Mean field theories and differential identities for multispecies Ising models and exponential random graph models

Senkevich, Oleg (2022) Mean field theories and differential identities for multispecies Ising models and exponential random graph models. Doctoral thesis, Northumbria University.

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Abstract

This work is concerned with the mean field theories (MFTs) of multispecies Ising models and various probabilistic ensembles of graphs known as exponential random graph models (ERGMs). The MFT is a universal approximation, in which the true Hamiltonian of the model is linearised by introducing self-consistent mean fields, and it turns out that the mean field self-consistency equations can be obtained as low viscosity solutions of certain viscous partial differential equations (PDEs) that arise from differential identities obeyed by the Helmholtz free energies of the socalled mean field models, whose exact thermodynamic solutions coincide with the mean field self-consistency equations.

Thermodynamic equations of state are obtained for the multi component analogue of the Curie- Weiss (CW) model and analysed in detail for the 2-component case. This analysis largely extends the preceding works by providing a good orientation in the parameter space of the 2-component CW model, which reveals that, unlike the original CW model, the 2-component model admits critical points in nonzero fields and can exhibit any number from one to four of (meta)stable macrostates. The results are confirmed by Monte-Carlo (MC) simulations, and some applications of the model are discussed.

The above discussion is followed by the mean field analysis of a particular class of ERGMs, known as homogeneousMarkov random graphs. Such models are precisely described by the MFT at large sizes due to their infinite-dimensional nature, and this work provides a simple unified approach to study such models at the macroscopic level, reveals a possibility of (meta)stable macrostates of moderate connectance at arbitrarily low temperatures, and gives a general result relating the order of the interactions with the maximum number of (meta)stable macrostates. MC simulations show that, as expected, the theory seems to be exact for macroscopic observables of large homogeneous ERGMs, but often fails at the microscopic level or for the models of small size. The results suggest that the long-tailed distributions, common in real-world networks, cannot be fully explained by the spontaneous symmetry breaking in homogeneous ERGMs, and for this reason, the heterogeneous ERGM, based on the multicomponent CW model, is introduced and discussed.

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