For community detection problem, spectral clustering is a widely used method for detecting clusters in networks. In this paper, we propose an improved spectral clustering (ISC) approach under the degree corrected stochastic block model (DCSBM). ISC is designed based on the k-means clustering algorithm on the weighted leading K + 1 eigenvectors of a regularized Laplacian matrix where the weights are their corresponding eigenvalues. Theoretical analysis of ISC shows that under mild conditions the ISC yields stable consistent community detection. Numerical results show that ISC outperforms classical spectral clustering methods for community detection on both simulated and eight empirical networks. Especially, ISC provides a significant improvement on two weak signal networks Simmons and Caltech, with error rates of 121/1137 and 96/590, respectively.
Imperial College London（伦敦帝国理工学院）；University of Bristol（布里斯托尔大学）
Spectral clustering is a popular method for community detection in network graphs: starting from a matrix representation of the graph, the nodes are clustered on a low dimensional projection obtained from a truncated spectral decomposition of the matrix. Estimating correctly the number of communities and the dimension of the reduced latent space is critical for good performance of spectral clustering algorithms. Furthermore, many real-world graphs, such as enterprise computer networks studied in cyber-security applications, often display heterogeneous within-community degree distributions. Suc…
Spectral clustering is a fast and popular algorithm for finding clusters in networks. Recently, Chaudhuri et al. (2012) and Amini et al.(2012) proposed inspired variations on the algorithm that artificially inflate the node degrees for improved statistical performance. The current paper extends the previous statistical estimation results to the more canonical spectral clustering algorithm in a way that removes any assumption on the minimum degree and provides guidance on the choice of the tuning parameter. Moreover, our results show how the "star shape" in the eigenvectors--a common feature of…
Many approaches have been proposed to discover clusters within networks. Community finding field encompasses approaches which try to discover clusters where nodes are tightly related within them but loosely related with nodes of other clusters. However, a community network configuration is not the only possible latent structure in a graph. Core-periphery and hierarchical network configurations are valid structures to discover in a relational dataset. On the other hand, a network is not completely explained by only knowing the membership of each node. A high level view of the inter-cluster rela…
University of Alberta（阿尔伯塔大学）；University of New South Wales（新南威尔士大学）；University of Social Sciences, Poland（波兰社会科学大学）
The Mixed-Membership Stochastic Blockmodel~(MMSB) is proposed as one of the state-of-the-art Bayesian relational methods suitable for learning the complex hidden structure underlying the network data. However, the current formulation of MMSB suffers from the following two issues: (1), the prior information~(e.g. entities' community structural information) can not be well embedded in the modelling; (2), community evolution can not be well described in the literature. Therefore, we propose a non-parametric fragmentation coagulation based Mixed Membership Stochastic Blockmodel (fcMMSB). Our model…
Vertex clustering in a stochastic blockmodel graph has wide applicability and has been the subject of extensive research. In thispaper, we provide a short proof that the adjacency spectral embedding can be used to obtain perfect clustering for the stochastic blockmodel and the degree-corrected stochastic blockmodel. We also show an analogous result for the more general random dot product graph model.
