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

localization and centrality in networks eigenvector centrality is a common measure of the importance of nodes in a network here we show that under common conditions the eigenvector centrality displays a localization transition that causes most of the weight of the centrality to concentrate on a small number of nodes in the network in this regime the measure is no longer useful for distinguishing among the remaining nodes and its efficacy as a network metric is impaired as a remedy we propose an alternative centrality measure based on the nonbacktracking matrix which gives results closely similar to the standard eigenvector centrality in dense networks where the latter is well behaved but avoids localization and gives useful results in regimes where the standard centrality fails

minimax rates of community detection in stochastic block models in statistics as well as in computer science probability and applied mathematics community detection for the stochastic block model sbm is probably the most studied topic in network analysis many methodologies have been proposed some beautiful and signicant phase transition results are obtained in various settings in this paper we provide a general minimax theory for community detection it gives minimax rates of the mis match ratio for a wide rage of settings including homogeneous and inhomogeneous sbms dense and sparse networks nite and growing number of communities the minimax rates are exponential dierent from polynomial rates we often see in statistical literature an immediate consequence of the result is to establish threshold phenomenon for strong consistency exact recovery as well as weak consistency partial recovery we obtain the upper bound by a range of penalized likelihood type approaches the lower bound is achieved by a novel reduction from a global mis match ratio to a local clustering problem for one node through an exchangeability property

fast community detection by score consider a network where the nodes split into k different communities the community labels for the nodes are unknown and it is of major interest to estimate them i e community detection degree corrected block model dcbm is a popular network model how to detect communities with the dcbm is an interesting problem where the main challenge lies in the degree heterogeneity we propose a new approach to community detection which we call the spectral clustering on ratios of eigenvectors score compared to classical spectral methods the main innovation is to use the entry wise ratios between the first leading eigenvector and each of the other leading eigenvectors for clustering let a be the adjacency matrix of the network we first obtain the k leading eigenvectors of a say hat eta 1 ldots hat eta k and let hat r be the n times k 1 matrix such that hat r i k hat eta k 1 i hat eta 1 i 1 leq i leq n 1 leq k leq k 1 we then use hat r for clustering by applying the k means method the central surprise is the effect of degree heterogeneity is largely ancillary and can be effectively removed by taking entry wise ratios between hat eta k 1 and hat eta 1 1 leq k leq k 1 the method is successfully applied to the web blogs data and the karate club data with error rates of 58 1222 and 1 34 respectively these results are more satisfactory than those by the classical spectral methods additionally compared to modularity methods score is easier to implement computationally faster and also has smaller error rates we develop a theoretic framework where we show that under mild conditions the score stably yields consistent community detection in the core of the analysis is the recent development on random matrix theory rmt where the matrix form bernstein inequality is especially helpful

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

pseudo likelihood methods for community detection in large sparse networks many algorithms have been proposed for fitting network models with communities but most of them do not scale well to large networks and often fail on sparse networks here we propose a new fast pseudo likelihood method for fitting the stochastic block model for networks as well as a variant that allows for an arbitrary degree distribution by conditioning on degrees we show that the algorithms perform well under a range of settings including on very sparse networks and illustrate on the example of a network of political blogs we also propose spectral clustering with perturbations a method of independent interest which works well on sparse networks where regular spectral clustering fails and use it to provide an initial value for pseudo likelihood we prove that pseudo likelihood provides consistent estimates of the communities under a mild condition on the starting value for the case of a block model with two communities

