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Mean Field Theory For Non-Equilibrium Network Reconstruction

There has been recent progress on the problem of inferring the structure of interactions in complex networks when they are in stationary states satisfying detailed balance, but little has been done for non-equilibrium systems. Here we introduce an approach to this problem, considering, as an example, the question of recovering the interactions in an asymmetrically-coupled, synchronously-updated Sherrington-Kirkpatrick model. We derive an exact iterative inversion algorithm and develop efficient approximations based on dynamical mean-field and Thouless-Anderson-Palmer equations that express the interactions in terms of equal-time and one time step-delayed correlation functions.