Introduction to papers on the modeling and analysis of network data---II

11/2010; DOI: 10.1214/10-AOAS365
Source: arXiv

ABSTRACT Introduction to papers on the modeling and analysis of network data---II Comment: Published in at the Annals of Applied Statistics ( by the Institute of Mathematical Statistics (

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    ABSTRACT: The exponential family of random graphs is among the most widely-studied 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.
    Journal of Statistical Mechanics Theory and Experiment 02/2012; 2012(05). · 1.87 Impact Factor
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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.
    The Annals of Applied Probability 08/2011; · 1.37 Impact Factor
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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" ill-posed. 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 Erdos-Renyi model. We also find classes of models where the limiting graphs differ from Erdos-Renyi graphs. A limitation of our approach, inherited from the limitation of graph limit theory, is that it works only for dense graphs.


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