October  2018, 23(8): 3461-3482. doi: 10.3934/dcdsb.2018234

Synchronization in networks with strongly delayed couplings

1. 

Associate Laboratory of Applied Computing and Mathematics - LAC, National Institute for Space Research - INPE, Av. dos Astronautas, 1758, São José dos Campos-SP, 12227-010, Brazil

2. 

Institute of Mathematics and Computer Sciences - ICMC, University of São Paulo - USP, Av. do Trabalhador São-Carlense 400, São Carlos-SP, 13566-590, Brazil

3. 

Institute of Mathematics, Technical University of Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany

1Current address: Department of Mathematics and Statistics, State University of Rio Grande do Norte - UERN, Mossoró-RN, 59610-210 Brazil

Received  September 2017 Revised  April 2018 Published  October 2018 Early access  August 2018

We investigate the stability of synchronization in networks of dynamical systems with strongly delayed connections. We obtain strict conditions for synchronization of periodic and equilibrium solutions. In particular, we show the existence of a critical coupling strength $κ_{c}$, depending only on the network structure, isolated dynamics and coupling function, such that for large delay and coupling strength $κ<κ_{c}$, the network possesses stable synchronization. The critical coupling $κ_{c}$ can be chosen independently of the delay for the case of equilibria, while for the periodic solution, $κ_{c}$ depends essentially on the delay and vanishes as the delay increases. We observe that, for random networks, the synchronization interval is maximal when the network is close to the connectivity threshold. We also derive scaling of the coupling parameter that allows for a synchronization of large networks for different network topologies.

Citation: Daniel M. N. Maia, Elbert E. N. Macau, Tiago Pereira, Serhiy Yanchuk. Synchronization in networks with strongly delayed couplings. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3461-3482. doi: 10.3934/dcdsb.2018234
References:
[1]

A. ArenasA. Díaz-GuileraJ. KurthsY. Moreno and C. Zhou, Synchronization in complex networks, Physics Reports, 469 (2008), 93-153.  doi: 10.1016/j.physrep.2008.09.002.

[2]

A. ArgyrisD. SyvridisL. LargerV. Annovazzi-LodiP. ColetI. FischerJ. Garcia-OjalvoC. R. MirassoL. Pesquera and K. A. Shore, Chaos-based communications at high bit rates using commercial fibre-optic links, Nature, 438 (2005), 343-346.  doi: 10.1038/nature04275.

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L. W. Beineke and R. J. Wilson, Topics in Algebraic Graph Theory, Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2004. doi: 10.1017/CBO9780511529993.

[4]

S. BoccalettiV. LatoraY. MorenoM. Chavez and D. Hwang, Complex networks: Structure and dynamics, Physics Reports, 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.

[5]

B. Bollobás, Random graphs, Combinatorics (Swansea, 1981), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, 52 (1981), 80-102.

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A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext. Springer New York, 2011. doi: 10.1007/978-1-4614-1939-6.

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S. A. CampbellI. Ncube and J. Wu, Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system, Physica D: Nonlinear Phenomena, 214 (2006), 101-119.  doi: 10.1016/j.physd.2005.12.008.

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P. Colet and R. Roy, Digital communication with synchronized chaotic lasers, Opt. Lett., 19 (1994), 2056-2058.  doi: 10.1364/OL.19.002056.

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T. Dahms, J. Lehnert and E. Schöll, Cluster and group synchronization in delay-coupled networks, Phys. Rev. E, 86 (2012), 016202. doi: 10.1103/PhysRevE.86.016202.

[10]

T. Erneux, Applied Delay Differential Equations, volume 3 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, 2009.

[11]

B. Fiedler, S. Yanchuk, V. Flunkert, P. Hövel, H. -J. Wünsche and E. Schöll, Delay stabilization of rotating waves near fold bifurcation and application to all-optical control of a semiconductor laser, Phys. Rev. E, 77 (2008), 066207, 9pp. doi: 10.1103/PhysRevE.77.066207.

