Kinetic Ising models are powerful tools for studying the non-equilibrium dynamics of complex , discrete systemsand for analyzing their experimental recordings. However, the behaviour of the model is in general not tractable for large networks ; therefore, mean field theories are frequently used to approximate its statistical properties. Many variants of the classical naive and TAP (i.e., second-order) mean field approximations have been proposed , each of which makes unique assumptions about the temporal evolution of the system's correlation structure . This disparity of methods makes it difficult to systematically advance mean field approaches beyond previous contributions. Here, we propose a unified framework for mean field theories of the dynamics of asymmetric kinetic Ising systems based on information geometry . The framework is built on Plefka expansions of the model around a simplified model obtained by an orthogonal projection to a sub-manifold of tractable probability distributions. This view not only unifies previous methods but also allows us to propose novel methods that, in contrast with traditional mean-field approaches, preserve correlations of the system, both in stationary and transient states. By comparing analytical approximations and exact numerical simulations, we show that the proposed methods provide more accurate estimates of covariance evolution in the system than classical equations, and consequently outperform previous mean field theories for solving the inverse Ising problem. In sum, our framework unifies and extends current mean-field approximations of kinetic Ising model, constituting a powerful tool for studying non-equilibrium dynamics of complex systems .
Kinetic Ising models are powerful tools for studying the non-equilibrium dynamics of complex systems. As their behavior is not tractable for large networks , many mean-field methods have been proposed for their analysis, each based on unique assumptions about the system's temporal evolution . This disparity of approaches makes it challenging to systematically advance mean-field methods beyond previous contributions. Here, we propose a unifying framework for mean-field theories of asymmetric kinetic Ising systems from an information geometry perspective . The framework is built on Plefka expansions of a system around a simplified model obtained by an orthogonal projection to a sub-manifold of tractable probability distributions. This view not only unifies previous methods but also allows us to develop novel methods that, in contrast with traditional approaches, preserve the system's correlations. We show that these new methods can outperform previous ones in predicting and assessing network properties near maximally fluctuating regimes .
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