In this framework, the entries in Z are within [-1, 1]. In the context of stochastic block model, there is also an assignment matrix Z with entries {0, 1} (see Rohe et al 2011 Spectral clustering and the high-dimensional stochastic blockmodel). How does Z here (in the special binary case) compare with the Z matrix in Rohe et al. 2011? Why is it not allowed for all variables to be assigned to one block? It's possible that one wants to analyze the interaction network among genes from the same pathway (the same block).

Network regression models, where the outcome comprises the valued edge in a network and the predictors are actor or dyad-level covariates, are used extensively in the social and biological sciences. Valid inference relies on accurately modeling the residual dependencies among the relations. Frequently homogeneity assumptions are placed on the errors which are commonly incorrect and ignore critical, natural clustering of the actors. In this work, we present a novel regression modeling framework that models the errors as resulting from a community-based dependence structure and exploits the subsequent exchangeability properties of the error distribution to obtain parsimonious standard errors for regression parameters.

Dynamic text networks have been widely studied in recent years, primarily because the Internet stores textual data in a way that allows links between different documents. Articles on the Wikipedia (Hoffman et al., 2010), citation networks in journal articles (Moody, 2004), and linked blog posts (Latouche et al., 2011) are examples of dynamic text networks, or networks of documents that are generated over time. But each application has idiosyncratic features, such as the structure of the links and the nature of the time varying documents, so analysis typically requires bespoke models that directly address those aspects.

The stochastic blockmodel (SBM) is a generative model for network data introduced in Holland et al. (1983). The SBM is a member of the general class of latent position random graph models introduced in Hoff et al. (2002). These models have been used in various application domains as diverse as social networks (vertices may represent people with edges indicating social interaction), citation networks (who cites whom), connectomics (brain connectivity networks; vertices may represent neurons with edges indicating axon-synapse-dendrite connections, or vertices may represent brain regions with edges indicating connectivity between regions), and many others. For comprehensive reviews of statistical models and applications, see Fienberg (2010), Goldenberg et al. (2010), Fienberg (2012). In general, statistical inference on graphs is becoming essential in many areas of science, engineering, and business. The SBM supposes that each of n vertices is assigned to one of K blocks. The probability of an 1 edge between two vertices depends only on their respective block memberships, and the presence of edges are conditionally independent given block memberships.

Online advertising is an important and huge industry. Having knowledge of the website attributes can contribute greatly to business strategies for ad-targeting, content display, inventory purchase or revenue prediction. In this paper, we introduce a stochastic blockmodeling for the website relations induced by the event of online user visitation. We propose two clustering algorithms to discover the intrinsic structures of websites, and compare the performance with a goodness-of-fit method and a deterministic graph partitioning method. We demonstrate the effectiveness of our algorithms on both simulation and AOL website dataset.

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