Semidefinite programs have recently been developed for the problem of community detection, which may be viewed as a special case of the stochastic blockmodel. Here, we develop a semidefinite program that can be tailored to other instances of the blockmodel, such as non-assortative networks and overlapping communities. We establish label recovery in sparse settings, with conditions that are analogous to recent results for community detection. In settings where the data is not generated by a blockmodel, we give an oracle inequality that bounds excess risk relative to the best blockmodel approxim…
We propose a robust, scalable, integrated methodology for community detection and community comparison in graphs. In our procedure, we first embed a graph into an appropriate Euclidean space to obtain a low-dimensional representation, and then cluster the vertices into communities. We next employ nonparametric graph inference techniques to identify structural similarity among these communities. These two steps are then applied recursively on the communities, allowing us to detect more fine-grained structure. We describe a hierarchical stochastic blockmodel---namely, a stochastic blockmodel wit…
University of Waterloo（滑铁卢大学）；Cornell University（康奈尔大学）
Much of the data being created on the web contains interactions between users and items. Stochastic blockmodels, and other methods for community detection and clustering of bipartite graphs, can infer latent user communities and latent item clusters from this interaction data. These methods, however, typically ignore the items' contents and the information they provide about item clusters, despite the tendency of items in the same latent cluster to share commonalities in content. We introduce content-augmented stochastic blockmodels (CASB), which use item content together with user-item intera…
Spectral embedding of network adjacency matrices often produces node representations living approximately around low-dimensional submanifold structures. In particular, hidden substructure is expected to arise when the graph is generated from a latent position model. Furthermore, the presence of communities within the network might generate community-specific submanifold structures in the embedding, but this is not explicitly accounted for in most statistical models for networks. In this article, a class of models called latent structure block models (LSBM) is proposed to address such scenarios…
In recent years there has been an increased interest in statistical analysis of data with multiple types of relations among a set of entities. Such multi-relational data can be represented as multi-layer graphs where the set of vertices represents the entities and multiple types of edges represent the different relations among them. For community detection in multi-layer graphs, we consider two random graph models, the multi-layer stochastic blockmodel (MLSBM) and a model with a restricted parameter space, the restricted multi-layer stochastic blockmodel (RMLSBM). We derive consistency results…
Spectral embedding of adjacency or Laplacian matrices of undirected graphs is a common technique for representing a network in a lower dimensional latent space, with optimal theoretical guarantees. The embedding can be used to estimate the community structure of the network, with strong consistency results in the stochastic blockmodel framework. One of the main practical limitations of standard algorithms for community detection from spectral embeddings is that the number of communities and the latent dimension of the embedding must be specified in advance. In this article, a novel Bayesian mo…
Stochastic blockmodels (SBM) and their variants, $e.g.$, mixed-membership and overlapping stochastic blockmodels, are latent variable based generative models for graphs. They have proven to be successful for various tasks, such as discovering the community structure and link prediction on graph-structured data. Recently, graph neural networks, $e.g.$, graph convolutional networks, have also emerged as a promising approach to learn powerful representations (embeddings) for the nodes in the graph, by exploiting graph properties such as locality and invariance. In this work, we unify these two di…
Amazon, Inc.（亚马逊公司）；The University of Texas at Austin（得克萨斯大学奥斯汀分校）
Stochastic blockmodels allow us to represent networks in terms of a latent community structure, often yielding intuitions about the underlying social structure. Typically, this structure is inferred based only on a binary network representing the presence or absence of interactions between nodes, which limits the amount of information that can be extracted from the data. In practice, many interaction networks contain much more information about the relationship between two nodes. For example, in an email network, the volume of communication between two users and the content of that communicati…
University of Colorado（科罗拉多大学）；BioFrontiers Institute（生物前沿研究所）
In bipartite networks, community structures are restricted to being disassortative, in that nodes of one type are grouped according to common patterns of connection with nodes of the other type. This makes the stochastic block model (SBM), a highly flexible generative model for networks with block structure, an intuitive choice for bipartite community detection. However, typical formulations of the SBM do not make use of the special structure of bipartite networks. Here we introduce a Bayesian nonparametric formulation of the SBM and a corresponding algorithm to efficiently find communities in…
Johns Hopkins University（约翰斯·霍普金斯大学）；University of Pittsburgh（匹兹堡大学）
