p-star models, mean field random networks and the heat hierarchy

Gino Biondini; Antonio Moro; Barbara Prinari; Oleg Senkevich

doi:10.1103/physreve.105.014306

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arXiv · 2105.09479

p-star models, mean field random networks and the heat hierarchy

Gino Biondini·Antonio Moro·Barbara Prinari·Oleg Senkevich

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Abstract

We consider the mean field analog of the p-star model for homogeneous random networks, and compare its behaviour with that of the p-star model and its classical mean field approximation in the thermodynamic regime. We show that the partition function of the mean field model satisfies a sequence of partial differential equations known as the heat hierarchy, and the models connectance is obtained as a solution of a hierarchy of nonlinear viscous PDEs. In the thermodynamic limit, the leading order solution develops singularities in the space of parameters that evolve as classical shocks regularised by a viscous term. Shocks are associated with phase transitions and stable states are automatically selected consistently with the Maxwell construction. The case p = 3 is studied in detail. Monte Carlo simulations show an excellent agreement between the p-star model and its mean field analog at the macroscopic level, although significant discrepancies arise when local features are compared.

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Gino Biondini, Antonio Moro, Barbara Prinari, Oleg Senkevich. 2021-05-19. p-star models, mean field random networks and the heat hierarchy. https://doi.org/10.1103/physreve.105.014306

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