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Dynamical systems associated with adjacency matrices
Explosive synchronization in mono and multilayer networks
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 |
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.
References:
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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./
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[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. |
[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. |









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