In network inference applications, it is often desirable to detect community structure, namely to cluster vertices into groups, or blocks, according to some measure of similarity. Beyond mere adjacency matrices, many real networks also involve vertex covariates that carry key information about underlying block structure in graphs. To assess the effects of such covariates on block recovery, we present a comparative analysis of two model-based spectral algorithms for clustering vertices in stochastic blockmodel graphs with vertex covariates. The first algorithm uses only the adjacency matrix, an…
Indian Institute of Technology, Kanpur（印度理工学院坎普尔分校）
The latent feature relational model (LFRM) is a generative model for graph-structured data to learn a binary vector representation for each node in the graph. The binary vector denotes the node's membership in one or more communities. At its core, the LFRM miller2009nonparametric is an overlapping stochastic blockmodel, which defines the link probability between any pair of nodes as a bilinear function of their community membership vectors. Moreover, using a nonparametric Bayesian prior (Indian Buffet Process) enables learning the number of communities automatically from the data. However, des…
Inference for the stochastic blockmodel is currently of burgeoning interest in the statistical community, as well as in various application domains as diverse as social networks, citation networks, brain connectivity networks (connectomics), etc. Recent theoretical developments have shown that spectral embedding of graphs yields tractable distributional results; in particular, a random dot product latent position graph formulation of the stochastic blockmodel informs a mixture of normal distributions for the adjacency spectral embedding. We employ this new theory to provide an empirical Bayes …
This chapter provides a self-contained introduction to the use of Bayesian inference to extract large-scale modular structures from network data, based on the stochastic blockmodel (SBM), as well as its degree-corrected and overlapping generalizations. We focus on nonparametric formulations that allow their inference in a manner that prevents overfitting, and enables model selection. We discuss aspects of the choice of priors, in particular how to avoid underfitting via increased Bayesian hierarchies, and we contrast the task of sampling network partitions from the posterior distribution with …
In Stochastic blockmodels, which are among the most prominent statistical models for cluster analysis of complex networks, clusters are defined as groups of nodes with statistically similar link probabilities within and between groups. A recent extension by Karrer and Newman incorporates a node degree correction to model degree heterogeneity within each group. Although this demonstrably leads to better performance on several networks it is not obvious whether modelling node degree is always appropriate or necessary. We formulate the degree corrected stochastic blockmodel as a non-parametric Ba…
Directed graphs have asymmetric connections, yet the current graph clustering methodologies cannot identify the potentially global structure of these asymmetries. We give a spectral algorithm called di-sim that builds on a dual measure of similarity that correspond to how a node (i) sends and (ii) receives edges. Using di-sim, we analyze the global asymmetries in the networks of Enron emails, political blogs, and the c elegans neural connectome. In each example, a small subset of nodes have persistent asymmetries; these nodes send edges with one cluster, but receive edges with another cluster.…
Random-walk based network embedding algorithms like DeepWalk and node2vec are widely used to obtain Euclidean representation of the nodes in a network prior to performing downstream inference tasks. However, despite their impressive empirical performance, there is a lack of theoretical results explaining their large-sample behavior. In this paper, we study node2vec and DeepWalk through the perspective of matrix factorization. In particular, we analyze these algorithms in the setting of community detection for stochastic blockmodel graphs (and their degree-corrected variants). By exploiting the…
In many applications of network analysis, it is important to distinguish between observed and unobserved factors affecting network structure. To this end, we develop spectral estimators for both unobserved blocks and the effect of covariates in stochastic blockmodels. On the theoretical side, we establish asymptotic normality of our estimators for the subsequent purpose of performing inference. On the applied side, we show that computing our estimator is much faster than standard variational expectation--maximization algorithms and scales well for large networks. Monte Carlo experiments sugges…
Fudan University（复旦大学）；Shanghai Center for Mathematical Sciences（上海数学中心）；University of Pennsylvania（宾夕法尼亚大学）；Shanghai University of Finance and Economics（上海财经大学）
We show that a simple community detection algorithm originated from stochastic blockmodel literature achieves consistency, and even optimality, for a broad and flexible class of sparse latent space models. The class of models includes latent eigenmodels (arXiv:0711.1146). The community detection algorithm is based on spectral clustering followed by local refinement via normalized edge counting.
The mixed membership stochastic blockmodel (MMSB) is a popular framework for community detection and network generation. It learns a low-rank mixed membership representation for each node across communities by exploiting the underlying graph structure. MMSB assumes that the membership distributions of the nodes are independently drawn from a Dirichlet distribution, which limits its capability to model highly correlated graph structures that exist in real-world networks. In this paper, we present a flexible richly structured MMSB model, \textit{Struct-MMSB}, that uses a recently developed stati…
