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The Kuramoto Model on Power Law Graphs: Synchronization and Contrast States


The relation between the structural properties of the network and its dynamics is a central question in the analysis of dynamical networks. It is especially relevant for complex networks found in real-world applications. This work presents mathematically rigorous analysis of coupled dynamical systems on power law graphs. Specifically, we study large systems of coupled Kuramoto phase oscillators. In the limit as the size of the network tends to infinity, we derive analytically tractable mean field partial differential equation for the probability density function describing the state of the coupled system. The mean field limit is used to establish an explicit formula for the synchronization threshold for coupled phase oscillators with randomly distributed intrinsic frequencies. Furthermore, we study stable spatial patterns generated by the Kuramoto model with repulsive coupling. In particular, we identify a family of stable steady-state solutions having multiple regions with distinct statistical properties. We call these solutions contrast states. Like chimera states, contrast states exhibit coexisting regions of highly localized (coherent) behavior and highly irregular (incoherent) distribution of phases. We provide a detailed mathematical analysis of contrast states in the KM using the Ott–Antonsen ansatz. The analysis of synchronization and contrast states provides new insights into the role of power law connectivity in shaping dynamics of coupled dynamical systems. In particular, we show that despite sparse connectivity, power law networks possess remarkable synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law distribution.

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  1. Throughout this paper, we use \(a\wedge b\) and \(a\vee b\) to denote \(\min \{a,b\}\) and \(\max \{a,b\}\), respectively.

  2. Here and below, \({\mathbb E~}_\omega \) denotes the mathematical expectation with respect to the probability space (2.4) underlying the random graph model.

  3. Equation (2.6) is derived from a system of weakly coupled oscillators (Kuramoto 1975; Hoppensteadt and Izhikevich 1997).

  4. See Guckenheimer and Holmes (1990) for the definition of the \(\omega \)–limit set.

  5. This order parameter is used in the analysis of the bifurcation underlying the transition to synchrony in the Kuramoto model on graphs (Chiba and Medvedev 2016, 2017). A similar order parameter was used by Laing for the analysis of chimera states in a ring of coupled oscillators (Laing 2009).


  • Abrams, D.M., Strogatz, S.H.: Chimera states in a ring of nonlocally coupled oscillators. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16(1), 21–37 (2006)

    MathSciNet  Article  Google Scholar 

  • Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)

    MathSciNet  Article  Google Scholar 

  • Barbašin, E.A., Krasovskiĭ, N.N.: On stability of motion in the large. Doklady Akad. Nauk SSSR (N.S.) 86, 453–456 (1952)

    MathSciNet  Google Scholar 

  • Borgs, C., Chayes, J., Lovász, L., Sós, V., Vesztergombi, K.: Limits of randomly grown graph sequences. Eur. J. Combin. 32(7), 985–999 (2011)

    MathSciNet  Article  Google Scholar 

  • Borgs, C., Chayes, J.T., Cohn, H., Zhao, Y.: An \(L^p\) theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions. (2014). arXiv:1401.2906

  • Chiba, H.: A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model. Ergod. Theory Dyn. Syst. 35(3), 762–834 (2015)

    MathSciNet  Article  Google Scholar 

  • Chiba, H., Medvedev, G.S.: The mean field analysis for the Kuramoto model on graphs I. The mean field equation and transition point formulas. (2016). arXiv:1612.06493

  • Chiba, H., Medvedev, G.S.: The mean field analysis of the Kuramoto model on graphs II. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. (2017). arXiv:1709.08305

  • Chiba, H., Nishikawa, I.: Center manifold reduction for large populations of globally coupled phase oscillators. Chaos 21(4), 043103–043110 (2011)

    MathSciNet  Article  Google Scholar 

  • Chiba, H., Medvedev, G.S., Mizuhara, M.S.: Bifurcations in the Kuramoto model on graphs. Chaos. 28, 073109 (2018). (in press)

  • Chung, F., Lu, L.: The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA 99(25), 15879–15882 (2002)

    MathSciNet  Article  Google Scholar 

  • Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge (2002). Revised reprint of the 1989 original

    Book  Google Scholar 

  • Golse, F.: On the dynamics of large particle systems in the mean field limit. In: Muntean, A., Rademacher, J., Zagaris, A. (eds.) Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Lect. Notes Appl. Math. Mech., vol. 3, pp. 1–144. Springer, Cham (2016)

    Google Scholar 

  • Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, vol. 42. Springer, New York (1990). Revised and corrected reprint of the 1983 original

    MATH  Google Scholar 

  • Hartman, P.: Ordinary Differential Equations. S. M. Hartman, Baltimore (1973). (Corrected reprint)

    MATH  Google Scholar 

  • Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks. Applied Mathematical Sciences, vol. 126. Springer, New York (1997)

