Fast Detection of Overlapping Communities via Online Tensor Methods

We present a fast tensor-based approach for detecting hidden overlapping communities under the mixed membership stochastic block (MMSB) model. We present two implementations, viz., a GPU-based implementation which exploits the parallelism of SIMD architectures and a CPU-based implementation for larger datasets, where the GPU memory does not suffice. Our GPU-based implementation involves a careful optimization of storage and communication, while in our CPU-based implementation, we perform sparse linear algebraic operations to exploit the data sparsity. We use stochastic gradient descent for multilinear spectral optimization and this allows for flexibility in the tradeoff between node sub-sampling and accuracy of the results. We validate our results on datasets from Facebook, Yelp and DBLP, where ground truth is available, using notions of p-values and false discovery rates, and obtain high accuracy for membership recovery. We compare our results, both in terms of execution time and accuracy, to the state-of-the-art algorithms such as the variational method, and report many orders of magnitude gain in the execution time. For instance, for DBLP dataset with about a million nodes and 16 million edges, the execution time is about two minutes. 1 Summary of Contributions Studying community formation is an important problem in social networks. A community generally refers to a group of individuals with shared interests or beliefs (e.g. music, sports, religion), or relationships (e.g. friends, co-workers). In a social network, we can typically observe and measure the interactions among the actors, but not the communities they belong to. A challenging problem is then to estimate the communities of the actors using only their observed interactions. In general, actors can participate in multiple communities, and detecting overlapping communities is even more challenging. Our goal is to design algorithms which can accurately detect overlapping communities, and yet be easily parallelizable for fast and scalable implementation on large graphs with millions of nodes. Moreover, we learn a probabilistic community model which allows us to carry out prediction tasks such as link classification. In this work, we present a fast approach for detecting overlapping communities under the mixed membership stochastic block model (MMSB) [1]. It is based on estimating tensors from subgraph counts such as 3-stars in the observed network, and then performing linear algebraic operations (e.g. SVD), and an iterative stochastic gradient descent method for tensor decomposition using implicit trilinear operations. We present two implementations, viz., a GPU-