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

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