Stochastic blockmodels and variants thereof are among the most widely used approaches to community detection for social networks and relational data. A stochastic blockmodel partitions the nodes of a network into disjoint sets, called communities. The approach is inherently related to clustering with mixture models; and raises a similar model selection problem for the number of communities. The Bayesian information criterion (BIC) is a popular solution, however, for stochastic blockmodels, the conditional independence assumption given the communities of the endpoints among different edges is u…
Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social networks, 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 feasible method to discover these communities. The stochastic blockmodel [Social Networks 5 (1983) 109--137] is a social network model with well-defined communities; each node is a member of one community. For …
We introduce a novel model for multilayer weighted networks that accounts for global noise in addition to local signals. The model is similar to a multilayer stochastic blockmodel (SBM), but the key difference is that between-block interactions independent across layers are common for the whole system, which we call ambient noise. A single block is also characterized by these fixed ambient parameters to represent members that do not belong anywhere else. This approach allows simultaneous clustering and typologizing of blocks into signal or noise in order to better understand their roles in the…
This paper considers the problem of modeling and estimating community memberships of nodes in a directed network where every row (column) node is associated with a vector determining its membership in each row (column) community. To model such directed network, we propose directed degree corrected mixed membership (DiDCMM) model by considering degree heterogeneity. DiDCMM is identifiable under popular conditions for mixed membership network when considering degree heterogeneity. Based on the cone structure inherent in the normalized version of the left singular vectors and the simplex structur…
Hypergraphs are a natural modeling paradigm for a wide range of complex relational systems. A standard analysis task is to identify clusters of closely related or densely interconnected nodes. Many graph algorithms for this task are based on variants of the stochastic blockmodel, a random graph with flexible cluster structure. However, there are few models and algorithms for hypergraph clustering. Here, we propose a Poisson degree-corrected hypergraph stochastic blockmodel (DCHSBM), a generative model of clustered hypergraphs with heterogeneous node degrees and edge sizes. Approximate maximum-…
We propose a dynamic edge exchangeable network model that can capture sparse connections observed in real temporal networks, in contrast to existing models which are dense. The model achieved superior link prediction accuracy on multiple data sets when compared to a dynamic variant of the blockmodel, and is able to extract interpretable time-varying community structures from the data. In addition to sparsity, the model accounts for the effect of social influence on vertices' future behaviours. Compared to the dynamic blockmodels, our model has a smaller latent space. The compact latent space r…
Clustering and community detection with multiple graphs have typically focused on aligned graphs, where there is a mapping between nodes across the graphs (e.g., multi-view, multi-layer, temporal graphs). However, there are numerous application areas with multiple graphs that are only partially aligned, or even unaligned. These graphs are often drawn from the same population, with communities of potentially different sizes that exhibit similar structure. In this paper, we develop a joint stochastic blockmodel (Joint SBM) to estimate shared communities across sets of heterogeneous non-aligned g…
Real-world networks usually have community structure, that is, nodes are grouped into densely connected communities. Community detection is one of the most popular and best-studied research topics in network science and has attracted attention in many different fields, including computer science, statistics, social sciences, among others. Numerous approaches for community detection have been proposed in literature, from ad-hoc algorithms to systematic model-based approaches. The large number of available methods leads to a fundamental question: whether a certain method can provide consistent e…
We consider community detection in Degree-Corrected Stochastic Block Models (DC-SBM). We propose a spectral clustering algorithm based on a suitably normalized adjacency matrix. We show that this algorithm consistently recovers the block-membership of all but a vanishing fraction of nodes, in the regime where the lowest degree is of order log$(n)$ or higher. Recovery succeeds even for very heterogeneous degree-distributions. The used algorithm does not rely on parameters as input. In particular, it does not need to know the number of communities.
Community detection for unweighted networks has been widely studied in network analysis, but the case of weighted networks remains a challenge. This paper proposes a general Distribution-Free Model (DFM) for weighted networks in which nodes are partitioned into different communities. DFM can be seen as a generalization of the famous stochastic blockmodels from unweighted networks to weighted networks. DFM does not require prior knowledge of a specific distribution for elements of the adjacency matrix but only the expected value. In particular, signed networks with latent community structures c…
