The discovery of community structures in social networks has gained significant attention since it is a fundamental problem in understanding the networks' topology and functions. However, most social network data are collected from partially observable networks with both missing nodes and edges. In this paper, we address a new problem of detecting overlapping community structures in the context of such an incomplete network, where communities in the network are allowed to overlap since nodes belong to multiple communities at once. To solve this problem, we introduce KroMFac, a new framework that conducts community detection via regularized nonnegative matrix factorization (NMF) based on the Kronecker graph model. Specifically, from a generative parameter matrix acquired by the expectation-maximization (EM) algorithm, we first estimate the missing part of the network. As our major contribution to the proposed framework, to improve community detection accuracy, we then characterize and select influential nodes (which tend to have high degrees) by ranking, and add them to the existing graph. Finally, we uncover the community structures by solving the regularized NMF-aided optimization problem in terms of maximizing the likelihood of the underlying graph. Furthermore, adopting normalized mutual information (NMI), we empirically show superiority of our KroMFac approach over two baseline schemes.
Network structures, such as social networks, web graphs and networks from systems biology, play important roles in many areas of science and our everyday lives. In order to study the networks one needs to first collect reliable large scale network data. While the social and information networks have become ubiquitous, the challenge of collecting complete network data still persists. Many times the collected network data is incomplete with nodes and edges missing. Commonly, only a part of the network can be observed and we would like to infer the unobserved part of the network. We address this issue by studying the Network Completion Problem: Given a network with missing nodes and edges, can we complete the missing part? We cast the problem in the Expectation Maximization (EM) framework where we use the observed part of the network to fit a model of network structure, and then we estimate the missing part of the network using the model, re-estimate the parameters and so on. We combine the EM with the Kronecker graphs model and design a scalable Metropolized Gibbs sampling approach that allows for the estimation of the model parameters as well as the inference about missing nodes and edges of the network. Experiments on synthetic and several real-world networks show that our approach can effectively recover the network even when about half of the nodes in the network are missing. Our algorithm outperforms not only classical link-prediction approaches but also the state of the art Stochastic block modeling approach. Furthermore, our algorithm easily scales to networks with tens of thousands of nodes. Distinct social networks are interconnected via membership overlap, which plays a key role when crossing information is investigated in the context of multiple-social-network analysis. Unfortunately, users do not always make their membership to two distinct social networks explicit, by specifying the so-called me edge (practically, corresponding to a link between the two accounts), thus missing a potentially very useful information. As a consequence, discovering missing me edges is an important problem to address in this context with potential powerful applications. In this paper, we propose a common-neighbor approach to detecting missing me edges, which returns good results in real-life settings. Indeed, an experimental campaign shows both that the state-of-the-art common-neighbor approaches cannot be effectively applied to our problem and, conversely, that our approach returns precise and complete results. We introduce the graph vertex similarity measure, Relation Strength Similarity (RSS), that utilizes a network's topology to discover and capture similar vertices. The RSS has the advantage that it is asymmetric; can be used in a weighted network; and has an adjustable "discovery range" parameter that enables exploration of friend of friend connections in a social network. To evaluate RSS we perform experiments on a coauthorship network from the CiteSeerX database. Our method significantly outperforms other vertex similarity measures in terms of the ability to predict future coauthoring behavior among authors in the CiteSeerX database for the near future 0 to 4 years out and reasonably so for 4 to 6 years out. Our personal social networks are big and cluttered, and currently there is no good way to organize them. Social networking sites allow users to manually categorize their friends into social circles (e.g. 'circles' on Google+, and 'lists' on Facebook and Twitter), however they are laborious to construct and must be updated whenever a user's network grows. We define a novel machine learning task of identifying users' social circles. We pose the problem as a node clustering problem on a user's ego-network, a network of connections between her friends. We develop a model for detecting circles that combines network structure as well as user profile information. For each circle we learn its members and the circle-specific user profile similarity metric. Modeling node membership to multiple circles allows us to detect overlapping as well as hierarchically nested circles. Experiments show that our model accurately identifies circles on a diverse set of data from Facebook, Google+, and Twitter for all of which we obtain hand-labeled ground-truth.