Network inference in the nonequilibrium steady state

Nonequilibrium systems lack an explicit characterization of their steady state like the Boltzmann distribution for equilibrium systems. This has drastic consequences for the inference of the parameters of a model when its dynamics lacks detailed balance. Such nonequilibrium systems occur naturally in applications like neural networks and gene regulatory networks. Here, we focus on the paradigmatic asymmetric Ising model and show that we can learn its parameters from independent samples of the nonequilibrium steady state. We present both an exact inference algorithm and a computationally more efficient, approximate algorithm for weak interactions based on a systematic expansion around mean-field theory. Obtaining expressions for magnetizations and two- and three-point spin correlations, we establish that these observables are sufficient to infer the model parameters. Further, we discuss the symmetries characterizing the different orders of the expansion around the mean field and show how different types of dynamics can be distinguished on the basis of samples from the nonequilibrium steady state.

Figure 1

Couplings inferred from the NESS. We consider a system of $N=10$ spins with random asymmetric couplings under Glauber dynamics, (1); see the text. We plot the relative root-mean-squared reconstruction error $\epsilon$ between the inferred and the true couplings versus the coupling strength $\beta$ for different numbers of samples $M$. Reconstruction errors (a, b) for the exact inference and (c, d) for the mean-field inference. The symmetric part of the couplings ${\mathit{J}}^{\text{sym}}=(\mathit{J}+{\mathit{J}}^T)/2$ (a, c) generally has a lower reconstruction error than the antisymmetric part ${\mathit{J}}^{\text{asym}}=(\mathit{J}-{\mathit{J}}^T)/2$ (b, d).

Figure 2

External fields inferred from the NESS. We consider a system of $N=10$ spins with random asymmetric couplings under Glauber dynamics, (1); see the text. We plot the relative root-mean-squared reconstruction error $\epsilon$ between the inferred and the true external fields versus the coupling and external field strength $\beta$ for the exact inference (a) and the mean-field inference (b).

Figure 3

Detecting sequential dynamics on the basis of three-point correlations. We sample configurations from the NESS of Glauber dynamics with sequential updates; see the text. Next, we reconstruct the model parameters from the magnetizations and two- and three-point correlations, assuming that the data were generated by Glauber dynamics with sequential updates (circles), parallel updates (triangles), or equilibrium dynamics (squares). The deviations $\mathrm{\Delta }{C}_{ijk}=C_{ijk}(\mathbf{h},\mathit{J})-C_{ijk}^{\text{sampled}}$ of the three-point correlations predicted by these three models from the corresponding correlations seen in the original samples are plotted versus the sampled correlations. The relative root-mean-squared prediction errors $\parallel \mathrm{\Delta }{C}_{ijk}{\parallel }_2/{\parallel C_{ijk}^{\text{sampled}}\parallel }_2$ are $0.003,0.06$, and 0.13 for the sequential, parallel, and equilibrium dynamics, respectively, clearly favoring the dynamics with sequential updates. The horizontal line is a guide for the eye, representing a perfect fit. We used $N=10$ spins, a coupling and external field strength of $\beta =0.2$ and $M={10}^{10}N$ samples.

Figure 4

Detecting parallel dynamics on the basis of three-point correlations. We sample magnetizations and correlations from the NESS of Glauber dynamics with parallel updates, (5), in the same way as done for sequential updates; see the text. Again, model parameters are obtained under the three alternative models with parallel (circles), sequential (triangles), and equilibrium dynamics (squares). Their predictions for the three-point correlations $C_{ijk}$ are compared to those observed in the original data. Shown are the deviations $\mathrm{\Delta }{C}_{ijk}=C_{ijk}(\mathbf{h},\mathit{J})-C_{ijk}^{\text{sampled}}$ of the predicted correlations from the sampled correlations versus the sampled correlations. The relative prediction errors $\parallel \mathrm{\Delta }{C}_{ijk}{\parallel }_2/{\parallel C_{ijk}^{\text{sampled}}\parallel }_2$ are $0.10,0.25$, and 0.41 for parallel, sequential, and equilibrium dynamics, respectively, showing that indeed parallel dynamics best describes the data. The horizontal line is a guide to the eye representing a perfect fit. We used a coupling and external field strength of $\beta =0.2$ and $M={10}^{10}N$ samples.