p-star models, mean field random networks and the heat hierarchy
arXiv:2105.09479 · doi:10.1103/PhysRevE.105.014306
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.
arXiv comment12 pages, 9 figures
References in corpus (13)
- Epidemic processes in complex networks
- The statistical mechanics of networks
- On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour
- Solution for the properties of a clustered network
- A mechanical approach to mean field spin models
- Shock dynamics of phase diagrams
- On quantum and relativistic mechanical analogues in mean field spin models
- The two-star model: exact solution in the sparse regime and condensation transition
- Complete integrability of information processing by biochemical reactions
- Thermodynamic phase transitions and shock singularities
- An exact study of phase transitions in mean field Potts models
- Thermodynamic Limit and Dispersive Regularisation in Matrix Models
- Symmetric Matrix Ensemble and Integrable Hydrodynamic Chains