[12]

M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Mathematical Journal, 23 (1973), 298-305. 

[13]

V. Flunkert, S. Yanchuk, T. Dahms and E. Schöll, Synchronizing distant nodes: A universal classification of networks, Phys. Rev. Lett. , 105 (2010), 254101. doi: 10.1103/PhysRevLett.105.254101.

[14]

S. Fortunato, Community detection in graphs, Physics Reports, 486 (2010), 75-174.  doi: 10.1016/j.physrep.2009.11.002.

[15]

J. Foss and J. Milton, Multistability in recurrent neural loops arising from delay, J Neurophysiol, 84 (2000), 975-985.  doi: 10.1152/jn.2000.84.2.975.

[16]

E. Fridman, Tutorial on lyapunov-based methods for time-delay systems, European Journal of Control, 20 (2014), 271-283.  doi: 10.1016/j.ejcon.2014.10.001.

[17]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.

[18]

J. D. Hart, J. P. Pade, T. Pereira, T. E. Murphy and R. Roy, Adding connections can hinder network synchronization of time-delayed oscillators, Phys. Rev. E, 92 (2015), 022804. doi: 10.1103/PhysRevE.92.022804.

[19]

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll and W. Kinzel, Strong and weak chaos in nonlinear networks with time-delayed couplings, Phys. Rev. Lett. , 107 (2011), 234102. doi: 10.1103/PhysRevLett.107.234102.

[20]

E. M. Izhikevich, Polychronization: Computation with spikes, Neural Computation, 18 (2006), 245-282.  doi: 10.1162/089976606775093882.

[21]

J. Javaloyes, P. Mandel and D. Pieroux, Dynamical properties of lasers coupled face to face, Phys. Rev. E, 67 (2003), 036201. doi: 10.1103/PhysRevE.67.036201.

[22]

W. Kinzel, A. Englert, G. Reents, M. Zigzag, and I. Kanter, Synchronization of networks of chaotic units with time-delayed couplings, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 79 (2009), 056207. doi: 10.1103/PhysRevE.79.056207.

[23]

M. LichtnerM. Wolfrum and S. Yanchuk, The spectrum of delay differential equations with large delay, SIAM J. Math. Anal., 43 (2011), 788-802.  doi: 10.1137/090766796.

[24]

L. Lücken, J. P. Pade, K. Knauer and S. Yanchuk, Reduction of interaction delays in networks, EPL (Europhysics Letters), 103 (2013), 10006.

[25]

D. N. M. MaiaE. E. N. Macau and T. Pereira, Persistence of network synchronization under nonidentical coupling functions, SIAM J. Appl. Dyn. Syst., 15 (2016), 1563-1580.  doi: 10.1137/15M1049786.

[26]

T. F. Móri, The maximum degree of the barabási-albert random tree, Comb. Probab. Comput., 14 (2005), 339-348.  doi: 10.1017/S0963548304006133.

[27]

M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167-256.  doi: 10.1137/S003614450342480.

[28]

G. OroszR. E. Wilson and G. Stepan, Traffic jams: dynamics and control, Phil. Trans. R. Soc. A, 368 (2010), 4455-4479.  doi: 10.1098/rsta.2010.0205.

[29]

T. PereiraJ. ElderingM. Rasmussen and A. Veneziani, Towards a theory for diffusive coupling functions allowing persistent synchronization, Nonlinearity, 27 (2014), 501-525.  doi: 10.1088/0951-7715/27/3/501.

[30]

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy and R. Roy, Robustness of optimal synchronization in real networks, Phys. Rev. Lett. , 107 (2011), 034102. doi: 10.1103/PhysRevLett.107.034102.

[31]

O. Riordan and A. Selby, The maximum degree of a random graph, Comb. Probab. Comput., 9 (2000), 549-572.  doi: 10.1017/S0963548300004491.

[32]

F. A. RodriguesT. K. DM. PeronP. Ji and J. Kurths, The kuramoto model in complex networks, Physics Reports, 610 (2016), 1-98.  doi: 10.1016/j.physrep.2015.10.008.

