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Invariance and stability conditions of interlayer synchronization manifold

Sarbendu Rakshit, Bidesh K. Bera, and Dibakar Ghosh
Phys. Rev. E 101, 012308 – Published 23 January 2020
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  • INTRODUCTION
  • MATHEMATICAL MODEL OF STOCHASTIC…
  • ANALYTICAL RESULTS
  • NUMERICAL ILLUSTRATIONS
  • GLOBAL STABILITY OF ILS STATE
  • EXISTENCE OF INTRALAYER SYNCHRONIZATION…
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We investigate interlayer synchronization in a stochastic multiplex hypernetwork which is defined by the two types of connections, one is the intralayer connection in each layer with hypernetwork structure and the other is the interlayer connection between the layers. Here all types of interactions within and between the layers are allowed to vary with a certain rewiring probability. We address the question about the invariance and stability of the interlayer synchronization state in this stochastic multiplex hypernetwork. For the invariance of interlayer synchronization manifold, the adjacency matrices corresponding to each tier in each layer should be equal and the interlayer connection should be either bidirectional or the interlayer coupling function should vanish after achieving the interlayer synchronization state. We analytically derive a necessary-sufficient condition for local stability of the interlayer synchronization state using master stability function approach and a sufficient condition for global stability by constructing a suitable Lyapunov function. Moreover, we analytically derive that intralayer synchronization is unattainable for this network architecture due to stochastic interlayer connections. Remarkably, our derived invariance and stability conditions (both local and global) are valid for any rewiring probabilities, whereas most of the previous stability conditions are only based on a fast switching approximation.

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    • Received 2 April 2019
    • Revised 2 January 2020

    DOI:https://doi.org/10.1103/PhysRevE.101.012308

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Coupled oscillatorsSynchronization
    Networks

    Authors & Affiliations

    Sarbendu Rakshit1, Bidesh K. Bera2,3, and Dibakar Ghosh1,*

    • 1Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
    • 2Department of Mathematics, Indian Institute of Technology Ropar, Punjab 140001, India
    • 3Department of Solar Energy and Environmental Physics, BIDR, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion, 8499000, Israel

    • *dibakar@isical.ac.in

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    Issue

    Vol. 101, Iss. 1 — January 2020

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    Images

    • Figure 1
      Figure 1

      Schematic of a multiplex hypernetwork consisting of two layers, each of size N=10 for two different time instants: (a) t=t1 and (b) t=t2. Colored solid circles denote each individual nodes and black doted lines represent the interlayer connections. Each layer consists of two independent tiers, which are represented by blue dashed and green solid lines, and refer to different types of interaction modes. Identical color of a node and its replica node signifies the ILS state.

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    • Figure 2
      Figure 2

      Effect of intralayer coupling values ε1 and ε2 on the transition of interlayer synchronization characterize through the interlayer synchronization error E for (a) pr=10−6 and (b) pr=0.01. The black lines denote the critical curves for Λ=0. Yellow and white portions represent the region of stable and unstable ILS state, respectively. Other parameters are fixed at psw=0.1,prand=0.015,pinter=0.05, and k=2.

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    • Figure 3
      Figure 3

      Variation of interlayer synchronization error in (ε1, prand) plane for (a) pinter=0.05 and (b) pinter=0.1 with fixed other parameter values at psw=0.1,pr=0.001,ε2=0.2, and k=2. The black line represents the critical curve for which Λ=0.0; above and below this curve denotes the regions for Λ>0.0 (unstable ILS state) and Λ<0.0 (stable ILS state), respectively.

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    • Figure 4
      Figure 4

      Dependency of interlayer synchronization on the mean degrees in the parameter plane of psw and ε2 for (a) k=2 and (b) k=5. The black line represents the critical curve where Λ=0.0. Other parameter values are prand=0.015,pr=0.001,pinter=0.06, and ε1=0.025.

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    • Figure 5
      Figure 5

      Variation of the interlayer synchronization error with respect to pinter and pr for (a) ε1=0.01,ε2=0.1 and (b) ε1=0.07,ε2=0.7. Other parameter values are psw=0.1,k=2 and prand=0.015. Black curves represent the critical curve for which Λ=0.

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    • Figure 6
      Figure 6

      Transition of interlayer synchrony in the (pinter, ε2) plane for (a) ε1=0.0, (b) ε1=0.025, (c) ε1=0.05, and (d) ε1=0.075. Other parameters: psw=0.1,prand=0.015,pr=0.001, and k=2. The black lines represent the critical curves for which Λ=0.0; the curves to the left and right denote the regions for Λ>0.0 (unstable ILS state) and Λ<0.0 (stable ILS state), respectively.

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    • Figure 7
      Figure 7

      Time evolution of the intralayer synchronization errors for layer-1 (E1(t), blue curve) and layer-2 (E2(t), red curve), respectively, at (a) pinter=0.05 and (b) η=1.0. Other parameters values: rp=10−6,psw=0.1,k=2, and prand=0.015. Here, E1(t) and E2(t) are the respective instantaneous errors for layer-1 and layer-2 at time t.

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    • Figure 8
      Figure 8

      Variation of the intralayer synchronization error Eintra in the (ε1,ε2) parameter plane for (a) pr=10−6 and (b) pr=10−2. Other parameter values are chosen as pinter=0.05,psw=0.1,k=2, and prand=0.015. The black portion in the color bar represents Eintra=0.0. Since in these parameter spaces, no black region appears, so intralayer synchronization does not emerge in the considered network.

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