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

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

the impact of a systematic reduction in shoe floor friction on heel contact walking kinematics a gait simulation approach falls initiated by slips and trips are a serious health hazard to older adults experimental studies have provided important descriptions of postural responses to slipping but it is difficult to determine why some slips result in falls from experiments alone computational modeling and simulation techniques can complement experimental approaches by identifying causes of failed recovery attempts the purpose of this study was to develop a method to determine the impact of a systematic reduction in the foot floor friction coefficient mu on the kinematics of walking shortly after heel contact approximately 200ms a walking model that included foot floor interactions was utilized to find the set of moments that best tracked the joint angles and measured ground reaction forces obtained from a non slipping dry trial a passive slip was simulated by driving the model with the joint moments from the dry simulation and by reducing mu slip simulations with values of mu greater than the subject specific peak required coefficient of friction rcof an experimental measure of slip resistant gait resulted in only minor deviations in gait kinematics from the dry condition in contrast slip simulations run in environments characterized by mu language en

a numerical scheme to couple subdomains with different time steps for predominantly linear transient analysis abstract this paper generalizes the explicit implicit time integration algorithms pioneered by belytschko hughes and their respective co workers and the feti domain decomposition methods introduced by farhat and his co workers to the case where the same newmark scheme but different and coefficients and different time steps are specified in each subdomain building upon the work of farhat crivelli and geradin it considers various interface boundary constraints and performs a stability analysis of the proposed time integration algorithms

statistical computational tradeoffs in planted problems and submatrix localization with a growing number of clusters and submatrices we consider two closely related problems planted clustering and submatrix localization in the planted clustering problem a random graph is generated based on an underlying cluster structure of the nodes the task is to recover these clusters given the graph the submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real valued random matrix of particular interest is the setting where the number of clusters submatrices is allowed to grow unbounded with the problem size these formulations cover several classical models such as planted clique planted densest subgraph planted partition planted coloring and the stochastic block model which are widely used for studying community detection graph clustering and bi clustering for both problems we show that the space of the model parameters cluster submatrix size edge probabilities and the mean of the submatrices can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities 1 the impossible regime where all algorithms fail 2 the hard regime where the computationally expensive maximum likelihood estimator mle succeeds 3 the easy regime where the polynomial time convexified mle succeeds 4 the simple regime where a local counting thresholding procedure succeeds moreover we show that each of these algorithms provably fails in the harder regimes our results establish the minimax recovery limits which are tight up to universal constants and hold even with a growing number of clusters submatrices and provide order wise stronger performance guarantees for polynomial time algorithms than previously known our study demonstrates the tradeoffs between statistical and computational considerations and suggests that the minimax limits may not be achievable by polynomial time algorithms

strong consistency of spectral clustering for stochastic block models in this paper we prove the strong consistency of several methods based on the spectral clustering techniques that are widely used to study the community detection problem in stochastic block models sbms we show that under some weak conditions on the minimal degree the number of communities and the eigenvalues of the probability block matrix the k means algorithm applied to the eigenvectors of the graph laplacian associated with its first few largest eigenvalues can classify all individuals into the true community uniformly correctly almost surely extensions to both regularized spectral clustering and degree corrected sbms are also considered we illustrate the performance of different methods on simulated networks

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications in this paper we extend our previous work on the stochastic block model a commonly used generative model for social and biological networks and the problem of inferring functional groups or communities from the topology of the network we use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram we describe in detail properties of the detectability undetectability phase transition and the easy hard phase transition for the community detection problem our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way i e that maximizes the overlap with the underlying group memberships and learning the underlying parameters of the block model finally we apply the algorithm to two examples of real world networks and discuss its performance

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

likelihood based model selection for stochastic block models the stochastic block model sbm provides a popular framework for modeling community structures in networks however more attention has been devoted to problems concerning estimating the latent node labels and the model parameters than the issue of choosing the number of blocks we consider an approach based on the log likelihood ratio statistic and analyze its asymptotic properties under model misspecification we show the limiting distribution of the statistic in the case of underfitting is normal and obtain its convergence rate in the case of overfitting these conclusions remain valid when the average degree grows at a polylog rate the results enable us to derive the correct order of the penalty term for model complexity and arrive at a likelihood based model selection criterion that is asymptotically consistent our analysis can also be extended to a degree corrected block model dcsbm in practice the likelihood function can be estimated using more computationally efficient variational methods or consistent label estimation algorithms allowing the criterion to be applied to large networks

