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July 2018, 23(5): 1931-1944. doi: 10.3934/dcdsb.2018189

Explosive synchronization in mono and multilayer networks

Inmaculada Leyva 1,2,, , Irene Sendiña-Nadal 1,2, and  Stefano Boccaletti 3,

1. 

Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
2. 

Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Madrid, Spain
3. 

CNR-Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy

* Corresponding author: I. Leyva

Received  April 2017 Revised  August 2017 Published  May 2018

Fund Project: Work partly supported by the Spanish Ministry of Economy under project FIS2013-41057-P and by GARECOM, Group of Research Excelence URJC-Banco de Santander. Authors acknowledge the computational resources and assistance provided by CRESCO, the supercomputing center of ENEA in Portici, Italy

Explosive synchronization, an abrupt transition to a collective coherent state, has been the focus of an extensive research since its first observation in scale-free networks with degree-frequency correlations. In this work, we report several scenarios where a first-order transition to synchronization occurs driven by the presence of a dependence between dynamics and network structure. Therefore, different mechanisms are shown to be able to prevent the formation of a giant synchronization cluster for sufficient large values of the coupling constant in both mono and multilayer networks. Using the Kuramoto model as a reference, we show how for an arbitrary network topology and frequency distribution, a very general weighting procedure acting on the weight of the links delays the synchronization transition forming independent synchronization clusters which suddenly merge above a critical threshold of the coupling constant. A completely different scenario in adaptive and multilayer networks is introduced which gives rise to the emergence of an explosive synchronization when a feedback between the dynamics and structure is operating by means of dependence links weighted through the order parameter.

Keywords: Complex networks, synchronization, chaos, phase transitions, coherence.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Inmaculada Leyva, Irene Sendiña-Nadal, Stefano Boccaletti. Explosive synchronization in mono and multilayer networks. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1931-1944. doi: 10.3934/dcdsb.2018189
References:
[1]

D. Achlioptas, R. M. D'Souza and J. Spencer, Explosive percolation in random networks, Science, 323 (2009), 1453–1455, URL http://www.sciencemag.org/content/323/5920/1453. abstract. doi: 10.1126/science.1167782.

[2]

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93–153, URL http://www.sciencedirect.com/science/ article/pii/S0370157308003384. doi: 10.1016/j.physrep.2008.09.002.

[3]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509–512, URL http://science.sciencemag.org/content/286/5439/509./ doi: 10.1126/science.286.5439.509.

[4]

S. Boccaletti, J. Almendral, S. Guan, I. Leyva, Z. Liu, I. Sendiña-Nadal, Z. Wang and Y. Zou, Explosive transitions in complex networks' structure and dynamics: Percolation and synchronization, Physics Reports, 660 (2016), 1–94, URL https://www.sciencedirect.com/ journal/physics-reports/vol/660. doi: 10.1016/j.physrep.2016.10.004.

[5]

S. Boccaletti, G. Bianconi, R. Criado, C. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang and M. Zanin, The structure and dynamics of multilayer networks, Phys. Rep., 544 (2014), 1–122, URL http://linkinghub.elsevier.com/retrieve/pii/S0370157314002105. doi: 10.1016/j.physrep.2014.07.001.

[6]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175–308, URL http://www.sciencedirect.com/science/article/pii/S037015730500462X. doi: 10.1016/j.physrep.2005.10.009.

[7]

M. M. Danziger, O. I. Moskalenko, S. A. Kurkin, X. Zhang, S. Havlin and S. Boccaletti, Explosive synchronization coexists with classical synchronization in the Kuramoto model Chaos, 26 (2016), 065307, 6 pp, URL http://scitation.aip.org/content/aip/journal/chaos/26/6/10.1063/1.4953345. doi: 10.1063/1.4953345.

[8]

P. Erdős and A. Rényi, On random graphs. i, Publicationes Mathematicae Debrecen, 6 (1959), 290-297.

[9]

J. Gómez-Gardeñes, S. Gómez, A. Arenas and Y. Moreno, Explosive synchronization transitions in scale-free networks, Phys. Rev. Lett., 106 (2011), 1–4, URL http://link.aps.org/doi/10.1103/PhysRevLett.106.128701
[10]

M. Kim, G. A. Mashour, S. Blain-Moraes, G. Vanini, V. Tarnal, E. Janke, A. G. Hudetz and U. Lee, Functional and topological conditions for explosive synchronization develop in human brain networks with the onset of anesthetic-induced unconsciousness, Front. Comput. Neurosci., 10 (2016), p1, URL https://www.ncbi.nlm.nih.gov/pubmed/26834616.

10.3389/fncom.2016.00001
[11]

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag, 1984. doi: 10.1007/978-3-642-69689-3.

