Abstract

An analogy between crowd synchrony and multi-layer neural network architectures is proposed. It indicates that many non-identical dynamical elements (oscillators) communicating indirectly via a few mediators (hubs) can synchronize when the number of delayed couplings to the hubs or the strength of the couplings is large enough. This phenomenon is modeled using a system of semiconductor lasers optically delay-coupled in either a fully connected or a diluted manner to a fixed number of non-identical central hub lasers. A universal phase transition to crowd synchrony with hysteresis is observed, where the time to achieve synchronization diverges near the critical coupling independent of the number of hubs.

© 2012 OSA

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2012

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(16), 168702 (2012).
[CrossRef] [PubMed]

2011

J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, “Explosive synchronization transitions in scale-free networks,” Phys. Rev. Lett.106(12), 128701 (2011).
[CrossRef] [PubMed]

2010

T. Danino, O. Mondragón-Palomino, L. Tsimring, and J. Hasty, “A synchronized quorum of genetic clocks,” Nature463(7279), 326–330 (2010).
[CrossRef] [PubMed]

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett.105(26), 264101 (2010).
[CrossRef] [PubMed]

2009

A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science323(5914), 614–617 (2009).
[CrossRef] [PubMed]

2007

S. De Monte, F. d’Ovidio, S. Danø, and P. G. Sørensen, “Dynamical quorum sensing: Population density encoded in cellular dynamics,” Proc. Natl. Acad. Sci. U.S.A.104(47), 18377–18381 (2007).
[CrossRef] [PubMed]

2004

J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, “Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing,” Proc. Natl. Acad. Sci. U.S.A.101(30), 10955–10960 (2004).
[CrossRef] [PubMed]

2001

P. Dallard, A. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. Ridsdill Smith, and M. Willford, “The London millennium footbridge,” Structural Engineer79, 17–21 (2001).

1994

A. Priel, M. Blatt, T. Grossmann, E. Domany, and I. Kanter, “Computational capabilities of restricted two-layered perceptrons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics50(1), 577–595 (1994).
[CrossRef] [PubMed]

1990

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, “Observation of antiphase states in a multimode laser,” Phys. Rev. Lett.65(14), 1749–1752 (1990).
[CrossRef] [PubMed]

1989

M. Opper, “Learning in neural networks: Solvable dynamics,” Europhys. Lett.8(4), 389–392 (1989).
[CrossRef]

1987

T. R. Kirkpatrick and D. Thirumalai, “Dynamics of the structural glass transition and the p-spin-interaction spin-glass model,” Phys. Rev. Lett.58(20), 2091–2094 (1987).
[CrossRef] [PubMed]

1985

D. J. Gross, I. Kanter, and H. Sompolinsky, “Mean-field theory of the Potts glass,” Phys. Rev. Lett.55(3), 304–307 (1985).
[CrossRef] [PubMed]

1958

F. Rosenblatt, “The perceptron: A probabilistic model for information storage and organization in the brain,” Psychol. Rev.65(6), 386–408 (1958).
[CrossRef] [PubMed]

Arenas, A.

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(16), 168702 (2012).
[CrossRef] [PubMed]

J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, “Explosive synchronization transitions in scale-free networks,” Phys. Rev. Lett.106(12), 128701 (2011).
[CrossRef] [PubMed]

Blatt, M.

A. Priel, M. Blatt, T. Grossmann, E. Domany, and I. Kanter, “Computational capabilities of restricted two-layered perceptrons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics50(1), 577–595 (1994).
[CrossRef] [PubMed]

Boccaletti, S.

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(16), 168702 (2012).
[CrossRef] [PubMed]

Bracikowski, C.

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, “Observation of antiphase states in a multimode laser,” Phys. Rev. Lett.65(14), 1749–1752 (1990).
[CrossRef] [PubMed]

Buldú, J. M.

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(16), 168702 (2012).
[CrossRef] [PubMed]

d’Ovidio, F.

S. De Monte, F. d’Ovidio, S. Danø, and P. G. Sørensen, “Dynamical quorum sensing: Population density encoded in cellular dynamics,” Proc. Natl. Acad. Sci. U.S.A.104(47), 18377–18381 (2007).
[CrossRef] [PubMed]

Dallard, P.

P. Dallard, A. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. Ridsdill Smith, and M. Willford, “The London millennium footbridge,” Structural Engineer79, 17–21 (2001).

Danino, T.

T. Danino, O. Mondragón-Palomino, L. Tsimring, and J. Hasty, “A synchronized quorum of genetic clocks,” Nature463(7279), 326–330 (2010).
[CrossRef] [PubMed]

Danø, S.

S. De Monte, F. d’Ovidio, S. Danø, and P. G. Sørensen, “Dynamical quorum sensing: Population density encoded in cellular dynamics,” Proc. Natl. Acad. Sci. U.S.A.104(47), 18377–18381 (2007).
[CrossRef] [PubMed]

De Monte, S.

S. De Monte, F. d’Ovidio, S. Danø, and P. G. Sørensen, “Dynamical quorum sensing: Population density encoded in cellular dynamics,” Proc. Natl. Acad. Sci. U.S.A.104(47), 18377–18381 (2007).
[CrossRef] [PubMed]

Domany, E.

A. Priel, M. Blatt, T. Grossmann, E. Domany, and I. Kanter, “Computational capabilities of restricted two-layered perceptrons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics50(1), 577–595 (1994).
[CrossRef] [PubMed]

Elowitz, M. B.

