Research ArticleNONLINEAR DYNAMICS

Revealing physical interaction networks from statistics of collective dynamics

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Science Advances  10 Feb 2017:
Vol. 3, no. 2, e1600396
DOI: 10.1126/sciadv.1600396
  • Fig. 1 Strategy of network reconstruction from responses of time invariants.

    (A) A networked dynamical system with unknown topology (gray) is perturbed by external driving signals Im, m ∈ {1, …, M}. (B) Potentially noisy, disordered, low-resolution data are collected from several different experiments. (C) The centers of mass z(m) of each of these distributions of points sampled in state space are calculated. (D) The changes Embedded Image in response to driving signals I(m) yield the network topology.

  • Fig. 2 Evaluation scheme illustrating robust reconstruction.

    (A) Representative adjacency matrix indicating network connectivity as defined by present (black) and absent (white) links. (B) ROC curve obtained by varying a threshold Jc separating links classified as existing Embedded Image from those classified as absent Embedded Image (see note S3). The AUC increases with decreasing noise level, with perfect ranking of reconstructed links in the limit of noiseless dynamics. Inset: The quality of network reconstruction, as specified by the AUC, increases with the number of driving-response experiments. (C) The number of experiments required for high-quality reconstruction (here, AUC > 0.95) increases sublinearly (compared to the dotted line) with network size and (inset) changes only weakly with the noise level. Data are shown for random networks of (default size) N = 50 Goodwin oscillators with a regular incoming degree of 4, a default noise level of 0.5, a default number of experiments of 25, and a number of sampled time points of 100; shading indicates SD across ensembles of network realizations.

  • Fig. 3 Revealing interaction types.

    (A) Beyond distinguishing existing from missing interactions (schematically represented by the medium gray and white adjacency matrix), activating and inhibiting interactions may be separately detected (dark gray, light gray, and white matrix). (B) The reconstruction quality (AUC) benefits from the separate reconstruction of different types of interactions (green curve) compared to joint reconstruction of existing and missing interactions (gray curve), and increases with the number of driving-response experiments. Data are shown for random networks of N = 50 Goodwin oscillators with a regular incoming degree of 4 and a number of sampled time points of 100; shading indicates SD across ensembles of network realizations.

  • Fig. 4 Robust reconstruction from one-dimensional sampling of multidimensional unit dynamics.

    (A) Scheme illustrates three-dimensional units (encircled), coupled through one observed variable (colored), whereas the other two variables are unobserved (gray). (B) The reconstruction quality (AUC) stays robust and reduces only slightly for reconstruction based on partial, one-dimensional measurements (green curve) relative to reconstruction based on three-dimensional measurements (gray curve), and increases with the number of driving-response experiments. Data are shown for random networks of N = 50 Goodwin oscillators with a regular incoming degree of 4 and a number of sampled time points of 100; shading indicates SD across ensembles of network realizations.

  • Fig. 5 Reconstruction of the circadian clock network in Drosophila.

    The quality of reconstruction (AUC) of the circadian clock (A) increases with the number of driving-response experiments, for both noiseless (gray curve) and noisy (green curve) dynamics (B), and changes only weakly with noise level (inset). Number of sampled time points, 300; default number of experiments, 20; noise level for noisy case, 0.01; shading indicates SD across ensembles of network realizations. (A) Modified with permission from Leloup and Goldbeter (39) (Fig. 1).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/2/e1600396/DC1

    note S1. Details of the derivation of invariant-based reconstruction.

    note S2. Error estimates for observables from sampled invariant density.

    note S3. Reconstruction evaluation.

    note S4. Moderate influence of link density.

    note S5. Reconstructing homogeneous and heterogeneous networks.

    note S6. Reconstruction of systems near fixed points.

    note S7. Reconstruction of chaotic systems.

    note S8. Performance compared with available standard baselines.

    note S9. Distinguishing activating from inhibiting interactions.

    note S10. The effect of missing information.

    note S11. Model descriptions.

    note S12. The effect of various driving conditions on reconstruction quality.

    note S13. Compressed sensing.

    fig. S1. Approximating the center of mass of invariant densities by the sample mean.

    fig. S2. Sparser networks require fewer experiments for robust reconstruction.

    fig. S3. Reconstruction is robust across network topologies.

    fig. S4. The quality of reconstruction increases with the number of experiments for a network of genetic regulators.

    fig. S5. Reconstruction of a network of Rössler oscillators exhibiting chaotic dynamics.

    fig. S6. Comparison of reconstruction quality across different approaches.

    fig. S7. Comparison of reconstruction quality against transfer entropy.

    fig. S8. Separate reconstruction of activating and inhibiting interactions enhances the quality of reconstruction.

    fig. S9. Quality of reconstruction (AUC score) decreases gradually with the fraction of hidden units in the network.

    fig. S10. Quality of reconstruction increases as driving signals overcome noise and finite sampling effects.

    References (4850)

  • Supplementary Materials

    This PDF file includes:

    • note S1. Details of the derivation of invariant-based reconstruction.
    • note S2. Error estimates for observables from sampled invariant density.
    • note S3. Reconstruction evaluation.
    • note S4. Moderate influence of link density.
    • note S5. Reconstructing homogeneous and heterogeneous networks.
    • note S6. Reconstruction of systems near fixed points.
    • note S7. Reconstruction of chaotic systems.
    • note S8. Performance compared with available standard baselines.
    • note S9. Distinguishing activating from inhibiting interactions.
    • note S10. The effect of missing information.
    • note S11. Model descriptions.
    • note S12. The effect of various driving conditions on reconstruction quality.
    • note S13. Compressed sensing.
    • fig. S1. Approximating the center of mass of invariant densities by the sample mean.
    • fig. S2. Sparser networks require fewer experiments for robust reconstruction.
    • fig. S3. Reconstruction is robust across network topologies.
    • fig. S4. The quality of reconstruction increases with the number of experiments
      for a network of genetic regulators.
    • fig. S5. Reconstruction of a network of Rössler oscillators exhibiting chaotic dynamics.
    • fig. S6. Comparison of reconstruction quality across different approaches.
    • fig. S7. Comparison of reconstruction quality against transfer entropy.
    • fig. S8. Separate reconstruction of activating and inhibiting interactions enhances the quality of reconstruction.
    • fig. S9. Quality of reconstruction (AUC score) decreases gradually with the fraction of hidden units in the network.
    • fig. S10. Quality of reconstruction increases as driving signals overcome noise and finite sampling effects.
    • References (48–50)

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