Effect of coupling asymmetry on mean-field solutions of direct and inverse Sherrington-Kirkpatrick model

We study how the degree of symmetry in the couplings influences the performance of three mean-field methods used for solving the direct and inverse problems for generalized Sherrington-Kirkpatrick models. In this context, the direct problem predicts the potentially time-varying magnetizations. The three theories include the first- and second-order Plefka expansions, referred to as naive mean field (nMF) and TAP, respectively, and a mean-field theory which is exact for fully asymmetric couplings. We call the last of these simply MF theory. We show that for the direct problem, nMF performs worse than the other two approximations, TAP outperforms MF when the coupling matrix is nearly symmetric, while MF works better when it is strongly asymmetric. For the inverse problem, MF performs better than both TAP and nMF, although an ad hoc adjustment of TAP can make it comparable to MF. For high temperatures the performance of TAP and MF approach each other.