Article
Introduction to papers on the modeling and analysis of network dataII
The Annals of Applied Statistics (Impact Factor: 1.46). 11/2010; 4(2). DOI: 10.1214/10AOAS365
Source: arXiv
ABSTRACT
Introduction to papers on the modeling and analysis of network dataII Comment: Published in at http://dx.doi.org/10.1214/10AOAS365 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)
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 "Random graphs have been widely studied (see [1] [2] for surveys of recent work) since the pioneering work on the independent case. The first serious attempt was made by Solomonoff and Rapoport [3] in the early 1950s, who proposed the 'random net' model in their investigation into mathematical biology. "
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ABSTRACT: The exponential family of random graphs is among the most widelystudied network models. We show that any exponential random graph model may alternatively be viewed as a lattice gas model with a finite Banach space norm. The system may then be treated by cluster expansion methods from statistical mechanics. In particular, we derive a convergent power series expansion for the limiting free energy in the case of small parameters. Since the free energy is the generating function for the expectations of other random variables, this characterizes the structure and behavior of the limiting network in this parameter region. 
 "More specifically we will consider exponential random graphs in which dependence between the random edges is defined through some finite graph, in imitation of the use of potential energy to provide dependence between particle states in a grand canonical ensemble of statistical physics. Exponential random graphs have been widely studied (see [3] [4] for a range of recent work) since the pioneering work on the independent case by Erd˝ os and Rényi [2]. We will concentrate on the phenomenon of phase transitions which can emerge for dependent variables. "
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ABSTRACT: We derive the full phase diagram for a large family of exponential random graph models, each containing a first order transition curve ending in a critical point. 
 "Graph and network data are increasingly common and a host of statistical methods have emerged in recent years. Entry to this large literature may be had from the research papers and surveys in Fienberg [21] [22]. One mainstay of the emerging theory are the 2000 Mathematics Subject Classification. "
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ABSTRACT: We introduce a method for the theoretical analysis of exponential random graph models. The method is based on a large deviations approximation to the normalizing constant shown to be consistent using theory developed by Chatterjee and Varadhan. The theory explains a host of difficulties encountered by applied workers: many distinct models have essentially the same MLE, rendering the problems "practically" illposed. We give the first rigorous proofs of "degeneracy" observed in these models. Here, almost all graphs have essentially no edges or are essentially complete. We supplement recent work of Bhamidi, Bresler and Sly showing that for many models, the extra sufficient statistics are useless: most realizations look like the results of a simple ErdosRenyi model. We also find classes of models where the limiting graphs differ from ErdosRenyi graphs. A limitation of our approach, inherited from the limitation of graph limit theory, is that it works only for dense graphs.