest in the statistics literature with a strong emphasis on the study of block models we study an important network feature called node popularity which is closely associated with community structure neither the classical stochastic block model nor its degree corrected extension can satisfactorily capture the dynamics of node popularity as observed in empirical networks we propose a popularity adjusted block model for flexible and realistic modelling of node popularity we establish consistency of likelihood modularity for community detection as well as estimation of node popularities and model parameters and demonstrate the advantages of the new modularity over the degree corrected block model modularity in simulations by analysing the political blogs network the british members of parliament network and the digital bibliography and library project bibliographical network we illustrate that improved empirical insights can be gained through this methodology

matrix estimation by universal singular value thresholding consider the problem of estimating the entries of a large matrix when the observed entries are noisy versions of a small random fraction of the original entries this problem has received widespread attention in recent times especially after the pioneering works of emmanuel candes and collaborators this paper introduces a simple estimation procedure called universal singular value thresholding usvt that works for any matrix that has a little bit of structure surprisingly this simple estimator achieves the minimax error rate up to a constant factor the method is applied to solve problems related to low rank matrix estimation blockmodels distance matrix completion latent space models positive definite matrix completion graphon estimation and generalized bradley terry models for pairwise comparison

control of markov chains with long run average cost criterion the dynamic programming equations the long run average cost control problem for discrete time markov chains on a countable state space is studied in a very general framework necessary and sufficient conditions for optimality in terms of the dynamic programming equations are given when an optimal stable stationary strategy is known to exist e g for the situations studied in stochastic differential systems stochastic control theory and applications ima vol math app 10 springer verlag new york berlin 1988 pp 57 77 a characterization of the desired solution of the dynamic programming equations is given in a special case also included is a novel convex analytic argument for deducing the existence of an optimal stable stationary strategy when that of a randomized one is known

pseudolikelihood estimation for social networks abstract interest in log linear modeling for social network data has grown steadily since holland and leinhardt 1981 proposed their p 1 model that model was designed for a single binary relationship directed graph representing interactions between individuals it assumed that interactions between pairs of individuals are mutually independent subsequent work has extended the model in various ways including block modeling and the case of dependence between pairs of individuals in empirical work it would often be desirable to fit a wide variety of these models as the differences in predictions or goodness of fit are likely to provide insights into the data this has not been common practice however because estimation for some of the models has been difficult and the maximum likelihood schemes developed for others involve different computer programs not always available in standard packages the focus of this article is on a general estimation technique that maximizes the pseudolikelihood the pro

consistent latent position estimation and vertex classification for random dot product graphs in this work we show that using the eigen decomposition of the adjacency matrix we can consistently estimate latent positions for random dot product graphs provided the latent positions are i i d from some distribution if class labels are observed for a number of vertices tending to infinity then we show that the remaining vertices can be classified with error converging to bayes optimal using the k nearest neighbors classification rule we evaluate the proposed methods on simulated data and a graph derived from

latent surface models for networks using aggregated relational data despite increased interest across a range of scientific applications in modeling and understanding social network structure collecting complete network data remains logistically and financially challenging especially in the social sciences this article introduces a latent surface representation of social network structure for partially observed network data we derive a multivariate measure of expected latent distance between an observed actor and unobserved actors with given features we also draw novel parallels between our work and dependent data in spatial and ecological statistics we demonstrate the contribution of our model using a random digit dial telephone survey and a multiyear prospective study of the relationship between network structure and the spread of infectious disease the model proposed here is related to previous network models which represents high dimensional structure through a projection to a low dimensional latent geometric surface encoding dependence as distance in the spa

spectral clustering and the high dimensional stochastic blockmodel networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors social ne tworks representing people who communicate with each other are one example communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks spectral clustering is a popular and computationally feasi ble method to discover these communities the stochastic block model holland et al 1983 is a social network model with well defined communities each node is a member of one community for a network generated from the stochastic block model we bound the number of nodes misclus tered by spectral clustering the asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes hence the name high dimensional in order to study spectral clustering under the stochastic block model we first show that under the more general latent space model the eigenvectors of the normalized graph laplacian asymptotically converge to the eigenvectors of a population normal ized graph laplacian aside from the implication for spectral clustering this provides insight into a graph visualization technique our method of studying the eigenvectors of random matrices is original ams 2000 subject classifications primary 62h30 62h25 secondary 60b20

optimal infinite horizon undiscounted control of finite probabilistic systems a finite input finite state finite output stochastic control problem with imperfect state observation and classical information pattern is shown to be meaningful as the horizon increases without bound and the discount rate approaches unity the plant model a finite probabilistic system includes the markov decision and partially observed markov decision problems as special cases under conditions resembling controllability and observability in linear systems we show that an optimal strategy exists it may be realized by a stationary policy on the state estimate its performance does not depend on the initial state distribution and convergence rates for its finite horizon and discounted performances are readily established