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

comment ridge regression and regularization of large matrices we view ridge regression through the lens of eigenvalue shrinkage and consider its influence on two modern problems in high dimensional statistical inference covariance estimation and community d

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

consistency of community detection in networks under degree corrected stochastic block models community detection is a fundamental problem in network analysis with applications in many diverse areas the stochastic block model is a common tool for model based community detection and asymptotic tools for checking consistency of community detection under the block model have been recently developed however the block model is limited by its assumption that all nodes within a community are stochastically equivalent and provides a poor fit to networks with hubs or highly varying node degrees within communities which are common in practice the degree corrected stochastic block model was proposed to address this shortcoming and allows variation in node degrees within a community while preserving the overall block community structure in this paper we establish general theory for checking consistency of community detection under the degree corrected stochastic block model and compare several community detection criteria under both the standard and the degree corrected models we show which criteria are consistent under which models and constraints as well as compare their relative performance in practice we find that methods based on the degree corrected block model which includes the standard block model as a special case are consistent under a wider class of models and that modularity type methods require parameter constraints for consistency whereas likelihood based methods do not on the other hand in practice the degree correction involves estimating many more parameters and empirically we find it is only worth doing if the node degrees within communities are indeed highly variable we illustrate the methods on simulated networks and on a network of political blogs

on semidefinite relaxations for the block model the stochastic block model sbm is a popular tool for community detection in networks but fitting it by maximum likelihood mle involves a computationally infeasible optimization problem we propose a new semidefinite programming sdp solution to the problem of fitting the sbm derived as a relaxation of the mle we put ours and previously proposed sdps in a unified framework as relaxations of the mle over various subclasses of the sbm which also reveals a connection to the well known problem of sparse pca our main relaxation which we call sdp 1 is tighter than other recently proposed sdp relaxations and thus previously established theoretical guarantees carry over however we show that sdp 1 exactly recovers true communities over a wider class of sbms than those covered by current results in particular the assumption of strong assortativity of the sbm implicit in consistency conditions for previously proposed sdps can be relaxed to weak assortativity for our approach thus significantly broadening the class of sbms covered by the consistency results we also show that strong assortativity is indeed a necessary condition for exact recovery for previously proposed sdp approaches and not an artifact of the proofs our analysis of sdps is based on primal dual witness constructions which provides some insight into the nature of the solutions of various sdps in particular we show how to combine features from sdp 1 and already available sdps to achieve the most flexibility in terms of both assortativity and block size constraints as our relaxation has the tendency to produce communities of similar sizes this tendency makes it the ideal tool for fitting network histograms a method gaining popularity in the graphon estimation literature as we illustrate on an example of a social networks of dolphins we also provide empirical evidence that sdps outperform spectral methods for fitting sbms with a large number of blocks

estimation of the number of communities in the stochastic block model in this article we introduce an estimator for the number of communities in the stochastic block model sbm based on the maximization of a penalized version of the so called krichevsky trofimov mixture distribution we prove its eventual almost sure convergence to the underlying number of communities without assuming a known upper bound on that quantity our results apply to both the dense and the sparse regimes to our knowledge this is the first consistency result for the estimation of the number of communities in the sbm in the unbounded case that is when the number of communities is allowed to grow with the same size

finite element formulation for modelling large deformations in elasto viscoplastic polycrystals anisotropic elasto viscoplastic behaviour in polycrystalline materials is modelled using a new updated lagrangian formulation based on a three field form of the hu washizu variational principle to create a stable finiteelement method in the context of nearly incompressible behaviour the meso scale is characterized by a representative volume element which contains grains governed by single crystal behaviour a new fully implicit two level backward euler integration scheme together with an efficient finite element formulation including consistent linearization is presented the proposed finite element model is capable of predicting non homogeneous meso fields which for example may impact subsequent recrystallization finally simple deformations involving an aluminium alloy are considered in order to demonstrate the algorithm