[33]

J. Schlesner, A. Amann, N. B. Janson, W. Just and E. Schöll, Self-stabilization of high-frequency oscillations in semiconductor superlattices by time-delay autosynchronization, Phys. Rev. E, 68 (2003), 066208. doi: 10.1103/PhysRevE.68.066208.

[34]

J. SieberM. WolfrumM. Lichtner and S. Yanchuk, On the stability of periodic orbits in delay equations with large delay, Discrete Contin. Dyn. Syst. A, 33 (2013), 3109-3134.  doi: 10.3934/dcds.2013.33.3109.

[35]

W. Singer, Synchronization of cortical activity and its putative role in information processing and learning, Annu. Rev. Physiol., 55 (1993), 349-374.  doi: 10.1146/annurev.ph.55.030193.002025.

[36]

H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, volume 57. Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

[37]

M. C. SorianoJ. Garcia-OjalvoC. R. Mirasso and I. Fischer, Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers, Rev. Mod. Phys., 85 (2013), 421-470.  doi: 10.1103/RevModPhys.85.421.

[38]

E. SteurW. MichielsH. Huijberts and H. Nijmeijer, Networks of diffusively time-delay coupled systems: Conditions for synchronization and its relation to the network topology, Physica D, 277 (2014), 22-39.  doi: 10.1016/j.physd.2014.03.004.

[39]

M. WolfrumS. YanchukP. Hövel and E. Schöll, Complex dynamics in delay-differential equations with large delay, Eur. Phys. J. Special Topics, 191 (2010), 91-103.  doi: 10.1140/epjst/e2010-01343-7.

[40]

J. Wu, Symmetric functional differential equations and neural networks with memory, Transactions of the American Mathematical Society, 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.

[41]

S. Yanchuk and G. Giacomelli, Spatio-temporal phenomena in complex systems with time delays, Journal of Physics A: Mathematical and Theoretical, 50 (2017), 103001, 56pp.

[42]

S. Yanchuk and P. Perlikowski, Delay and periodicity, Phys. Rev. E, 79 (2009), 046221, 9pp. doi: 10.1103/PhysRevE.79.046221.

[43]

S. Yanchuk and M. Wolfrum, Instabilities of equilibria of delay-differential equations with large delay, In D. H. van Campen, M. D. Lazurko, and W. P. J. M. van der Oever, editors, Proceedings of ENOC-2005, pages 1060-1065, Eindhoven, Netherlands, August 2005.

show all references

References:
[1]

A. ArenasA. Díaz-GuileraJ. KurthsY. Moreno and C. Zhou, Synchronization in complex networks, Physics Reports, 469 (2008), 93-153.  doi: 10.1016/j.physrep.2008.09.002.

[2]

A. ArgyrisD. SyvridisL. LargerV. Annovazzi-LodiP. ColetI. FischerJ. Garcia-OjalvoC. R. MirassoL. Pesquera and K. A. Shore, Chaos-based communications at high bit rates using commercial fibre-optic links, Nature, 438 (2005), 343-346.  doi: 10.1038/nature04275.

[3]

L. W. Beineke and R. J. Wilson, Topics in Algebraic Graph Theory, Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2004. doi: 10.1017/CBO9780511529993.

[4]

S. BoccalettiV. LatoraY. MorenoM. Chavez and D. Hwang, Complex networks: Structure and dynamics, Physics Reports, 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.

[5]

B. Bollobás, Random graphs, Combinatorics (Swansea, 1981), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, 52 (1981), 80-102.

[6]

A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext. Springer New York, 2011. doi: 10.1007/978-1-4614-1939-6.

[7]

S. A. CampbellI. Ncube and J. Wu, Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system, Physica D: Nonlinear Phenomena, 214 (2006), 101-119.  doi: 10.1016/j.physd.2005.12.008.

[8]

P. Colet and R. Roy, Digital communication with synchronized chaotic lasers, Opt. Lett., 19 (1994), 2056-2058.  doi: 10.1364/OL.19.002056.

