Network inference using asynchronously updated kinetic Ising Model

Network structures are reconstructed from dynamical data by respectively naive mean field (nMF) and Thouless-Anderson-Palmer (TAP) approximations. For TAP approximation, we use two methods to reconstruct the network: a) iteration method; b) casting the inference formula to a set of cubic equations and solving it directly. We investigate inference of the asymmetric Sherrington- Kirkpatrick (S-K) model using asynchronous update. The solutions of the sets cubic equation depend of temperature T in the S-K model, and a critical temperature Tc is found around 2.1. For T < Tc, the solutions of the cubic equation sets are composed of 1 real root and two conjugate complex roots while for T > Tc there are three real roots. The iteration method is convergent only if the cubic equations have three real solutions. The two methods give same results when the iteration method is convergent. Compared to nMF, TAP is somewhat better at low temperatures, but approaches the same performance as temperature increase. Both methods behave better for longer data length, but for improvement arises, TAP is well pronounced.

💡 Research Summary

The paper addresses the problem of reconstructing the underlying interaction network of a system from time‑resolved dynamical data, focusing on a kinetic Ising model with asynchronous (random‑time) updates. The authors consider a non‑symmetric Sherrington‑Kirkpatrick (S‑K) spin glass as a testbed, where each spin flips according to a Poisson process with a rate that depends on the instantaneous local field generated by the weighted sum of its neighbors. The temperature parameter (T) controls the noise level of the dynamics.

Because exact maximum‑likelihood inference of the coupling matrix (J_{ij}) is computationally prohibitive, the study adopts two mean‑field approximations. The first is the naïve mean‑field (nMF) approach, which linearises the relationship between observed spin averages and correlations, yielding a simple linear system (\mathbf{C} = \mathbf{J}\mathbf{D}). While computationally cheap, nMF neglects higher‑order feedback and therefore performs poorly when the system is strongly coupled (low (T)).

The second approximation is the Thouless‑Anderson‑Palmer (TAP) scheme, which adds an Onsager reaction term to the mean‑field equation. For the asynchronous kinetic Ising model the TAP equation reads

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