[12]

I. Leyva, A. Navas, I. Sendiña-Nadal, J. A. Almendral, J. M. Buldú, M. Zanin, D. Papo and S. Boccaletti, Explosive transitions to synchronization in networks of phase oscillators, Sci. Rep., 3 (2013), 1281, URL http://www.nature.com/doifinder/10.1038/srep01281. doi: 10.1038/srep01281.

[13]

I. Leyva, I. Sendiña-Nadal, J. A. Almendral, A. Navas, S. Olmi and S. Boccaletti, Explosive synchronization in weighted complex networks, Phys. Rev. E, 88 (2013), 042808, URL http://link.aps.org/doi/10.1103/PhysRevE.88.042808. doi: 10.1103/PhysRevE.88.042808.

[14]

I. Leyva, R. Sevilla-Escoboza, J. M. Buldú, I. Sendiña-Nadal, J. Gómez-Gardeñes, A. Arenas, Y. Moreno, S. Gómez, R. Jaimes-Reátegui and S. Boccaletti, Explosive first-order transition to synchrony in networked chaotic oscillators, Phys. Rev. Lett., 108 (2012), 168702, URL http://link.aps.org/doi/10.1103/PhysRevLett.108.168702. doi: 10.1103/PhysRevLett.108.168702.

[15]

E. A. Martens, E. Barreto, S. H. Strogatz, E. Ott, P. So and T. M. Antonsen, Exact results for the Kuramoto model with a bimodal frequency distributions, Phys. Rev. E, 79 (2009), 026204, 11 pp, URL https://link.aps.org/doi/10.1103/PhysRevE.79.026204 doi: 10.1103/PhysRevE.79.026204.

[16]

A. Navas, J. A. Villacorta-Atienza, I. Leyva, J. A. Almendral, I. Sendiña Nadal and S. Boccaletti, Effective centrality and explosive synchronization in complex networks, Phys. Rev. E, 92 (2015), 062820, URL http://link.aps.org/doi/10.1103/PhysRevE.92.062820. doi: 10.1103/PhysRevE.92.062820.

[17]

D. Pazó, Thermodynamic limit of the first-order phase transition in the Kuramoto model, Phys. Rev. E, 72 (2005), 046211, 6pp, URL http://journals.aps.org/pre/abstract/10.1103/PhysRevE.72.046211. doi: 10.1103/PhysRevE.72.046211.

[18]

M. Rohden, A. Sorge, M. Timme and D. Witthaut, Self-organized synchronization in decentralized power grids, Phys. Rev. Lett., 109 (2012), 064101, URL http://link.aps.org/doi/10.1103/PhysRevLett.109.064101. doi: 10.1103/PhysRevLett.109.064101.

[19]

X. Zhang, S. Boccaletti, S. Guan and Z. Liu, Explosive synchronization in adaptive and multilayer networks, Phys. Rev. Lett., 114 (2015), 038701, URL http://link.aps.org/doi/10.1103/PhysRevLett.114.038701. doi: 10.1103/PhysRevLett.114.038701.

[20]

X. Zhang, Y. Zou, S. Boccaletti and Z. Liu, Explosive synchronization as a process of explosive percolation in dynamical phase space, Sci. Rep., 4 (2014), 5200, URL http://www.ncbi.nlm.nih.gov/pubmed/24903808. doi: 10.1038/srep05200.

show all references

References:
[1]

D. Achlioptas, R. M. D'Souza and J. Spencer, Explosive percolation in random networks, Science, 323 (2009), 1453–1455, URL http://www.sciencemag.org/content/323/5920/1453. abstract. doi: 10.1126/science.1167782.

[2]

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Phys. Rep., 469 (2008), 93–153, URL http://www.sciencedirect.com/science/ article/pii/S0370157308003384. doi: 10.1016/j.physrep.2008.09.002.

[3]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509–512, URL http://science.sciencemag.org/content/286/5439/509./ doi: 10.1126/science.286.5439.509.

[4]

S. Boccaletti, J. Almendral, S. Guan, I. Leyva, Z. Liu, I. Sendiña-Nadal, Z. Wang and Y. Zou, Explosive transitions in complex networks' structure and dynamics: Percolation and synchronization, Physics Reports, 660 (2016), 1–94, URL https://www.sciencedirect.com/ journal/physics-reports/vol/660. doi: 10.1016/j.physrep.2016.10.004.

[5]

S. Boccaletti, G. Bianconi, R. Criado, C. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang and M. Zanin, The structure and dynamics of multilayer networks, Phys. Rep., 544 (2014), 1–122, URL http://linkinghub.elsevier.com/retrieve/pii/S0370157314002105. doi: 10.1016/j.physrep.2014.07.001.

[6]

S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175–308, URL http://www.sciencedirect.com/science/article/pii/S037015730500462X. doi: 10.1016/j.physrep.2005.10.009.

