Research ArticleNETWORK SCIENCE

Synchronization in networks with multiple interaction layers

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Science Advances  16 Nov 2016:
Vol. 2, no. 11, e1601679
DOI: 10.1126/sciadv.1601679
  • Fig. 1 Schematic representation of a network with two layers of interaction.

    The two layers (corresponding here to solid violet and dashed orange links) are made of links of different types for the same nodes, such as different means of transport between two cities or chemical and electric connections between neurons. Note that the layers are fully independent, in that they are described by two different Laplacians L(1) and L(2), so that the presence of a connection between two nodes in one layer does not affect their connection status in the other.

  • Fig. 2 Maximum Lyapunov exponent for ER-ER topologies in case 1 (top panel) and case 2 (bottom panel).

    The darker blue lines mark the points in the (σ1, σ2) space where Λ vanishes, whereas the striped lines indicate the critical values of σ2 if layer 2 is considered in isolation (or, equivalently, if σ1 = 0).

  • Fig. 3 Maximum Lyapunov exponent in case 3 for ER-ER and SF-SF topologies (top and bottom panel, respectively).

    The darker blue lines mark the points in the (σ1, σ2) plane where the maximum Lyapunov exponent is 0, whereas the striped lines indicate the stability limits for the σ1 = 0 and σ2 = 0. The points marked in the top panel indicate the choices of coupling strengths used for the numerical validation of the model. Note that for SF networks in class III, the stability window disappears.

  • Fig. 4 Numerical validation of the stability analysis.

    The error of synchronization increases as long as the only active layer is the one predicted to be unstable. When the other layer is switched on, at time 100, the error of synchronization decays exponentially toward 0, as predicted by the model. With respect to Fig. 3, the top left panel corresponds to region II, where layer 1 is unstable and layer 2 is stable, and the interaction strengths used were σ1 = 0.04 and σ2 = 0.3. The bottom left panel corresponds to region IV, where layer 1 is stable and layer 2 is unstable, and the interaction strengths were σ1 = 0.15 and σ2 = 0.5. The top right and bottom right panels correspond to region VI, where both layers are unstable. The layer active from the beginning was layer 1 for the top right panel and layer 2 for the bottom right panel. In both cases, the interaction strengths were σ1 = 0.04 and σ2 = 0.5.

  • Fig. 5 Identification of the critical points.

    For a system with ER-ER topology in case 3 and fixed σ2 = 1, the synchronization error never vanishes if σ1 < σc ≈ 0.04. Conversely, as soon as σ1 > σc, the system is again able to synchronize (green line). One recovers the monolayer case by imposing σ2 = 0, for which similar results are found, with a critical coupling strength of approximately 0.08 (red line). Both results are in perfect agreement with the theoretical predictions (see Fig. 4).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/11/e1601679/DC1

    fig. S1. Maximum Lyapunov exponent Λ for systems falling into case 1 (layer 1 in stability class I and layer 2 in stability class II) for SF-SF, ER-SF, and SF-ER topologies (left, center, and right panels, respectively).

    fig. S2. Maximum Lyapunov exponent Λ for systems falling into case 2 (layer 1 in stability class I and layer 2 in stability class III) for SF-SF, ER-SF, and SF-ER topologies (left, center, and right panels, respectively).

    fig. S3. Maximum Lyapunov exponent Λ for systems falling into case 3 (layer 1 in stability class II and layer 2 in stability class III) for ER-SF and SF-ER topologies (left and right panels, respectively).

  • Supplementary Materials

    This PDF file includes:

    • fig. S1. Maximum Lyapunov exponent Λ for systems falling into case 1 (layer 1 in stability class I and layer 2 in stability class II) for SF-SF, ER-SF, and SF-ER topologies (left, center, and right panels, respectively).
    • fig. S2. Maximum Lyapunov exponent Λ for systems falling into case 2 (layer 1 in stability class I and layer 2 in stability class III) for SF-SF, ER-SF, and SF-ER topologies (left, center, and right panels, respectively).
    • fig. S3. Maximum Lyapunov exponent Λ for systems falling into case 3 (layer 1 in stability class II and layer 2 in stability class III) for ER-SF and SF-ER topologies (left and right panels, respectively).

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