We propose a dynamic network sampling scheme to optimize block recovery for stochastic blockmodel (SBM) in the case where it is prohibitively expensive to observe the entire graph. Theoretically, we provide justification of our proposed Chernoff-optimal dynamic sampling scheme via the Chernoff information. Practically, we evaluate the performance, in terms of block recovery, of our method on several real datasets from different domains. Both theoretically and practically results suggest that our method can identify vertices that have the most impact on block structure so that one can only chec…
Community detection is a central problem of network data analysis. Given a network, the goal of community detection is to partition the network nodes into a small number of clusters, which could often help reveal interesting structures. The present paper studies community detection in Degree-Corrected Block Models (DCBMs). We first derive asymptotic minimax risks of the problem for a misclassification proportion loss under appropriate conditions. The minimax risks are shown to depend on degree-correction parameters, community sizes, and average within and between community connectivities in an…
When searching for gene pathways leading to specific disease outcomes, additional information on gene characteristics is often available that may facilitate to differentiate genes related to the disease from irrelevant background when connections involving both types of genes are observed and their relationships to the disease are unknown. We propose method to single out irrelevant background genes with the help of auxiliary information through a logistic regression, and cluster relevant genes into cohesive groups using the adjacency matrix. Expectation-maximization algorithm is modified to ma…
China University of Mining and Technology（中国矿业大学）；Nankai University（南开大学）
Based on the classical Degree Corrected Stochastic Blockmodel (DCSBM) model for network community detection problem, we propose two novel approaches: principal component clustering (PCC) and normalized principal component clustering (NPCC). Without any parameters to be estimated, the PCC method is simple to be implemented. Under mild conditions, we show that PCC yields consistent community detection. NPCC is designed based on the combination of the PCC and the RSC method (Qin & Rohe 2013). Population analysis for NPCC shows that NPCC returns perfect clustering for the ideal case under DCSBM. P…
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. In this paper we propose to automatically determine k in a graph generated from a Stochastic Blockmodel. Our main contribution is twofold; first, we theoretically e…
We consider the problem of estimating a consensus community structure by combining information from multiple layers of a multi-layer network using methods based on the spectral clustering or a low-rank matrix factorization. As a general theme, these "intermediate fusion" methods involve obtaining a low column rank matrix by optimizing an objective function and then using the columns of the matrix for clustering. However, the theoretical properties of these methods remain largely unexplored. In the absence of statistical guarantees on the objective functions, it is difficult to determine if the…
China University of Mining and Technology（中国矿业大学）；Nankai University（南开大学）
Community detection in network analysis is an attractive research area recently. Here, under the degree-corrected mixed membership (DCMM) model, we propose an efficient approach called mixed regularized spectral clustering (Mixed-RSC for short) based on the regularized Laplacian matrix. Mixed-RSC is designed based on an ideal cone structure of the variant for the eigen-decomposition of the population regularized Laplacian matrix. We show that the algorithm is asymptotically consistent under mild conditions by providing error bounds for the inferred membership vector of each node. As a byproduc…
We consider the problem of estimating community memberships of nodes in a network, where every node is associated with a vector determining its degree of membership in each community. Existing provably consistent algorithms often require strong assumptions about the population, are computationally expensive, and only provide an overall error bound for the whole community membership matrix. This paper provides uniform rates of convergence for the inferred community membership vector of each node in a network generated from the Mixed Membership Stochastic Blockmodel (MMSB); to our knowledge, thi…
GIPSA-lab（吉卜赛实验室）；UGA（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；L2S（信号与系统实验室）；CentraleSupélec（中央理工-高等电力学院）；University of Paris Saclay（巴黎萨克雷大学）
This article considers the problem of community detection in sparse dynamical graphs in which the community structure evolves over time. A fast spectral algorithm based on an extension of the Bethe-Hessian matrix is proposed, which benefits from the positive correlation in the class labels and in their temporal evolution and is designed to be applicable to any dynamical graph with a community structure. Under the dynamical degree-corrected stochastic block model, in the case of two classes of equal size, we demonstrate and support with extensive simulations that our proposed algorithm is capab…
Cornell University（康奈尔大学）；University of California, Davis（加州大学戴维斯分校）；University of Pennsylvania（宾夕法尼亚大学）
The stochastic block model (SBM) is a popular framework for studying community detection in networks. This model is limited by the assumption that all nodes in the same community are statistically equivalent and have equal expected degrees. The degree-corrected stochastic block model (DCSBM) is a natural extension of SBM that allows for degree heterogeneity within communities. This paper proposes a convexified modularity maximization approach for estimating the hidden communities under DCSBM. Our approach is based on a convex programming relaxation of the classical (generalized) modularity max…