    Book  Google Scholar 

  • Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  • Kaliuzhnyi-Verbovetskyi, D., Medvedev, G.S.: The semilinear heat equation on sparse random graphs. SIAM J. Math. Anal. 49(2), 1333–1355 (2017)

    MathSciNet  Article  Google Scholar 

  • Kaliuzhnyi-Verbovetskyi, D., Medvedev, G.S.: The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit. SIAM J. Math. Anal. 50(3), 2441–2465 (2018)

    MathSciNet  Article  Google Scholar 

  • Krasovskiĭ, N.N.: Nekotorye zadachi teorii ustoichivosti dvizheniya. Gosudarstv. Izdat. Fiz.-Mat. Lit, Moscow (1959)

    Google Scholar 

  • Kuramoto, Y.: Self-entrainment of a population of coupled non-linear oscillators. In: International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975). Lecture Notes in Phys., vol. 39, pp. 420–422. Springer, Berlin (1975)

  • Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlocally coupled phase oscillators. Nonlinear Phenom. Complex Syst. 5, 380–385 (2002)

    Google Scholar 

  • Laing, C.R.: Chimera states in heterogeneous networks. Chaos 19(1), 013113–013118 (2009)

    MathSciNet  Article  Google Scholar 

  • Lancellotti, C.: On the Vlasov limit for systems of nonlinearly coupled oscillators without noise. Transp. Theory Stat. Phys. 34(7), 523–535 (2005)

    MathSciNet  Article  Google Scholar 

  • LaSalle, J.: Some extensions of Liapunov’s second method. IRE Trans. Circuit Theory 7(4), 520–527 (1960)

    MathSciNet  Article  Google Scholar 

  • Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Comb. Theory Ser. B 96(6), 933–957 (2006)

    MathSciNet  Article  Google Scholar 

  • Medvedev, G.S.: Stochastic stability of continuous time consensus protocols. SIAM J. Control Optim. 50(4), 1859–1885 (2012)

    MathSciNet  Article  Google Scholar 

  • Medvedev, G.S.: The nonlinear heat equation on W-random graphs. Arch. Ration. Mech. Anal. 212(3), 781–803 (2014a)

  • Medvedev, G.S.: Small-world networks of Kuramoto oscillators. Phys. D 266, 13–22 (2014b)

  • Medvedev, G.S.: The continuum limit for the Kuramoto model on sparse random graphs. (2018). arXiv:1802.03787

  • Medvedev, G.S., Tang, X.: Stability of twisted states in the Kuramoto model on Cayley and random graphs. J. Nonlinear Sci. 25(6), 1169–1208 (2015)

    MathSciNet  Article  Google Scholar 

  • Medvedev, G.S., Douglas Wright, J.: Stability of twisted states in the continuum Kuramoto model. SIAM J. Appl. Dyn. Syst. 16(1), 188–203 (2017)

    MathSciNet  Article  Google Scholar 

  • Motsch, S., Tadmor, E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56(4), 577–621 (2014)

    MathSciNet  Article  Google Scholar 

  • Neunzert, H.: Mathematical investigations on particle-in-cell methods Fluid Dyn. Trans. 9, 229–254 (1978)

  • Omelchenko, O.E.: Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators. Nonlinearity 26(9), 2469 (2013)

    MathSciNet  Article  Google Scholar 

  • Ott, E., Antonsen, T.M.: Low dimensional behavior of large systems of globally coupled oscillators. Chaos 18, 037113 (2008)

    MathSciNet  Article  Google Scholar 

  • Porter, M.A., Gleeson, J.P.: Dynamical Systems on Networks. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol. 4. Springer, Cham (2016). A tutorial

    Book  Google Scholar 

  • Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D 143(1–4), 1–20 (2000)

    MathSciNet  Article  Google Scholar 

  • Strogatz, S.H., Mirollo, R.E.: Stability of incoherence in a population of coupled oscillators. J. Stat. Phys. 63(3–4), 613–635 (1991)

    MathSciNet  Article  Google Scholar 

  • Wiley, D.A., Strogatz, S.H., Girvan, M.: The size of the sync basin. Chaos 16(1), 015103–015108 (2006)

    MathSciNet  Article  Google Scholar 

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This work was supported in part by the NSF DMS grants 1412066 and 1715161 (to GM).

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Correspondence to Georgi S. Medvedev.

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Communicated by Paul Newton.

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Medvedev, G.S., Tang, X. The Kuramoto Model on Power Law Graphs: Synchronization and Contrast States. J Nonlinear Sci 30, 2405–2427 (2020).

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  • Coupled oscillators
  • Synchronization
  • Chimera state
  • Scale free graph
  • Graph limit
  • Mean field limit