Accuracy of network reconstruction and fraction of hidden variables. The inference accuracy is plotted as a function of the fraction of visible variables, . Results for system size and data lengths (black circles) or (red triangles) are shown.

Estimation of hidden degrees of freedom. Mean square errors of observed interaction strengths (upper) and discrepancy of observed variables (lower, black circles) and discrepancy of total (observed and hidden) variables (lower, red triangles) are shown with various assumed numbers of hidden variables. The actual numbers of hidden variables are , 20, 30, and 40, from left to right. Results with system size and data length are shown.

Performance comparison between inference methods. Predicted interactions versus actual interactions, separately in observed-to-observed (a and e), hidden-to-observed (b and f), observed-to-hidden (c and g), and hidden-to-hidden (d and h), for two numbers of hidden variables (first row) and 40 (second row). The mean square errors between predicted interactions and actual interactions are shown as a function of the fraction () of observed variables over total variables (i–l). We compared five inference methods: naïve mean-field (nMF), Thouless-Anderson-Palmer (TAP), exact mean-field (eMF), maximum likelihood estimation (MLE), and free-energy minimization (FEM). Results with system size and data length are shown. For MLE, we used a learning rate .

Neural network reconstruction with hidden nodes. Activities of the 80 most active neurons are plotted with black dots representing active states and white dots representing silent states (a). Discrepancies between observed and expected neuronal activities (black circles) and discrepancies between entire neuronal activities and their expectations (red triangles) are shown as a functions of the number of hidden variables (b). A predicted neuronal network is visualized in which green nodes represent observed neurons, while white nodes represent hidden variables. The red and blue edges represent positive and negative couplings, respectively. Edge direction is clockwise. The node size scales with value of firing rate, and edge thickness scales with coupling weight (c). Given the reconstructed coupling strengths, external local fields, and configurations of hidden variables, the activities of 80 neurons are reconstructed (d). Inference accuracy of remaining neuronal activities are shown as a function of the number of input neurons for ignoring (blue) and including (black) the existence of hidden variables (e). Inferred covariances versus actual covariances when hidden variables are ignored (f) and included (g).

Stock-market network reconstruction with hidden nodes. The time series of the difference between opening and closing prices of 25 American companies are shown from January 2005 to July 2018 (a). The network reconstruction considered for different periods: from August 2014 to July 2016 (b, c, d), and from August 2016 to July 2018 (e, f, g). Discrepancies between observed and expected configurations (black circles) and discrepancies between entire variables and their expectations (red triangles) are computed to estimate the necessary number of hidden variables (b, e). Reconstructed stock-market networks are visualized (c, f) in which the red and blue edges represent positive and negative couplings, respectively. Edge direction is clockwise, and edge thickness scales with coupling strengths. Inferred covariances versus actual covariances (d, g). Cumulative profits are shown as a function of the time period for trade on everyday (fully trade) (h) and on certain days (alternative trade) (i) with different trade strategies: random trades (black), strategic trades ignoring (blue), and including hidden variables (red). Profit per transaction versus time window size for the network reconstruction (j), for everyday trade (dashed line) and certain-day trade (solid line).