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

consistent community detection in multi layer network data summarywe consider multi layer network data where the relationships between pairs of elements are reflected in multiple modalities and may be described by multivariate or even high dimensional vectors under the multi layer stochastic block model framework we derive consistency results for a least squares estimation of memberships our theorems show that as compared to single layer community detection a multi layer network provides much richer information that allows for consistent community detection from a much sparser network with required edge density reduced by a factor of the square root of the number of layers moreover the multi layer framework can detect cohesive community structure across layers which might be hard to detect by any single layer or simple aggregation simulations and a data example are provided to support the theoretical results

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

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

the geometry of kernelized spectral clustering clustering of data sets is a standard problem in many areas of science and engineering the method of spectral clustering is based on embedding the data set using a kernel function and using the top eigenvectors of the normalized laplacian to recover the connected components we study the performance of spectral clustering in recovering the latent labels of i i d samples from a finite mixture of nonparametric distributions the difficulty of this label recovery problem depends on the overlap between mixture components and how easily a mixture component is divided into two nonoverlapping components when the overlap is small compared to the indivisibility of the mixture components the principal eigenspace of the population level normalized laplacian operator is approximately spanned by the square root kernelized component densities in the finite sample setting and under the same assumption embedded samples from different components are approximately orthogonal with high probability when the sample size is large as a corollary we control the fraction of samples mislabeled by spectral clustering under finite mixtures with nonparametric components

community detection in sparse networks via grothendieck s inequality we present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs the method is based on grothendieck s inequality unlike the previous uses of this inequality that lead to constant relative accuracy we achieve any given relative accuracy by leveraging randomness we illustrate the method with the problem of community detection in sparse networks those with bounded average degrees we demonstrate that even in this regime various simple and natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices the method is general it can be applied to a variety of stochastic models of networks and semidefinite programs

a study of the stability of subcycling algorithms in structural dynamics algorithms for explicit integration of structural dynamics problems with multiple time steps subcycling are investigated only one such algorithm due to smolinski and sleith has proved to be stable in a classical sense a simplified version of this algorithm that retains its stability is presented however as with the original version it can be shown to sacrifice accuracy to achieve stability another algorithm in use is shown to be only statistically stable in that a probability of stability can be assigned if appropriate time step limits are observed this probability improves rapidly with the number of degrees of freedom in a finite element model the stability problems are shown to be a property of the central difference method itself which is modified to give the subcycling algorithm a related problem is shown to arise when a constraint equation in time is introduced into a time continuous space time finite element model

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

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

asynchronous multi domain variational integrators for nonlinear hyperelastic solids we present the asynchronous multi domain variational time integrators with a dual domain decomposition method for the initial hyperbolic boundary value problem in hyperelasticity variational time integration schemes based on the principle of minimal action within the lagrangian framework are constructed for the equation of motion and implemented into a variational finite element framework which is systematically derived from the three field de veubeke hu washizu variational principle to accommodate the incompressibility constraint present in an analysis of nearly incompressible materials for efficient parallel computing we use the dual domain decomposition method with local lagrange multipliers to ensure the continuity of the displacement field at the interface between subdomains the method for time discretization and the multi domain spatial decomposition enable us to use different types of integrators explicit vs implicit and different time steps on different parts of a computational domain and thus efficiently capture the underlying physics with less computational effort the energy conservation of our nonlinear midpoint asynchronous integration scheme is investigated using the energy method and both local and the global energy error estimates are derived we illustrate the performance of proposed variational multi domain time integrators by means of three examples first the method of manufactured solutions is used to examine the consistency of the formulation in the second example we investigate energy conservation and stability finally we apply the method to the motion of a heterogeneous plane domain where different integrators and time discretization steps are used accordingly with disparate material data of individual parts