[9]

T. Dahms, J. Lehnert and E. Schöll, Cluster and group synchronization in delay-coupled networks, Phys. Rev. E, 86 (2012), 016202. doi: 10.1103/PhysRevE.86.016202.

[10]

T. Erneux, Applied Delay Differential Equations, volume 3 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, 2009.

[11]

B. Fiedler, S. Yanchuk, V. Flunkert, P. Hövel, H. -J. Wünsche and E. Schöll, Delay stabilization of rotating waves near fold bifurcation and application to all-optical control of a semiconductor laser, Phys. Rev. E, 77 (2008), 066207, 9pp. doi: 10.1103/PhysRevE.77.066207.

[12]

M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Mathematical Journal, 23 (1973), 298-305. 

[13]

V. Flunkert, S. Yanchuk, T. Dahms and E. Schöll, Synchronizing distant nodes: A universal classification of networks, Phys. Rev. Lett. , 105 (2010), 254101. doi: 10.1103/PhysRevLett.105.254101.

[14]

S. Fortunato, Community detection in graphs, Physics Reports, 486 (2010), 75-174.  doi: 10.1016/j.physrep.2009.11.002.

[15]

J. Foss and J. Milton, Multistability in recurrent neural loops arising from delay, J Neurophysiol, 84 (2000), 975-985.  doi: 10.1152/jn.2000.84.2.975.

[16]

E. Fridman, Tutorial on lyapunov-based methods for time-delay systems, European Journal of Control, 20 (2014), 271-283.  doi: 10.1016/j.ejcon.2014.10.001.

[17]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.

[18]

J. D. Hart, J. P. Pade, T. Pereira, T. E. Murphy and R. Roy, Adding connections can hinder network synchronization of time-delayed oscillators, Phys. Rev. E, 92 (2015), 022804. doi: 10.1103/PhysRevE.92.022804.

[19]

S. Heiligenthal, T. Dahms, S. Yanchuk, T. Jüngling, V. Flunkert, I. Kanter, E. Schöll and W. Kinzel, Strong and weak chaos in nonlinear networks with time-delayed couplings, Phys. Rev. Lett. , 107 (2011), 234102. doi: 10.1103/PhysRevLett.107.234102.

[20]

E. M. Izhikevich, Polychronization: Computation with spikes, Neural Computation, 18 (2006), 245-282.  doi: 10.1162/089976606775093882.

[21]

J. Javaloyes, P. Mandel and D. Pieroux, Dynamical properties of lasers coupled face to face, Phys. Rev. E, 67 (2003), 036201. doi: 10.1103/PhysRevE.67.036201.

[22]

W. Kinzel, A. Englert, G. Reents, M. Zigzag, and I. Kanter, Synchronization of networks of chaotic units with time-delayed couplings, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 79 (2009), 056207. doi: 10.1103/PhysRevE.79.056207.

[23]

M. LichtnerM. Wolfrum and S. Yanchuk, The spectrum of delay differential equations with large delay, SIAM J. Math. Anal., 43 (2011), 788-802.  doi: 10.1137/090766796.

[24]

L. Lücken, J. P. Pade, K. Knauer and S. Yanchuk, Reduction of interaction delays in networks, EPL (Europhysics Letters), 103 (2013), 10006.

[25]

D. N. M. MaiaE. E. N. Macau and T. Pereira, Persistence of network synchronization under nonidentical coupling functions, SIAM J. Appl. Dyn. Syst., 15 (2016), 1563-1580.  doi: 10.1137/15M1049786.

[26]

T. F. Móri, The maximum degree of the barabási-albert random tree, Comb. Probab. Comput., 14 (2005), 339-348.  doi: 10.1017/S0963548304006133.

[27]

M. E. J. Newman, The structure and function of complex networks, SIAM Review, 45 (2003), 167-256.  doi: 10.1137/S003614450342480.

