Model

To demonstrate the interdependence between configurations of variables in dynamic stochastic systems, we use a kinetic model in which the state of variable \(i\) at the time point \(t+1\), \(\sigma_i(t+1)\) \((i = 1,N)\), depends on the state of all the variables at the previous time point \(t\), \(\vec{\sigma}(t)\), in the form of the following conditional probability

\[P[\sigma_i(t+1)|\vec{\sigma}(t)] = \frac{\exp [ \sigma_i(t+1) H_i(t)]}{\mathcal{N}}\]

where \(H_i(t) = \sum_j W_{ij} \sigma_j(t)\) represents the local field, and \(\mathcal{N} = \sum_{\sigma_i(t+1)} \exp[\sigma_i(t+1) H_i(t)]\) normalizing factor. Intuitively, the state \(\sigma_i(t+1)\) tends to align with the local field \(H_i(t)\).

In the inverse problem, we infer the coupling strength between variables \(W_{ij}\) from time series data of variable configurations \(\vec{\sigma}\).