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 statistical relational learning model, hinge-loss Markov random fields (HL-MRFs), as a structured prior to model complex dependencies among node attributes, multi-relational links, and their relationship with mixed-membership distributions. Our model is specified using a probabilistic programming templating language that uses weighted first-order logic rules, which enhances the model's interpretability. Further, our model is capable of learning latent characteristics in real-world networks via meaningful latent variables encoded as a complex combination of observed features and membership distributions. We present an expectation-maximization based inference algorithm that learns latent variables and parameters iteratively, a scalable stochastic variation of the inference algorithm, and a method to learn the weights of HL-MRF structured priors. We evaluate our model on six datasets across three different types of networks and corresponding modeling scenarios and demonstrate that our models are able to achieve an improvement of 15\% on average in test log-likelihood and faster convergence when compared to state-of-the-art network models.
Graduate School of System Informatics, Kobe University（神户大学系统信息学研究科）
Many kinds of data can be represented as a network or graph. It is crucial to infer the latent structure underlying such a network and to predict unobserved links in the network. Mixed Membership Stochastic Blockmodel (MMSB) is a promising model for network data. Latent variables and unknown parameters in MMSB have been estimated through Bayesian inference with the entire network; however, it is important to estimate them online for evolving networks. In this paper, we first develop online inference methods for MMSB through sequential Monte Carlo methods, also known as particle filters. We the…
University of Southern California（南加州大学）；University of Montreal（蒙特利尔大学）；University of Amsterdam（阿姆斯特丹大学）
We propose a stochastic gradient Markov chain Monte Carlo (SG-MCMC) algorithm for scalable inference in mixed-membership stochastic blockmodels (MMSB). Our algorithm is based on the stochastic gradient Riemannian Langevin sampler and achieves both faster speed and higher accuracy at every iteration than the current state-of-the-art algorithm based on stochastic variational inference. In addition we develop an approximation that can handle models that entertain a very large number of communities. The experimental results show that SG-MCMC strictly dominates competing algorithms in all cases.
Actors in realistic social networks play not one but a number of diverse roles depending on whom they interact with, and a large number of such role-specific interactions collectively determine social communities and their organizations. Methods for analyzing social networks should capture these multi-faceted role-specific interactions, and, more interestingly, discover the latent organization or hierarchy of social communities. We propose a hierarchical Mixed Membership Stochastic Blockmodel to model the generation of hierarchies in social communities, selective membership of actors to subset…
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…
Directional and pairwise measurements are often used to model inter-relationships in a social network setting. The Mixed-Membership Stochastic Blockmodel (MMSB) was a seminal work in this area, and many of its capabilities were extended since then. In this paper, we propose the \emph{Dynamic Infinite Mixed-Membership stochastic blockModel (DIM3)}, a generalised framework that extends the existing work to a potentially infinite number of communities and mixture memberships for each of the network's nodes. This model is in a dynamic setting, where additional model parameters are introduced to re…
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…
Researchers have extensively studied the social mechanisms that drive the formation of networks observed among preschool children. However, less attention has been given to global network structures in terms of blockmodels. A blockmodel is a network where the nodes are groups of equivalent units (according to links to others) from a studied network. Cugmas et al. (2019) showed that mutuality, popularity, assortativity, and different types of transitivity mechanisms can lead the global network structure to the proposed asymmetric core-cohesive blockmodel. Yet, they did not provide any evidence …
We consider the probabilistic analogue to neural network matrix factorization (Dziugaite & Roy, 2015), which we construct with Bayesian neural networks and fit with variational inference. We find that a linear model fit with variational inference can attain equivalent predictive performance to the regular neural network variants on the Movielens data sets. We discuss the implications of this result, which include some suggestions on the pros and cons of using the neural network construction, as well as the variational approach to inference. Such a probabilistic approach is required, however, w…
The \emph{Mixed-Membership Stochastic Blockmodel (MMSB)} is a popular framework for modeling social network relationships. It can fully exploit each individual node's participation (or membership) in a social structure. Despite its powerful representations, this model makes an assumption that the distributions of relational membership indicators between two nodes are independent. Under many social network settings, however, it is possible that certain known subgroups of people may have high or low correlations in terms of their membership categories towards each other, and such prior informati…
This paper addresses the question of whether it is possible to generate networks with a given global structure (defined by selected blockmodels, i.e., cohesive, core-periphery, hierarchical and transitivity), considering only different types of triads. Two methods are used to generate networks: (i) the method of relocating links; and (ii) the Monte Carlo Multi Chain algorithm implemented in the "ergm" package implemented in R. Although all types of triads can generate networks with the selected blockmodel types, the selection of only a subset of triads improves the generated networks' blockmod…
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…
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…
We propose a new dynamic stochastic blockmodel that focuses on the analysis of interaction lengths in networks. The model does not rely on a discretization of the time dimension and may be used to analyze networks that evolve continuously over time. The framework relies on a clustering structure on the nodes, whereby two nodes belonging to the same latent group tend to create interactions and non-interactions of similar lengths. We introduce a fast variational expectation-maximization algorithm to perform inference, and adapt a widely used clustering criterion to perform model choice. Finally,…
