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Sparse Network Optimization for Synchronization

Author

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  • Regina S. Burachik

    (University of South Australia)

  • Alexander C. Kalloniatis

    (Defence Science and Technology Group)

  • C. Yalçın Kaya

    (University of South Australia)

Abstract

We propose new mathematical optimization models for generating sparse dynamical graphs, or networks, that can achieve synchronization. The synchronization phenomenon is studied using the Kuramoto model, defined in terms of the adjacency matrix of the graph and the coupling strength of the network, modelling the so-called coupled oscillators. Besides sparsity, we aim to obtain graphs which have good connectivity properties, resulting in small coupling strength for synchronization. We formulate three mathematical optimization models for this purpose. Our first model is a mixed integer optimization problem, subject to ODE constraints, reminiscent of an optimal control problem. As expected, this problem is computationally very challenging, if not impossible, to solve, not only because it involves binary variables but also some of its variables are functions. The second model is a continuous relaxation of the first one, and the third is a discretization of the second, which is computationally tractable by employing standard optimization software. We design dynamical graphs that synchronize, by solving the relaxed problem and applying a practical algorithm for various graph sizes, with randomly generated intrinsic natural frequencies and initial phase variables. We test robustness of these graphs by carrying out numerical simulations with random data and constructing the expected value of the network’s order parameter and its variance under this random data, as a guide for assessment.

Suggested Citation

  • Regina S. Burachik & Alexander C. Kalloniatis & C. Yalçın Kaya, 2021. "Sparse Network Optimization for Synchronization," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 229-251, October.
    • Handle: RePEc:spr:joptap:v:191:y:2021:i:1:d:10.1007_s10957-021-01933-9
    DOI: 10.1007/s10957-021-01933-9
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    References listed on IDEAS

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    1. Kalloniatis, Alexander C. & McLennan-Smith, Timothy A. & Roberts, Dale O., 2020. "Modelling distributed decision-making in Command and Control using stochastic network synchronisation," European Journal of Operational Research, Elsevier, vol. 284(2), pages 588-603.
    2. C. Y. Kaya & J. M. Martínez, 2007. "Euler Discretization and Inexact Restoration for Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 134(2), pages 191-206, August.
    3. Anthony Dekker, 2007. "Studying Organisational Topology with Simple Computational Models," Journal of Artificial Societies and Social Simulation, Journal of Artificial Societies and Social Simulation, vol. 10(4), pages 1-6.
    4. Nahid Banihashemi & C. Yalçın Kaya, 2013. "Inexact Restoration for Euler Discretization of Box-Constrained Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 156(3), pages 726-760, March.
    5. C. Kaya & Helmut Maurer, 2014. "A numerical method for nonconvex multi-objective optimal control problems," Computational Optimization and Applications, Springer, vol. 57(3), pages 685-702, April.
    6. Walter Alt & C. Yalçın Kaya & Christopher Schneider, 2016. "Dualization and discretization of linear-quadratic control problems with bang–bang solutions," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 47-77, February.
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