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Global robustness versus local vulnerabilities in complex synchronous networks

Melvyn Tyloo and Philippe Jacquod
Phys. Rev. E 100, 032303 – Published 5 September 2019
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  • INTRODUCTION
  • RESISTANCE DISTANCES, CENTRALITIES, AND…
  • SYNCHRONIZED OSCILLATORS UNDER EXTERNAL…
  • NUMERICAL RESULTS
  • CONCLUSION
  • ACKNOWLEDGMENTS
    • References

    Abstract

    In complex network-coupled dynamical systems, two questions of central importance are how to identify the most vulnerable components and how to devise a network making the overall system more robust to external perturbations. To address these two questions, we investigate the response of complex networks of coupled oscillators to local perturbations. We quantify the magnitude of the resulting excursion away from the unperturbed synchronous state through quadratic performance measures in the angle or frequency deviations. We find that the most fragile oscillators in a given network are identified by centralities constructed from network resistance distances. Further defining the global robustness of the system from the average response over ensembles of homogeneously distributed perturbations, we find that it is given by a family of topological indices known as generalized Kirchhoff indices. Both resistance centralities and Kirchhoff indices are obtained from a spectral decomposition of the stability matrix of the unperturbed dynamics and can be expressed in terms of resistance distances. We investigate the properties of these topological indices in small-world and regular networks. In the case of oscillators with homogeneous inertia and damping coefficients, we find that inertia only has small effects on robustness of coupled oscillators. Numerical results illustrate the validity of the theory.

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    • Received 9 May 2019

    DOI:https://doi.org/10.1103/PhysRevE.100.032303

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Collective behavior in networksNetwork resilienceRobustness
    1. Physical Systems
    Coupled oscillatorsSynchronization
    NetworksNonlinear Dynamics

    Authors & Affiliations

    Melvyn Tyloo1,3 and Philippe Jacquod2,3

    • 1Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
    • 2Department of Quantum Matter Physics, University of Geneva, CH-1211 Geneva, Switzerland
    • 3School of Engineering, University of Applied Sciences of Western Switzerland HES-SO, CH-1951 Sion, Switzerland

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    Vol. 100, Iss. 3 — September 2019

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    Images

    • Figure 1
      Figure 1

      Ratio of the performance measures P1 for graph (a) vs. (e) of Fig. 2 (shown in the insets), for a quench perturbation of magnitude δP0=0.01 and duration γτ0=500 on node k=1,2,3,...20 (see text). On average, graph (e) is four times more robust to external perturbations than graph (a) (blue dashed line). However, some nodes of graph (a) can be more robust than those of graph (e) (red crosses correspond to quench perturbation applied on the red nodes shown in the inset). Both specific local vulnerabilities (crosses) and global averaged robustness (blue dashed lines) are well predicted by combinations of local centralities, and global topological indices (orange solid line, see text).

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    • Figure 2
      Figure 2

      Six networks with n=20 nodes obtained by the rewiring procedure of Ref. [48], starting from a cyclic graph and rewiring every edge of the network with a probability p=0.15 (a), p=0.3 (b), p=0.45 (c), p=0.6 (d), p=0.75 (e), and p=0.9 (f). The node numbering used in Fig. 3 is indicated in panel (a).

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    • Figure 3
      Figure 3

      Performances measures P1 (left) and P2 (right) for the graphs of Fig. 2 (top), Fig. 2 (middle), Fig. 2 (bottom), and a quench perturbation of magnitude δP0=0.01 on node k. Numerical results (circles) and analytical Eqs. (17) (solid lines) are plotted for different durations of perturbation γτ0=0.5 (black), 1 (blue), 10 (red), 100 (green). The asymptotic values of short and long τ0 given in Eqs. (20) (dotted line) and (21) (dashed line) are shown, vertically shifted by an arbitrary amount for clarity. The node numbering is given in Fig. 2.

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    • Figure 4
      Figure 4

      Resistance centralities C1(k) (top) and C2(k) (bottom), given in Eqs. (3) and (7), respectively, for the six graphs of Fig. 2.

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    • Figure 5
      Figure 5

      Comparison between C1(k) and C2(k) for the six graphs of Fig. 2 (see inset).

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    • Figure 6
      Figure 6

      Time-evolution of angles (left) and frequencies (right) following a quench perturbation applied on node 6 (top panels) and 11 (bottom panels) of graph (f) in Fig. 2 with γτ0=50. The trajectory of the perturbed oscillator is shown in red. Angles and frequencies spread more when the perturbation is applied on node 6 than on node 11, in agreement with predictions of Eqs. (21) since node 6 has the smallest, node 11 the largest centrality in this graph.

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    • Figure 7
      Figure 7

      Trajectories of angles and phases for the graphs of Fig. 2 with p=0.15 (top) and p=0.9 (bottom) obtained by numerically time-evolving Eq. (10) for the same quench perturbation with γτ0=50 applied on the node colored in red in the insets. In the four left panels, perturbed nodes are close to median value of C2(k), respectively, in graph with p=0.15 (top) and p=0.9 (bottom). In the four right panels, perturbed nodes are the most (top) and least (bottom) central ones according to C2(k), respectively, in graph with p=0.15 and p=0.9.

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    • Figure 8
      Figure 8

      Averaged performances measures 〈P1∞〉, 〈P2∞〉 for the graphs of Fig. 2 obtained numerically (circles) and predicted analytically, Eqs. (17) (solid lines) for perturbations with γτ0=0.5 (black), 1 (blue), 10 (red), 100 (green). The asymptotic values of short and long τ0 given in Eqs. (22) (dotted line) and (23) (dashed line) are shown, vertically shifted by an arbitrary amount for clarity.

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    • Figure 9
      Figure 9

      Left panel: clustering coefficient Cl, average geodesic distance l and generalized Kirchhoff indices Kf1 and Kf2, as a function of the rewiring probability p for Strogatz-Watts rewired networks [48]. Each data point corresponds to an average over 30 realizations of randomly rewired graphs, obtained from an initial cyclic graph with n=1000 nodes and nearest to 10th-nearest neighbor coupling, where each edge is randomly rewired with a probability p (parameters chosen similar as in Ref. [48], such that the graph remains connected while rewiring and is still sparse). Right panel: ratio of the Kirchhoff indices and of clustering coefficient vs. average geodesic distance. Small-world network are easily identified by the steepest slope of the orange line.

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    • Figure 10
      Figure 10

      Generalized Kirchhoff indices Kf1 (green) and Kf2 (purple) given in Eq. (25), for a cyclic network with n=50 nodes with nearest and qth-nearest-neighbor coupling. The inset sketches the model for q=17, 19, and 24 and with one path involving qth range coupling starting from node 1 (red). The addition of the qth-nearest-neighbor coupling does not reduce geodesic distance between the reference node (red) and the set of nodes colored in blue.

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