laplacian eigenmaps from sparse noisy similarity measurements manifold learning and dimensionality reduction techniques are ubiquitous in science and engineering but can be computationally expensive procedures when applied to large datasets or when similarities are expensive to compute to date little work has been done to investigate the tradeoff between computational resources and the quality of learned representations we present both theoretical and experimental explorations of this question in particular we consider laplacian eigenmaps embeddings based on a kernel matrix and explore how the embeddings behave when this kernel matrix is corrupted by occlusion and noise our main theoretical result shows that under modest noise and occlusion assumptions we can with high probability recover a good approximation to the laplacian eigenmaps embedding based on the uncorrupted kernel matrix our results also show how regularization can aid this approximation experimentally we explore the effects of noise and occlusion on laplacian eigenmaps embeddings of two real world datasets one from speech processing and one from neuroscience as well as a synthetic dataset

spectral methods for community detection and graph partitioning we consider three distinct and well studied problems concerning network structure community detection by modularity maximization community detection by statistical inference and normalized cut graph partitioning each of these problems can be tackled using spectral algorithms that make use of the eigenvectors of matrix representations of the network we show that with certain choices of the free parameters appearing in these spectral algorithms the algorithms for all three problems are in fact identical and hence that at least within the spectral approximations used here there is no difference between the modularity and inference based community detection methods or between either and graph partitioning

co clustering directed graphs to discover asymmetries and directional communities in directed graphs relationships are asymmetric and these asymmetries contain essential structural information about the graph directed relationships lead to a new type of clustering that is not feasible in undirected graphs we propose a spectral co clustering algorithm called di sim for asymmetry discovery and directional clustering a stochastic co blockmodel is introduced to show favorable properties of di sim to account for the sparse and highly heterogeneous nature of directed networks di sim uses the regularized graph laplacian and projects the rows of the eigenvector matrix onto the sphere a nodewise asymmetry score and di sim are used to analyze the clustering asymmetries in the networks of enron emails political blogs and the caenorhabditiselegans chemical connectome in each example a subset of nodes have clustering asymmetries these nodes send edges to one cluster but receive edges from another cluster such nodes yield insightful information e g communication bottlenecks about directed networks but are missed if the analysis ignores edge direction

impact of regularization on spectral clustering the performance of spectral clustering can be considerably improved via regularization as demonstrated empirically in amini et al ann statist 41 2013 2097 2122 here we provide an attempt at quantifying this improvement through theoretical analysis under the stochastic block model sbm and its extensions previous results on spectral clustering relied on the minimum degree of the graph being sufficiently large for its good performance by examining the scenario where the regularization parameter tau is large we show that the minimum degree assumption can potentially be removed as a special case for an sbm with two blocks the results require the maximum degree to be large grow faster than log n as opposed to the minimum degree more importantly we show the usefulness of regularization in situations where not all nodes belong to well defined clusters our results rely on a bias variance like trade off that arises from understanding the concentration of the sample laplacian and the eigengap as a function of the regularization parameter as a byproduct of our bounds we propose a data driven technique dkest standing for estimated davis kahan bounds for choosing the regularization parameter this technique is shown to work well through simulations and on a real data set

using maximum entry wise deviation to test the goodness of fit for stochastic block models the stochastic block model is widely used for detecting community structures in network data how to test the goodness of fit of the model is one of the fundamental problems and has gained growing

on the stability of symplectic and energy momentum algorithms for non linear hamiltonian systems with symmetry abstract this paper presents a detailed comparison of two implicit time integration schemes for a simple non linear hamiltonian system with symmetry the motion of a particle in a central force field the goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid point rule the proto typical implicit symplectic method and a particular energy momentum conserving scheme and to compare the two schemes with respect to accuracy while all results presented herein are within the context of a simple model problem the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non linear solid mechanics namely the presence of large and relatively slow overall motions together with high frequency internal motions