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Asymptotic Structure of Constrained Exponential Random Graph Models

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Abstract

In this paper, we study exponential random graph models subject to certain constraints. We obtain some general results about the asymptotic structure of the model. We show that there exists non-trivial regions in the phase plane where the asymptotic structure is uniform and there also exists non-trivial regions in the phase plane where the asymptotic structure is non-uniform. We will get more refined results for the star model and in particular the two-star model for which a sharp transition from uniform to non-uniform structure, a stationary point and phase transitions will be obtained.

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Acknowledgements

The author is very grateful to two anonymous referees and the editor for helpful comments and suggestions. The author also thanks David Aristoff for helpful discussions. The author is partially supported by NSF Grant DMS-1613164.

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  1. Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL, 32306, USA

    Lingjiong Zhu

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  1. Lingjiong Zhu
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Correspondence to Lingjiong Zhu.

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Zhu, L. Asymptotic Structure of Constrained Exponential Random Graph Models. J Stat Phys 166, 1464–1482 (2017). https://doi.org/10.1007/s10955-017-1733-y

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  • DOI: https://doi.org/10.1007/s10955-017-1733-y

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