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Critical Phenomena in Exponential Random Graphs

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Abstract

The exponential family of random graphs is one of the most promising class of network models. Dependence between the random edges is defined through certain finite subgraphs, analogous to the use of potential energy to provide dependence between particle states in a grand canonical ensemble of statistical physics. By adjusting the specific values of these subgraph densities, one can analyze the influence of various local features on the global structure of the network. Loosely put, a phase transition occurs when a singularity arises in the limiting free energy density, as it is the generating function for the limiting expectations of all thermodynamic observables. We derive the full phase diagram for a large family of 3-parameter exponential random graph models with attraction and show that they all consist of a first order surface phase transition bordered by a second order critical curve.

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Acknowledgements

The author gratefully acknowledges the support of the National Science Foundation through two international travel grants, which enabled her to attend the 8th World Congress on Probability and Statistics and the 17th International Congress on Mathematical Physics, where she had the opportunity to discuss this work. She is also thankful to the anonymous referees for their useful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas, Austin, TX, 78712, USA

    Mei Yin

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  1. Mei Yin
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Correspondence to Mei Yin.

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Mei Yin’s research was partially supported by NSF grant DMS-1308333.

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Yin, M. Critical Phenomena in Exponential Random Graphs. J Stat Phys 153, 1008–1021 (2013). https://doi.org/10.1007/s10955-013-0874-x

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