[28]

G. OroszR. E. Wilson and G. Stepan, Traffic jams: dynamics and control, Phil. Trans. R. Soc. A, 368 (2010), 4455-4479.  doi: 10.1098/rsta.2010.0205.

[29]

T. PereiraJ. ElderingM. Rasmussen and A. Veneziani, Towards a theory for diffusive coupling functions allowing persistent synchronization, Nonlinearity, 27 (2014), 501-525.  doi: 10.1088/0951-7715/27/3/501.

[30]

B. Ravoori, A. B. Cohen, J. Sun, A. E. Motter, T. E. Murphy and R. Roy, Robustness of optimal synchronization in real networks, Phys. Rev. Lett. , 107 (2011), 034102. doi: 10.1103/PhysRevLett.107.034102.

[31]

O. Riordan and A. Selby, The maximum degree of a random graph, Comb. Probab. Comput., 9 (2000), 549-572.  doi: 10.1017/S0963548300004491.

[32]

F. A. RodriguesT. K. DM. PeronP. Ji and J. Kurths, The kuramoto model in complex networks, Physics Reports, 610 (2016), 1-98.  doi: 10.1016/j.physrep.2015.10.008.

[33]

J. Schlesner, A. Amann, N. B. Janson, W. Just and E. Schöll, Self-stabilization of high-frequency oscillations in semiconductor superlattices by time-delay autosynchronization, Phys. Rev. E, 68 (2003), 066208. doi: 10.1103/PhysRevE.68.066208.

[34]

J. SieberM. WolfrumM. Lichtner and S. Yanchuk, On the stability of periodic orbits in delay equations with large delay, Discrete Contin. Dyn. Syst. A, 33 (2013), 3109-3134.  doi: 10.3934/dcds.2013.33.3109.

[35]

W. Singer, Synchronization of cortical activity and its putative role in information processing and learning, Annu. Rev. Physiol., 55 (1993), 349-374.  doi: 10.1146/annurev.ph.55.030193.002025.

[36]

H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, volume 57. Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

[37]

M. C. SorianoJ. Garcia-OjalvoC. R. Mirasso and I. Fischer, Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers, Rev. Mod. Phys., 85 (2013), 421-470.  doi: 10.1103/RevModPhys.85.421.

[38]

E. SteurW. MichielsH. Huijberts and H. Nijmeijer, Networks of diffusively time-delay coupled systems: Conditions for synchronization and its relation to the network topology, Physica D, 277 (2014), 22-39.  doi: 10.1016/j.physd.2014.03.004.

[39]

M. WolfrumS. YanchukP. Hövel and E. Schöll, Complex dynamics in delay-differential equations with large delay, Eur. Phys. J. Special Topics, 191 (2010), 91-103.  doi: 10.1140/epjst/e2010-01343-7.

[40]

J. Wu, Symmetric functional differential equations and neural networks with memory, Transactions of the American Mathematical Society, 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.

[41]

S. Yanchuk and G. Giacomelli, Spatio-temporal phenomena in complex systems with time delays, Journal of Physics A: Mathematical and Theoretical, 50 (2017), 103001, 56pp.

[42]

S. Yanchuk and P. Perlikowski, Delay and periodicity, Phys. Rev. E, 79 (2009), 046221, 9pp. doi: 10.1103/PhysRevE.79.046221.

[43]

S. Yanchuk and M. Wolfrum, Instabilities of equilibria of delay-differential equations with large delay, In D. H. van Campen, M. D. Lazurko, and W. P. J. M. van der Oever, editors, Proceedings of ENOC-2005, pages 1060-1065, Eindhoven, Netherlands, August 2005.