a note on the ross taylor theorem abstract in this note the condition used in proving a result due to s m ross and h m taylor are examined these results pertain to the existence of bounded solutions to the average cost optimality equation for controlled markov processes with an average cost criterion in particular we show how the use of commonly found convexity concavity properties of value functions can be employed to verify a seemingly rather restrictive equicontinuity condition in addition we remark that several results in the literature can be viewed as special cases of the result by ross and taylor contrary to claims otherwise

involutions acting on representations for a group g with order 2 automorphism we consider irreducible representations v of g such that where is the contragredient representation of this isomorphism gives rise to a nondegenerate bilinear form on v which is invariant unique up to scalar and consequently is either symmetric or skew symmetric for such a representation we study the question of when this form is symmetric and when it is skew symmetric we begin with recalling a method of prasad 13 in the case that is trivial and g is a finite group of lie type we apply a theorem of klyachko to obtain results for all representations of many of the groups sl n fq improving the results of prasad which were for generic representations in section 3 we start by generalizing lemma 1 to the case of any order 2 automorphism in lemma 2 we are able to apply this to get another proof of a theorem of gow 5 for the group gl n fq and the transposeinverse automorphism and in section 4 we are also able to apply lemma 2 to distinguished representations and gelfand pairs in section 5 we find a generating set for a finite group g given that we know certain information about the representations of g satisfying finally in section 6 we turn to the case of a locally compact totally disconnected group we adapt the methods of section 2 along with approximation by compact open subgroups to obtain an equivalent result of theorem 4 of gow for gl n f where f is a nonarchimedean local field

model for heterogeneous random networks using continuous latent variables and an application to a tree fungus network the mixture model is a method of choice for modeling heterogeneous random graphs because it contains most of the known structures of heterogeneity hubs hierarchical structures or community structure one of the weaknesses of mixture models on random graphs is that at the present time there is no computationally feasible estimation method that is completely satisfying from a theoretical point of view moreover mixture models assume that each vertex pertains to one group so there is no place for vertices being at intermediate positions the model proposed in this article is a grade of membership model for heterogeneous random graphs which assumes that each vertex is a mixture of extremal hypothetical vertices the connectivity properties of each vertex are deduced from those of the extreme vertices in this new model the vector of weights of each vertex are fixed continuous parameters a model with a vector of parameters for each vertex is tractable because the number of observations is proportional to the square of the number of vertices of the network the estimation of the parameters is given by the maximum likelihood procedure the model is used to elucidate some of the processes shaping the heterogeneous structure of a well resolved network of host parasite interactions

the existence of minimum pair of state and policy for markov decision processes under the hypothesis of doeblin this paper studies the average cost markov decision process with compact state and action spaces and bounded lower semicontinuous cost functions following the idea of borkar s excellent papers siamj control optim 21 1983 pp 652 666 22 1984 pp 965 978 the general case where irreducibility is not assumed is considered under the hypothesis of doeblin and the existence of a minimum pair of state and policy which attains the infimum of the average expected cost over all initial states and policies is established further it is proved that under additional weak conditions there exists an optimal stationary policy in the usual sense

self dual representations of division algebras and weil groups a contrast irreducible selfdual representations of any group fall into two classes those which carry a symmetric bilinear form and the others which carry an alternating bilinear form the langlands correspondence which matches the irreducible representations of the weil group of a local field k of dimension n with the irreducible representations of the invertible elements of a division algebra d over k of index n takes selfdual representations to selfdual representations in this paper we use global methods to study how the langlands correspondence behaves relative to this distinction among selfdual representations we prove in particular that for n even is symplectic if and only if is orthogonal more generally we treat the case of gl m b for b a division algebra over k of index r and n mr

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

universally consistent vertex classification for latent positions graphs in this work we show that using the eigen decomposition of the adjacency matrix we can consistently estimate feature maps for latent position graphs with positive definite link function provided that the latent positions are i i d from some distribution f we then consider the exploitation task of vertex classification where the link

testing for nodal dependence in relational data matrices relational data are often represented as a square matrix the entries of which record the relationships between pairs of objects many statistical methods for the analysis of such data assume some degree of similarity or dependence between objects in terms of the way they relate to each other however formal tests for such dependence have not been developed we provide a test for such dependence using the framework of the matrix normal model a type of multivariate normal distribution parameterized in terms of row and column specific covariance matrices we develop a likelihood ratio test lrt for row and column dependence based on the observation of a single relational data matrix we obtain a reference distribution for the lrt statistic thereby providing an exact test for the presence of row or column correlations in a square relational data matrix additionally we provide extensions of the test to accommodate common features of such data such as undefined diagonal entries a nonzero mean multipl

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