A Survey of Statistical Network Models

Abstract

Bibliography 65ii Preface 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.

Exchangeable Random Measures for Sparse and Modular Graphs with Overlapping Communities

“…Of particular importance is the elaboration of random-graph models, which can capture the salient properties of real world graphs. Following the seminal work of Erdös and Rényi (1959), various network models have been proposed; see the overviews of Newman (2009), Kolaczyk (2009) or Goldenberg et al (2010). In particular, a large body of the literature has concentrated on models that can capture some modular or community structure within the network.…”

Review of statistical network analysis: models, algorithms, and software

“…Other reviews and collections that give excellent overviews of social network analysis and statistical modeling of network data include refs [12][13][14][15][16]. While these offer summaries of statistical network modeling that are possibly more comprehensive than our own, we offer this document as an alternative and more concise paper with a focus on brevity and on the relationships between various statistical network models.…”

Random-Walk Models of Network Formation and Sequential Monte Carlo Methods for Graphs

“…These hold for the random-walk models but also for many other network formation models, such as PA graphs, fitness models and vertex copying models (e.g. Newman (2009) and Goldenberg et al (2010)). Despite the attention that these models have received in the literature, little work exists on inference (see Section 4.5 for references).…”

“…Tools that have been developed for such problems include, for example, graphon models (Borgs et al, 2008;Hoff et al, 2002;Bickel et al, 2011;Ambroise and Matias, 2012;Wolfe and Olhede, 2013;Gao et al, 2015), the subfamily of stochastic block models (e.g. Goldenberg et al (2010)), preferential attachment graphs (Barabási and Albert, 1999) and interacting particle models (Liggett, 2005). In this context, a statistical model is a family P = {P θ , θ ∈ T} of probability distributions on graphs, and sample data are explained either as a single graph drawn from the model, or as a subgraph of such a graph.…”

“…(a) graphon models (as mentioned above) (these have recently received considerable attention in statistics and include stochastic block models (Goldenberg et al, 2010), and many models that are popular in machine learning (see Orbanz and Roy (2015)); (b) models that randomize a graph to match a given degree sequence, such as the configuration model (Durrett, 2006); (c) models describing a formation process, such as PA models (Barabási and Albert, 1999).…”