[7]

M. M. Danziger, O. I. Moskalenko, S. A. Kurkin, X. Zhang, S. Havlin and S. Boccaletti, Explosive synchronization coexists with classical synchronization in the Kuramoto model Chaos, 26 (2016), 065307, 6 pp, URL http://scitation.aip.org/content/aip/journal/chaos/26/6/10.1063/1.4953345. doi: 10.1063/1.4953345.

[8]

P. Erdős and A. Rényi, On random graphs. i, Publicationes Mathematicae Debrecen, 6 (1959), 290-297.

[9]

J. Gómez-Gardeñes, S. Gómez, A. Arenas and Y. Moreno, Explosive synchronization transitions in scale-free networks, Phys. Rev. Lett., 106 (2011), 1–4, URL http://link.aps.org/doi/10.1103/PhysRevLett.106.128701
[10]

M. Kim, G. A. Mashour, S. Blain-Moraes, G. Vanini, V. Tarnal, E. Janke, A. G. Hudetz and U. Lee, Functional and topological conditions for explosive synchronization develop in human brain networks with the onset of anesthetic-induced unconsciousness, Front. Comput. Neurosci., 10 (2016), p1, URL https://www.ncbi.nlm.nih.gov/pubmed/26834616.

10.3389/fncom.2016.00001
[11]

Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer-Verlag, 1984. doi: 10.1007/978-3-642-69689-3.

[12]

I. Leyva, A. Navas, I. Sendiña-Nadal, J. A. Almendral, J. M. Buldú, M. Zanin, D. Papo and S. Boccaletti, Explosive transitions to synchronization in networks of phase oscillators, Sci. Rep., 3 (2013), 1281, URL http://www.nature.com/doifinder/10.1038/srep01281. doi: 10.1038/srep01281.

[13]

I. Leyva, I. Sendiña-Nadal, J. A. Almendral, A. Navas, S. Olmi and S. Boccaletti, Explosive synchronization in weighted complex networks, Phys. Rev. E, 88 (2013), 042808, URL http://link.aps.org/doi/10.1103/PhysRevE.88.042808. doi: 10.1103/PhysRevE.88.042808.

[14]

I. Leyva, R. Sevilla-Escoboza, J. M. Buldú, I. Sendiña-Nadal, J. Gómez-Gardeñes, A. Arenas, Y. Moreno, S. Gómez, R. Jaimes-Reátegui and S. Boccaletti, Explosive first-order transition to synchrony in networked chaotic oscillators, Phys. Rev. Lett., 108 (2012), 168702, URL http://link.aps.org/doi/10.1103/PhysRevLett.108.168702. doi: 10.1103/PhysRevLett.108.168702.

[15]

E. A. Martens, E. Barreto, S. H. Strogatz, E. Ott, P. So and T. M. Antonsen, Exact results for the Kuramoto model with a bimodal frequency distributions, Phys. Rev. E, 79 (2009), 026204, 11 pp, URL https://link.aps.org/doi/10.1103/PhysRevE.79.026204 doi: 10.1103/PhysRevE.79.026204.

[17]

D. Pazó, Thermodynamic limit of the first-order phase transition in the Kuramoto model, Phys. Rev. E, 72 (2005), 046211, 6pp, URL http://journals.aps.org/pre/abstract/10.1103/PhysRevE.72.046211. doi: 10.1103/PhysRevE.72.046211.

[18]

M. Rohden, A. Sorge, M. Timme and D. Witthaut, Self-organized synchronization in decentralized power grids, Phys. Rev. Lett., 109 (2012), 064101, URL http://link.aps.org/doi/10.1103/PhysRevLett.109.064101. doi: 10.1103/PhysRevLett.109.064101.

[19]

X. Zhang, S. Boccaletti, S. Guan and Z. Liu, Explosive synchronization in adaptive and multilayer networks, Phys. Rev. Lett., 114 (2015), 038701, URL http://link.aps.org/doi/10.1103/PhysRevLett.114.038701. doi: 10.1103/PhysRevLett.114.038701.

[20]

X. Zhang, Y. Zou, S. Boccaletti and Z. Liu, Explosive synchronization as a process of explosive percolation in dynamical phase space, Sci. Rep., 4 (2014), 5200, URL http://www.ncbi.nlm.nih.gov/pubmed/24903808. doi: 10.1038/srep05200.

