Skip to main content
SpringerLink
Log in

Sparse Network Optimization for Synchronization

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data Availability

The datasets generated and analysed during the current study are available in the ArXiv repository, https://arxiv.org/src/2006.00428v1/anc—also see Reference [5].

References

  1. Alt, W., Kaya, C.Y., Schneider, C.: Dualization and discretization of linear-quadratic control problems with bang-bang solutions. EURO J. Comput. Optim. 4, 47–77 (2016)

    Article  MathSciNet  Google Scholar 

  2. Banihashemi, N., Kaya, C.Y.: Inexact restoration for Euler discretization of box-constrained optimal control problems. J. Optim. Theory Appl. 156, 726–760 (2013)

    Article  MathSciNet  Google Scholar 

  3. Barahona, M., Pecora, L.M.: Synchronization in small world systems. Phys. Rev. Lett. 89, 054101 (2002)

    Article  Google Scholar 

  4. Brede, M.: Local versus global synchronization in networks of non-identical Kuramoto oscillators. Eur. Phys. J. B. 62, 87 (2008)

    Article  Google Scholar 

  5. Burachik, R. S., Kalloniatis, A. C., Kaya, C. Y.: Ancillary files for the preprint arXiv:2006.00428v1 (2020)

  6. Burachik, R.S., Kaya, C.Y., Majeed, S.N.: A duality approach for solving control-constrained linear-quadratic optimal control problems. SIAM J. Control Optim. 52, 1771–1782 (2014)

    Article  MathSciNet  Google Scholar 

  7. Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 3rd edn. Wiley, Chichester (2016)

    Book  Google Scholar 

  8. Corless, R.M., Kaya, C.Y., Moir, R.H.C.: Optimal residuals and the Dahlquist test problem. Numer. Algorithms 81, 1253–1274 (2019)

    Article  MathSciNet  Google Scholar 

  9. Dekker, A.H.: Studying organisational topology with simple computational models. J. Artif. Soc. Simul. 10, 6 (2007)

    Google Scholar 

  10. Dekker, A.H., Taylor, R.: Synchronization properties of trees in the Kuramoto model. SIAM J. Appl. Dyn. Syst. 12, 596–617 (2013)

    Article  MathSciNet  Google Scholar 

  11. Donetti, L., Hurtado, P.I., Munoz, M.A.: Entangled networks, synchronization, and optimal network topology. Phys. Rev. Lett. 95, 188701 (2005)

    Article  Google Scholar 

  12. Donetti, L., Neri, F., Munoz, M.A.: Optimal network topologies: expanders, cages, Ramanujan graphs, entangled networks and all that. J. Stat. Mech. P08007 (2006)

  13. Estrada, E., Gago, S., Caporossi, G.: Design of highly synchronizable and robust networks. Automatica 46, 1835 (2010)

    Article  MathSciNet  Google Scholar 

  14. Fazlyab, M., Doerfler, F., Preciado, V.M.: Optimal network design for synchronization of coupled oscillators. Automatica 84, 181 (2017)

    Article  MathSciNet  Google Scholar 

  15. Forger, D.B.: Biological Clocks, Rhythms, and Oscillations. The MIT Press, Cambridge (2017)

    MATH  Google Scholar 

  16. Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming, 2nd ed. (Brooks/Cole Publishing Company/Cengage Learning, 2003)

  17. Hong, H., Choi, M.Y., Kim, B.J.: Synchronization on small-world networks. Phys. Rev. E 65, 026139 (2002)

    Article  Google Scholar 

  18. Hoory, S., Linial, N., Widgerson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43(4), 439–561 (2006)

    Article  MathSciNet  Google Scholar 

  19. Ichinomiya, T.: Frequency synchronization in a random oscillator network. Phys. Rev. E 70(2), 026116 (2004)

    Article  Google Scholar 

  20. Kalloniatis, A.C., McLennan-Smith, T.A., Roberts, D.O.: Modelling distributed decision-making in command and control using stochastic network synchronisation. Eur. J. Oper. Res. 284, 588–603 (2020)

    Article  Google Scholar 

  21. Kaya, C.Y.: Inexact restoration for Runge–Kutta discretization of optimal control problems. SIAM J. Numer. Anal. 48(4), 1492–1517 (2010)

    Article  MathSciNet  Google Scholar 

  22. Kaya, C.Y., Martínez, J.M.: Euler discretization for inexact restoration and optimal control. J. Optim. Theory Appl. 134, 191–206 (2007)

    Article  MathSciNet  Google Scholar 

  23. Kaya, C.Y., Maurer, H.: A numerical method for nonconvex multi-objective optimal control problems. Comput. Optim. Appl. 57(3), 685–702 (2014)