We propose and analyze a generic method for community recovery in stochastic block models and degree corrected block models. This approach can exactly recover the hidden communities with high probability when the expected node degrees are of order $\log n$ or higher. Starting from a roughly correct community partition given by some conventional community recovery algorithm, this method refines the partition in a cross clustering step. Our results simplify and extend some of the previous work on exact community recovery, discovering the key role played by sample splitting. The proposed method i…
China University of Mining and Technology（中国矿业大学）；Nankai University（南开大学）
Spectral clustering methods are widely used for detecting clusters in networks for community detection, while a small change on the graph Laplacian matrix could bring a dramatic improvement. In this paper, we propose a dual regularized graph Laplacian matrix and then employ it to three classical spectral clustering approaches under the degree-corrected stochastic block model. If the number of communities is known as $K$, we consider more than $K$ leading eigenvectors and weight them by their corresponding eigenvalues in the spectral clustering procedure to improve the performance. Three improv…
GIPSA-lab（GIPSA实验室）；UGA（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立高等物理、数学与信息学院）；L2S（信号与系统实验室）；CentraleSupélec（中央理工-高等电力学院）；University of Paris Saclay（巴黎萨克雷大学）
Spectral clustering is one of the most popular, yet still incompletely understood, methods for community detection on graphs. This article studies spectral clustering based on the Bethe-Hessian matrix $H_r = (r^2-1)I_n + D-rA$ for sparse heterogeneous graphs (following the degree-corrected stochastic block model) in a two-class setting. For a specific value $r = ζ$, clustering is shown to be insensitive to the degree heterogeneity. We then study the behavior of the informative eigenvector of $H_ζ$ and, as a result, predict the clustering accuracy. The article concludes with an overview of the …
Community detection or clustering is a fundamental task in the analysis of network data. Many real networks have a bipartite structure which makes community detection challenging. In this paper, we consider a model which allows for matched communities in the bipartite setting, in addition to node covariates with information about the matching. We derive a simple fast algorithm for fitting the model based on variational inference ideas and show its effectiveness on both simulated and real data. A variation of the model to allow for degree-correction is also considered, in addition to a novel ap…
The Ohio State University（俄亥俄州立大学）；University of Georgia（佐治亚大学）
Community detection in large social networks is affected by degree heterogeneity of nodes. The D-SCORE algorithm for directed networks was introduced to reduce this effect by taking the element-wise ratios of the singular vectors of the adjacency matrix before clustering. Meaningful results were obtained for the statistician citation network, but rigorous analysis on its performance was missing. First, this paper establishes theoretical guarantee for this algorithm and its variants for the directed degree-corrected block model (Directed-DCBM). Second, this paper provides significant improvemen…
We propose a goodness-of-fit test for degree-corrected stochastic block models (DCSBM). The test is based on an adjusted chi-square statistic for measuring equality of means among groups of $n$ multinomial distributions with $d_1,\dots,d_n$ observations. In the context of network models, the number of multinomials, $n$, grows much faster than the number of observations, $d_i$, corresponding to the degree of node $i$, hence the setting deviates from classical asymptotics. We show that a simple adjustment allows the statistic to converge in distribution, under null, as long as the harmonic mean …
Microsoft Research - INRIA Joint Centre（微软研究院-INRIA联合中心）；École Normale Supérieure（巴黎高等师范学院）；INRIA Paris（INRIA巴黎）
Motivated by community detection, we characterise the spectrum of the non-backtracking matrix $B$ in the Degree-Corrected Stochastic Block Model. Specifically, we consider a random graph on $n$ vertices partitioned into two equal-sized clusters. The vertices have i.i.d. weights $\{ ϕ_u \}_{u=1}^n$ with second moment $Φ^{(2)}$. The intra-cluster connection probability for vertices $u$ and $v$ is $\frac{ϕ_u ϕ_v}{n}a$ and the inter-cluster connection probability is $\frac{ϕ_u ϕ_v}{n}b$. We show that with high probability, the following holds: The leading eigenvalue of the non-backtracking matrix …
Microsoft Research - INRIA Joint Centre（微软研究院-法国国家信息与自动化研究所联合中心）；INRIA Paris（法国国家信息与自动化研究所巴黎分部）；École Normale Supérieure（巴黎高等师范学院）
We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random graph on $n$ nodes, having i.i.d. weights $(ϕ_u)_{u=1}^n$ (possibly heavy-tailed), partitioned into $q \geq 2$ asymptotically equal-sized clusters. The model parameters are two constants $a,b > 0$ and the finite second moment of the weights $Φ^{(2)}$. Vertices $u$ and $v$ are connected by an edge with probability $\frac{ϕ_u ϕ_v}{n}a$ when they are in the same class and with probability $\frac{ϕ_u ϕ_v}{n}b$ otherwise. We prove that it is information-theoretically impossible to estimate the clusters in a way positively co…
We consider the problem of multiway clustering in the presence of unknown degree heterogeneity. Such data problems arise commonly in applications such as recommendation system, neuroimaging, community detection, and hypergraph partitions in social networks. The allowance of degree heterogeneity provides great flexibility in clustering models, but the extra complexity poses significant challenges in both statistics and computation. Here, we develop a degree-corrected tensor block model with estimation accuracy guarantees. We present the phase transition of clustering performance based on the no…
