  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.