Dáil Éireann is the principal chamber of the Irish parliament. The 31st Dáil Éireann is the principal chamber of the Irish parliament. The 31st Dáil was in session from March 11th, 2011 to February 6th, 2016. Many of the members of the Dáil were active on social media and many were Twitter users who followed other members of the Dáil. The pattern of following amongst these politicians provides insights into political alignment within the Dáil. We propose a new model, called the generalized latent space stochastic blockmodel, which extends and generalizes both the latent space model and the sto…
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…
Many networks are complex dynamical systems, where both attributes of nodes and topology of the network (link structure) can change with time. We propose a model of co-evolving networks where both node at- tributes and network structure evolve under mutual influence. Specifically, we consider a mixed membership stochastic blockmodel, where the probability of observing a link between two nodes depends on their current membership vectors, while those membership vectors themselves evolve in the presence of a link between the nodes. Thus, the network is shaped by the interaction of stochastic proc…
Collective classification models attempt to improve classification performance by taking into account the class labels of related instances. However, they tend not to learn patterns of interactions between classes and/or make the assumption that instances of the same class link to each other (assortativity assumption). Blockmodels provide a solution to these issues, being capable of modelling assortative and disassortative interactions, and learning the pattern of interactions in the form of a summary network. The Supervised Blockmodel provides good classification performance using link struct…
We discuss two views on extending existing methods for complex network modeling which we dub the communities first and the networks first view, respectively. Inspired by the networks first view that we attribute to White, Boorman, and Breiger (1976)[1], we formulate the multiple-networks stochastic blockmodel (MNSBM), which seeks to separate the observed network into subnetworks of different types and where the problem of inferring structure in each subnetwork becomes easier. We show how this model is specified in a generative Bayesian framework where parameters can be inferred efficiently usi…
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 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…
DaSciM, LIX, École Polytechnique, Institute Polytechnique de Paris（巴黎综合理工学院数据科学与机器学习中心、计算机科学实验室、巴黎理工学院）；Noah’s Ark Lab, Huawei Technologies France（华为技术法国诺亚方舟实验室）；AUEB（雅典经济与商业大学）
In many domains data is currently represented as graphs and therefore, the graph representation of this data becomes increasingly important in machine learning. Network data is, implicitly or explicitly, always represented using a graph shift operator (GSO) with the most common choices being the adjacency, Laplacian matrices and their normalisations. In this paper, a novel parametrised GSO (PGSO) is proposed, where specific parameter values result in the most commonly used GSOs and message-passing operators in graph neural network (GNN) frameworks. The PGSO is suggested as a replacement of the…
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…
Mean-field variational inference (MFVI) has been widely applied in large scale Bayesian inference. However MFVI, which assumes a product distribution on the latent variables, often leads to objective functions with many local optima, making optimization algorithms sensitive to initialization. In this paper, we study the advantage of structured variational inference for the two class Stochastic Blockmodel. The variational distribution is constructed to have pairwise dependency structure on the nodes of the network. We prove that, in a broad density regime and for general random initializations,…
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…
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…
In this paper we address the problem of modeling relational data, which appear in many applications such as social network analysis, recommender systems and bioinformatics. Previous studies either consider latent feature based models but disregarding local structure in the network, or focus exclusively on capturing local structure of objects based on latent blockmodels without coupling with latent characteristics of objects. To combine the benefits of the previous work, we propose a novel model that can simultaneously incorporate the effect of latent features and covariates if any, as well as …
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…
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…
Observations consisting of measurements on relationships for pairs of objects arise in many settings, such as protein interaction and gene regulatory networks, collections of author-recipient email, and social networks. Analyzing such data with probabilisic models can be delicate because the simple exchangeability assumptions underlying many boilerplate models no longer hold. In this paper, we describe a latent variable model of such data called the mixed membership stochastic blockmodel. This model extends blockmodels for relational data to ones which capture mixed membership latent relationa…
With the emergence of social networking services, researchers enjoy the increasing availability of large-scale heterogenous datasets capturing online user interactions and behaviors. Traditional analysis of techno-social systems data has focused mainly on describing either the dynamics of social interactions, or the attributes and behaviors of the users. However, overwhelming empirical evidence suggests that the two dimensions affect one another, and therefore they should be jointly modeled and analyzed in a multi-modal framework. The benefits of such an approach include the ability to build b…
College of William & Mary（威廉玛丽学院）；University of Oxford（牛津大学）
Discovering statistical structure from links is a fundamental problem in the analysis of social networks. Choosing a misspecified model, or equivalently, an incorrect inference algorithm will result in an invalid analysis or even falsely uncover patterns that are in fact artifacts of the model. This work focuses on unifying two of the most widely used link-formation models: the stochastic blockmodel (SBM) and the small world (or latent space) model (SWM). Integrating techniques from kernel learning, spectral graph theory, and nonlinear dimensionality reduction, we develop the first statistical…
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…
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…
University of Wisconsin（威斯康星大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
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 provide…