Figure 1.  (a) Phase synchronization level $r$ vs the coupling strength $\lambda$, for different values of the frequency gap $\gamma$ (see. Eq. (2)) at $\langle k\rangle = 40$ with network size N = 500. Frequency distribution $g(\omega)$ is chosen to be homogeneous in the [0, 1] interval. (b) Same as in (a), but for different values of the average degree $\langle k\rangle$ at $\gamma = 0.4$. In both panels, the legends report the color and symbol codes for the different plotted curves. In (c) and (d), the degree $k_i$ that each node achieves after the network growth is completed (upper plots) and the average of the natural frequencies $\langle \omega_j\rangle$ of the neighboring nodes ($j\in {\it N}(i)$, bottom plots) are reported vs. the node's natural frequency $\omega_i$, for $\langle k \rangle = 100$ and frequency gaps $\gamma = 0.0$ (c), and $\gamma = 0.4$ (d). The red solid line in (d) is a sketch of the theoretical prediction $f(\omega)$. Panels (e) and (f) show $r$ (color coded according to the color bar) in the parameter space ($\lambda, \gamma$) for (e) $\langle k \rangle = 20$ and (f) $\langle k \rangle = 60$. The horizontal dashed lines mark the separation between the region of the parameter space where a second-order transition occurs (below the line) and that in which the transition is instead of the first order type (above the line). The yellow striped area delimits the hysteresis region
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Figure 2.  (Top row) $r$ vs. $\lambda$. Synchronization transition schemes for different $g(\omega)$ or network construction rules, for system size N = 500. (Bottom row) The final node degree $k_i$ as a function of $\omega_i$. (a)-(b) Rayleigh distribution for $\gamma = 0.3$. In (b), the red solid line depicts the theoretical prediction $f(\omega)$, obtained with the same method of the red solid line in panel (d) of Fig. 1; (c)-(d) uniform frequency distribution, but network constructed accordingly to a local mean field condition (see text) for $\gamma = 0$, and $\gamma = 0.4$ (see legend for color code). In panel (d) $\gamma = 0.4$. In all cases, $\langle k\rangle = 60$
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Figure 3.  (a) Synchronization schemes ($r$ vs. $\lambda /\langle k\rangle$) for ER networks of size $N = 500$, $\langle k\rangle $ = 30, for the un-weighted case (dark blue squares), and weighted cases for several frequency distributions within the range $[0, 1]$. (b) Node strengths $s_i$ (see text for definition) vs. natural frequencies $\omega_i$, for the un-weighted (dark blue squares) and weighted (light blue circles) networks reported in (a). The solid line corresponds to the analytical prediction
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Figure 4.  (a) $I$ as a function of $\mu = \lambda r$ (solid curve) as given by Eq. (16). The dashed lines are those straight lines whose intersection with $I$ marks the backward ($\lambda_1^c$) and forward ($\lambda_2^c$) critical points of ES for a fully connected network and for a uniform frequency distribution. (b) The corresponding synchronization order parameter $r$ as a function of the coupling strength. Solid (dashed) curves correspond to the stable (unstable) solution. Dotted vertical lines mark the region of hysteresis defined by $\lambda_1^c$ and $\lambda_2^c$ in (a)
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Figure 5.  Plots of the matrix $r_{ij}$ (Eq. (17)) for fully connected (a)-(d) and ER (e-h) networks. Oscillators are labeled in ascending order of the frequency $\omega_i$. The critical coupling strengths where the ES takes place are $\lambda_c$ = 2.13 in $\lambda_c = 2.17$ respectively
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Figure 6.  Comparison between synchronization centrality $\Lambda^C_i$ (black squares) and topological centrality $\Lambda^A_i$ (red dots). (a) ER networks, $\langle k \rangle$ = 50, $\Lambda^C_i$ and $\Lambda^A_i$ are reported vs. the nodes' natural frequencies $\omega_i$; (b) SF networks, $\langle k \rangle$ = 12, $\Lambda^C_i$ and $\Lambda^A_i$ are plotted vs. the node degrees $k_i$. All data refer to ensemble averages over 100 different network realizations for N = 500 nodes
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Figure 7.  Forward (black line with squares) and backward (red line with circles) synchronization transitions for a single network with N = 1000 and f = 1. The insets report the dependence on $f$ of the average transition points $\lambda_F$ and $\lambda_B$ for ten realizations
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Figure 8.  Analytical solutions of Eq. (23) for the order parameter $r$ vs. $\lambda$ for the ER network with $f$ = 1
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Figure 9.  Synchronization parameters $r^{1}$ and $r^{2}$ vs. $\lambda$ for two inter-dependent networks with $N = 10^3$ and $f = 1$. Squares and circles (triangles and stars) denote the forward (backward) transitions, and the insets report the average width $\langle d\rangle$ of the hysteresis loop as a function of $f$. Here layer 1 is a ER network with $\langle k\rangle = 12$. From (a) to (b), layer 2 is ER network with $\langle k\rangle = 12$ and $\langle k\rangle = 6$, and $g(w_i^2)$ is a random homogeneous distribution in the range $[-1, 1]$. From (c) to (d), layer 2 is ER network and SF network with $\langle k\rangle = 12$, and $g(w_i^2)$ is Lorentzian distribution and random homogeneous fashion
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