Variational perturbation and extended Plefka approaches to dynamics on random networks: the case of the kinetic Ising model

TL;DR

This paper introduces novel variational and extended Plefka methods to analyze the dynamics of the kinetic Ising model on random networks, improving accuracy over traditional mean field approaches especially in complex coupling regimes.

Contribution

It develops a variational perturbation and an extended Plefka expansion that go beyond saddle point approximations for the kinetic Ising model, with demonstrated superior performance.

Findings

Extended Plefka expansion outperforms other approximations across regimes.

Unconstrained variational approach is less effective at small couplings.

Methods are validated on Sherrington-Kirkpatrick type couplings.

Abstract

We describe and analyze some novel approaches for studying the dynamics of Ising spin glass models. We first briefly consider the variational approach based on minimizing the Kullback-Leibler divergence between independent trajectories and the real ones and note that this approach only coincides with the mean field equations from the saddle point approximation to the generating functional when the dynamics is defined through a logistic link function, which is the case for the kinetic Ising model with parallel update. We then spend the rest of the paper developing two ways of going beyond the saddle point approximation to the generating functional. In the first one, we develop a variational perturbative approximation to the generating functional by expanding the action around a quadratic function of the local fields and conjugate local fields whose parameters are optimized. We derive analytical expressions for the optimal parameters and show that when the optimization is suitably restricted, we recover the mean field equations that are exact for the fully asymmetric random couplings (M\'ezard and Sakellariou, 2011). However, without this restriction the results are different. We also describe an extended Plefka expansion in which in addition to the magnetization, we also fix the correlation and response functions. Finally, we numerically study the performance of these approximations for Sherrington-Kirkpatrick type couplings for various coupling strengths, degrees of coupling symmetry and external fields. We show that the dynamical equations derived from the extended Plefka expansion outperform the others in all regimes, although it is computationally more demanding. The unconstrained variational approach does not perform well in the small coupling regime, while it approaches dynamical TAP equations of (Roudi and Hertz, 2011) for strong couplings.