Fig. 1. The scatter plots for the true tested couplings versus the reconstructed ones. (a) Reconstruction for the symmetric SK model with k = 0; (b) inference for the asymmetric SK model with k = 1. Red dots, inferred couplings with asynchronous nMF approximation; black dots, inferred ones with equilibrium nMF approximation. The recovered asynchronous J_ij’s in (a) are symmetrized while no symmetrization for them in (b). The other parameters for both panels are g = 0.3, N = 20, θ = 0, L = 20 × 10⁷.

Fig. 2. Mean square error (ε) versus (a) data length L, (b) system size N, (c) external field θ and (d) temperature 1/g. Black squares show nMF, red circles, TAP, blue up triangle SHO and pink down triangle AVE respectively. The parameters are g = 0.3, N = 20, θ = 0, L = 10⁷ except when varied in a panel.

Fig. 3. Inferred asynchronous versus equilibrium couplings for retinal data. Red open dots show the self-couplings which by convention are equal to zero for the equilibrium model.

Fig. 4. Traded volume data for the stock of Fannie Mae (FNM), a mortgage company. Black line for time series of traded volumes V_i(t), red for summed volumes during time interval Δt, blue for the threshold Vith=χViav×Δt. Parameters: Δt = 50 s and χ = 1.

Fig. 5. Histograms of inferred couplings by equilibrium nMF and re-scaled asynchronous nMF. Black squares for re-scaled N(J_asyn) to have the same standard deviation as N(J_eq). Here χ = 0.5 and Δt = 200 s for both the methods.

Fig. 6. Histograms of the eigenvalues of the equal time connected correlation matrix. Parameters: χ = 0.5 and Δt = 100 s.

Fig. 7. Inferred financial networks, showing only the largest interaction strengths (proportional to the width of links and arrows). Colors are indicative, and chosen by a modularity-based community detection algorithm.^[16] Parameters: χ = 0.5 and Δt = 100 s. (a) Equilibrium inference (the figure originates from Ref. [86]). (b) Asynchronous inference with τ = 20 s.

Fig. 8. (a) Temporal behavior of all allele frequencies defined as f_i[1]. Data recorded every 5 generations. (b) An example of pairwise correlation changing with time. With finite population size, there exists strong fluctuations in the system. (c) Scatter plot for the reconstructed against the tested fitness with DCA-nMF (red dots) and DCA-PLM (blue dots) algorithm for J_ij’s. Parameters: the number of loci L = 25, the number of individuals N = 200, mutation rate μ = 0.01, recombination rate r = 0.1, crossover rate ρ = 0.5, standard deviation of epistatic fitness σ = 0.002.

Fig. 9. Phase diagram for epistatic fitness recovery with DCA-nMF J_ij’s from the average of singletime data. (a) Mutation rate μ versus recombination rate r. For large recombination while low mutation, KNS inference does not work. However, for small r, the KNS inference theory does not satisfied. (b) Epistatic fitness strength σ with r. For large recombination and very small fitness, KNS inference does not work.

Fig. 10. Scatter-plots of inferred epistatic fitness against the true fitness based on the averaged results from singletime: (a) with sad face: r = 0.1, where the KNS theory cannot be satisfied here, (b) with smiling face: r = 0.5, where inference works, (c) with not-that-sad face: r = 0.9, where the KNS theory works but with very heavy fluctuations. Here the DCA-nMF algorithm for J_ij’s is utilized.

Fig. 11. Corresponding scatter plots by alltime averages with Fig. 10. The DCA-nMF algorithm for J_ij’s is also used here. The parameters for each sub-panel are the same as those in Fig. 10.