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 subsets of these communities, and the resultant networks due to within- and cross-community interactions. Furthermore, to automatically discover these latent structures from social networks, we develop a Gibbs sampling algorithm for our model. We conduct extensive validation of our model using synthetic networks, and demonstrate the utility of our model in real-world datasets such as predator-prey networks and citation networks.
While there has been a great deal of work on graph clustering and community structure inference @cite_11 @cite_10 @cite_17 @cite_18 @cite_6 , probabilistic generative models for hierarchical community formation and latent multi-role inference of every actor and link have just started to draw attention.
@cite_18 @cite_11 @cite_6 @cite_10 @cite_17
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We provide the first information theoretical tight analysis for inference of latent community structure given a sparse graph along with high dimensional node covariates, correlated with the same latent communities. Our work bridges recent theoretical breakthroughs in detection of latent community structure without nodes covariates and a large body of empirical work using diverse heuristics for combining node covariates with graphs for inference. The tightness of our analysis implies in particular, the information theoretic necessity of combining the different sources of information. Our analysis holds for networks of large degrees as well as for a Gaussian version of the model. There has been much interest in unsupervised learning of hierarchical generative models such as deep belief networks. Scaling such models to full-sized, high-dimensional images remains a difficult problem. To address this problem, we present the convolutional deep belief network, a hierarchical generative model which scales to realistic image sizes. This model is translation-invariant and supports efficient bottom-up and top-down probabilistic inference. Key to our approach is probabilistic max-pooling, a novel technique which shrinks the representations of higher layers in a probabilistically sound way. Our experiments show that the algorithm learns useful high-level visual features, such as object parts, from unlabeled images of objects and natural scenes. We demonstrate excellent performance on several visual recognition tasks and show that our model can perform hierarchical (bottom-up and top-down) inference over full-sized images. Graph clustering has been widely applied in exploring regularities emerging in relational data. Recently, the rapid development of network theory correlates graph clustering with the detection of community structure, a common and important topological characteristic of networks. Most existing methods investigate the community structure at a single topological scale. However, as shown by empirical studies, the community structure of real world networks often exhibits multiple topological descriptions, corresponding to the clustering at different resolutions. Furthermore, the detection of multiscale community structure is heavily affected by the heterogeneous distribution of node degree. It is very challenging to detect multiscale community structure in heterogeneous networks. In this paper, we propose a novel, unified framework for detecting community structure from the perspective of dimensionality reduction. Based on the framework, we first prove that the well-known Laplacian matrix for network partition and the widely-used modularity matrix for community detection are two kinds of covariance matrices used in dimensionality reduction. We then propose a novel method to detect communities at multiple topological scales within our framework. We further show that existing algorithms fail to deal with heterogeneous node degrees. We develop a novel method to handle heterogeneity of networks by introducing a rescaling transformation into the covariance matrices in our framework. Extensive tests on real world and artificial networks demonstrate that the proposed correlation matrices significantly outperform Laplacian and modularity matrices in terms of their ability to identify multiscale community structure in heterogeneous networks. There has been much interest in unsupervised learning of hierarchical generative models such as deep belief networks (DBNs); however, scaling such models to full-sized, high-dimensional images remains a difficult problem. To address this problem, we present the convolutional deep belief network, a hierarchical generative model that scales to realistic image sizes. This model is translation-invariant and supports efficient bottom-up and top-down probabilistic inference. Key to our approach is probabilistic max-pooling, a novel technique that shrinks the representations of higher layers in a probabilistically sound way. Our experiments show that the algorithm learns useful high-level visual features, such as object parts, from unlabeled images of objects and natural scenes. We demonstrate excellent performance on several visual recognition tasks and show that our model can perform hierarchical (bottom-up and top-down) inference over full-sized images.