Abstract
We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together with a complete characterization of the phase diagram of the model. We borrow tools from statistical mechanics to obtain limit theorems for the edge density. First, we investigate the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases. Then, we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a mean-field approximation of the model. Some of our results can be extended with no substantial changes to more general classes of exponential random graphs.
Funding Statement
A. Bianchi was supported in part by Università di Padova, through the BIRD project 239937 “Stochastic dynamics on graphs and random structures”. A. Bianchi and F. Collet are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM) and acknowledge partial support from the projects “Moment Problem techniques for particle systems and packing graph” and “Ferromagnetism versus synchronization: how does disorder destroy universality?”
Acknowledgments
The authors thank Diego Alberici, Marco Formentin and Richard C. Kraaij for useful discussions and suggestions. We also acknowledge the anonymous referees for carefully reading the first version of the manuscript and making several valuable comments.
Citation
Alessandra Bianchi. Francesca Collet. Elena Magnanini. "Limit theorems for exponential random graphs." Ann. Appl. Probab. 34 (5) 4863 - 4898, October 2024. https://doi.org/10.1214/24-AAP2084
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