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

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 (Mézard and Sakellariou 2011 J. Stat. Mech. 2011 L07001). 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 and the degrees of coupling symmetry, for both temporally constant but random, as well as time varying 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 J. Stat. Mech. 2011 P03031) for strong couplings.