    Article  MathSciNet  Google Scholar 

  24. Kelly, D., Gottwald, G.A.: On the topology of synchrony optimized networks of a Kuramoto-model with non-identical oscillators. Chaos 21, 025110 (2011)

    Article  MathSciNet  Google Scholar 

  25. Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, Berlin (1984)

    Book  Google Scholar 

  26. Lü, L., Chen, D., Ren, X.L., Zhang, Q.M., Zhang, Y.C., Zhou, T.: Vital nodes identification in complex networks. Phys. Rep. 650, 1–63 (2016)

    Article  MathSciNet  Google Scholar 

  27. Medvedev, G.S., Tang, X.: Stability of twisted states in the Kuramoto model on Cayley and Random graphs. J. Nonlinear Sci. 25, 1169–1208 (2015)

    Article  MathSciNet  Google Scholar 

  28. Medvedev, G.S., Tang, X.: Synchronization of coupled chaotic maps. Physica D 304–305, 42–51 (2015)

    Article  MathSciNet  Google Scholar 

  29. Mohar, B.: Some applications of Laplace eigenvalues of graphs. In: Hahn, G., Sabidussi, G. (eds.) Graph Symmetry: Algebraic Methods and Applications. NATO ASI Series C 497, pp. 225–275. Kluwer, Dordrecht (1997)

    Chapter  Google Scholar 

  30. Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (2006)

    MATH  Google Scholar 

  31. Ochab, J., Gora, P.F.: Synchronization of coupled oscillators in a local one-dimensional Kuramoto model. Acta Phys. Polonica B Proc. Suppl. 3, 453–462 (2010)

    Google Scholar 

  32. Oh, E., Lee, D.-S., Kahng, B., Kim, D.: Synchronization transition of heterogeneously coupled oscillators on scale-free networks. Phys. Rev. E 75, 011104 (2007)

    Article  MathSciNet  Google Scholar 

  33. Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109 (1998)

    Article  Google Scholar 

  34. Rodrigues, F.A., Peron, T.K.D.M., Ji, P., Kurths, J.: The Kuramoto model in complex networks. Phys. Rep. 610, 1–98 (2016)

    Article  MathSciNet  Google Scholar 

  35. Rogge, J.A., Aeyels, D.: Stability of phase locking in a ring of unidirectionally coupled oscillators. J. Phys. A Math. Gen. 37, 11135–11148 (2004)

    Article  MathSciNet  Google Scholar 

  36. Tanaka, T., Aoyagi, T.: Optimal weighted networks of phase oscillators for synchronization. Phys. Rev. E. 78, 046210 (2008)

    Article  MathSciNet  Google Scholar 

  37. Taylor, R., Kalloniatis, A., Hoek, K.: Organisational hierarchy constructions with easy Kuramoto synchronisation. J. Phys. A Math. Theor. 53(8), 085701 (2020)

    Article  MathSciNet  Google Scholar 

  38. Vossen, G., Maurer, H.: On \(L^1\)-minimization in optimal control and applications to robotics. Opt. Control Appl. Methods 27, 301–321 (2006)

    Article  Google Scholar 

  39. Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Progr. 106, 25–57 (2006)

    Article  Google Scholar 

  40. Yanagita, T., Mikhailov, A.S.: Design of oscillator networks with enhanced synchronization tolerance against noise. Phys. Rev. E. 85, 056206 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are indebted to the two anonymous reviewers for carefully reading the initial submission and making valuable suggestions and comments which improved the paper. They would also like to acknowledge valuable discussions with Subhra Dey at the initiation of this project and with Richard Taylor during its later stages. This research was a collaboration under the auspices of the Modelling Complex Warfighting initiative between the Commonwealth of Australia (represented by the Defence Science and Technology Group) and the University of South Australia through a Defence Science Partnerships agreement.

Author information

Authors and Affiliations

  1. Mathematics, UniSA STEM, University of South Australia, Mawson Lakes, S.A., 5095, Australia

    Regina S. Burachik & C. Yalçın Kaya

  2. Joint and Operations Analysis Division, Defence Science and Technology Group, Canberra, 2600, Australia

    Alexander C. Kalloniatis

Authors
  1. Regina S. Burachik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander C. Kalloniatis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Yalçın Kaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Yalçın Kaya.

Additional information

Communicated by Anita Schöbel.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burachik, R.S., Kalloniatis, A.C. & Kaya, C.Y. Sparse Network Optimization for Synchronization. J Optim Theory Appl 191, 229–251 (2021). https://doi.org/10.1007/s10957-021-01933-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-021-01933-9

Keywords

Mathematics Subject Classification

Search

Navigation