Figure 6.5.  Synchronization map in the $\sigma\times\tau$ parameter space. The color scale represents $\Re(\lambda)<0$ in which $\lambda$ is a solution of (6.7) with maximal real part. The white color stands for the instability region ($\Re(\lambda)>0$). The parameters used were $\alpha = 1$ and $\beta = \pi$.
Figure 6.2.  Numerically-computed-spectrum}Numerically computed spectrum for the equilibrium of two Stuart-Landau oscillators (see Sec. 6) coupled as in (1.1) with identity coupling function and parameters $\alpha = 1$, $\beta = \pi$, $\tau = 20$, and $\kappa = 0.7$. The points approaching the curve on the left side of the figure belongs to the pseudo-continuous spectrum and the isolated points on the right belongs to the strongly unstable spectrum. Solid lines show the re-scaled asymptotic continuous spectrum $\Gamma_{A}$. The gray strip represents a break on the figure, which is necessary due to the different scales of the two parts of the spectrum.
Figure 6.1.  A directed ring network with 4 nodes.
Figure 6.3.  Time series of the synchronization error for Eq. (3.3) and network given in Fig. 6.1 where $\kappa = 0.49$ for the left figure and $\kappa = 0.51$ for the right. Other parameters are $\alpha = -1$, $\beta = \pi$, $\tau = 100$. The history functions were taken as constant and non-zero.
Figure 6.4.  The asymptotic continuous spectrum (blue lines) and the pseudo-continuous spectrum (red dots) for the periodic solution of Stuart-Landau system given by Eqs. (6.6) and (6.7) respectively. The parameters are $\sigma = -0.08$ (with $\mu = \rho_{L} = 2$, the spectral radius of the Laplacian matrix of the network in Fig. 6.1, and $\kappa = 0.04$), $\alpha = 1$, $\beta = \pi$ and $\tau = 20$.
Figure 6.6.  Characteristic time for the synchronization of two Stuart-Landau coupled oscillators. The red curve is $t_{\text{tr}}(\kappa) = 20\ln^{-1}(2\kappa)$. The blue dots were obtained by fixing $\kappa$ and computing $\eta$, which stands for the angular coefficient of Eq. (6.8) in log scale in which $||\xi(t)|| = ||x_{1}(t)-x_{2}(t)||$, and then taking $t_{\text{tr}} = 1/\eta$. The parameters used were $\alpha = -1$, $\beta = \pi$ and $\tau = 20$. The history functions were taken as constant and non-zero.
Figure 7.1.  Illustrations of a BA network (left) and an ER network (right), both with $n = 100$. Some hubs in the BA network are highlighted with black color and bigger size.
Table 7.1.  Laplacian spectral radius $\rho_{L}$ and synchronization window for the coupling parameter $\kappa$ (for strong delay) of some regular graphs.
Graph $\rho_{L}$ Synchronization window
Complete $n$ $(0, r_{0}/n)$
Ring $\begin{array}{l} {\rm{4}}\;{\rm{if}}\;{n}\;{\rm{is}}\;{\rm{even}}\\ 2 + 2\cos \left( {2{\rm{\pi }}/n} \right)\;{\rm{if}}\;{n}\;{\rm{is}}\;{\rm{odd}} \end{array}$ $(0, r_{0}/4)$ or $\left(0, r_{0}/(2+2\cos\left({2\pi}/{n}\right))\right)$
Star $n$ $(0, r_{0}/n)$
Path $2+2\cos\left({\pi}/{n}\right)$ $\left(0, r_{0}/\left(2+2\cos\left({\pi}/{n}\right)\right)\right)$
Graph $\rho_{L}$ Synchronization window
Complete $n$ $(0, r_{0}/n)$
Ring $\begin{array}{l} {\rm{4}}\;{\rm{if}}\;{n}\;{\rm{is}}\;{\rm{even}}\\ 2 + 2\cos \left( {2{\rm{\pi }}/n} \right)\;{\rm{if}}\;{n}\;{\rm{is}}\;{\rm{odd}} \end{array}$ $(0, r_{0}/4)$ or $\left(0, r_{0}/(2+2\cos\left({2\pi}/{n}\right))\right)$
Star $n$ $(0, r_{0}/n)$
Path $2+2\cos\left({\pi}/{n}\right)$ $\left(0, r_{0}/\left(2+2\cos\left({\pi}/{n}\right)\right)\right)$
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