A survey of statistical network models

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference

The Discrete Infinite Logistic Normal Distribution

We present the discrete infinite logistic normal distribution (DILN), a Bayesian nonparametric prior for mixed membership models. DILN is a generalization of the hierarchical Dirichlet process (HDP) that models correlation structure between the weights of the atoms at the group level. We derive a representation of DILN as a normalized collection of gamma-distributed random variables, and study its statistical properties. We consider applications to topic modeling and derive a variational inference algorithm for approximate posterior inference. We study the empirical performance of the DILN topic model on four corpora, comparing performance with the HDP and the correlated topic model (CTM). To deal with large-scale data sets, we also develop an online inference algorithm for DILN and compare with online HDP and online LDA on the Nature magazine, which contains approximately 350,000 articles.Comment: This paper will appear in Bayesian Analysis. A shorter version of this paper appeared at AISTATS 2011, Fort Lauderdale, FL, US

The analysis of social network data: an exciting frontier for statisticians

The catalyst for this paper is the recent interest in the relationship between social networks and an individual's health, which has arisen following a series of papers by Nicholas Christakis and James Fowler on person- to-person spread of health behaviors. In this issue, they provide a detailed explanation of their methods that offers insights, justifications, and responses to criticisms [1]. In this paper, we introduce some of the key statistical methods used in social network analysis and indicate where those used by Christakis and Fowler (CF) fit into the general framework. The intent is to provide the background necessary for readers to be able to make their own evaluation of the work by CF and understand the challenges of research involving social networks. We entertain possible solutions to some of the difficulties encountered in accounting for confounding effects in analyses of peer effects and provide comments on the contributions of CF