J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, “Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing,” Proc. Natl. Acad. Sci. U.S.A.101(30), 10955–10960 (2004).
[CrossRef] [PubMed]

Fitzpatrick, A.

P. Dallard, A. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. Ridsdill Smith, and M. Willford, “The London millennium footbridge,” Structural Engineer79, 17–21 (2001).

Flint, A.

P. Dallard, A. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. Ridsdill Smith, and M. Willford, “The London millennium footbridge,” Structural Engineer79, 17–21 (2001).

Garcia-Ojalvo, J.

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett.105(26), 264101 (2010).
[CrossRef] [PubMed]

J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, “Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing,” Proc. Natl. Acad. Sci. U.S.A.101(30), 10955–10960 (2004).
[CrossRef] [PubMed]

Gómez, S.

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(16), 168702 (2012).
[CrossRef] [PubMed]

J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, “Explosive synchronization transitions in scale-free networks,” Phys. Rev. Lett.106(12), 128701 (2011).
[CrossRef] [PubMed]

Gómez-Gardeñes, J.

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(16), 168702 (2012).
[CrossRef] [PubMed]

J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, “Explosive synchronization transitions in scale-free networks,” Phys. Rev. Lett.106(12), 128701 (2011).
[CrossRef] [PubMed]

Gross, D. J.

D. J. Gross, I. Kanter, and H. Sompolinsky, “Mean-field theory of the Potts glass,” Phys. Rev. Lett.55(3), 304–307 (1985).
[CrossRef] [PubMed]

Grossmann, T.

A. Priel, M. Blatt, T. Grossmann, E. Domany, and I. Kanter, “Computational capabilities of restricted two-layered perceptrons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics50(1), 577–595 (1994).
[CrossRef] [PubMed]

Hasty, J.

T. Danino, O. Mondragón-Palomino, L. Tsimring, and J. Hasty, “A synchronized quorum of genetic clocks,” Nature463(7279), 326–330 (2010).
[CrossRef] [PubMed]

Huang, Z.

A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science323(5914), 614–617 (2009).
[CrossRef] [PubMed]

Jaimes-Reátegui, R.

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(16), 168702 (2012).
[CrossRef] [PubMed]

James, G.

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, “Observation of antiphase states in a multimode laser,” Phys. Rev. Lett.65(14), 1749–1752 (1990).
[CrossRef] [PubMed]

Kanter, I.

A. Priel, M. Blatt, T. Grossmann, E. Domany, and I. Kanter, “Computational capabilities of restricted two-layered perceptrons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics50(1), 577–595 (1994).
[CrossRef] [PubMed]

D. J. Gross, I. Kanter, and H. Sompolinsky, “Mean-field theory of the Potts glass,” Phys. Rev. Lett.55(3), 304–307 (1985).
[CrossRef] [PubMed]

Kirkpatrick, T. R.

T. R. Kirkpatrick and D. Thirumalai, “Dynamics of the structural glass transition and the p-spin-interaction spin-glass model,” Phys. Rev. Lett.58(20), 2091–2094 (1987).
[CrossRef] [PubMed]

Le Bourva, S.

P. Dallard, A. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. Ridsdill Smith, and M. Willford, “The London millennium footbridge,” Structural Engineer79, 17–21 (2001).

Leyva, I.

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(16), 168702 (2012).
[CrossRef] [PubMed]

Low, A.

P. Dallard, A. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. Ridsdill Smith, and M. Willford, “The London millennium footbridge,” Structural Engineer79, 17–21 (2001).

Masoller, C.

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett.105(26), 264101 (2010).
[CrossRef] [PubMed]

Mondragón-Palomino, O.

T. Danino, O. Mondragón-Palomino, L. Tsimring, and J. Hasty, “A synchronized quorum of genetic clocks,” Nature463(7279), 326–330 (2010).
[CrossRef] [PubMed]

Moreno, Y.

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(16), 168702 (2012).
[CrossRef] [PubMed]

J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, “Explosive synchronization transitions in scale-free networks,” Phys. Rev. Lett.106(12), 128701 (2011).
[CrossRef] [PubMed]

Opper, M.

M. Opper, “Learning in neural networks: Solvable dynamics,” Europhys. Lett.8(4), 389–392 (1989).
[CrossRef]

Priel, A.

A. Priel, M. Blatt, T. Grossmann, E. Domany, and I. Kanter, “Computational capabilities of restricted two-layered perceptrons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics50(1), 577–595 (1994).
[CrossRef] [PubMed]

Ridsdill Smith, R.

P. Dallard, A. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. Ridsdill Smith, and M. Willford, “The London millennium footbridge,” Structural Engineer79, 17–21 (2001).

Rosenblatt, F.

F. Rosenblatt, “The perceptron: A probabilistic model for information storage and organization in the brain,” Psychol. Rev.65(6), 386–408 (1958).
[CrossRef] [PubMed]

Roy, R.

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett.105(26), 264101 (2010).
[CrossRef] [PubMed]

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, “Observation of antiphase states in a multimode laser,” Phys. Rev. Lett.65(14), 1749–1752 (1990).
[CrossRef] [PubMed]

Sendiña-Nadal, I.

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(16), 168702 (2012).
[CrossRef] [PubMed]

Sevilla-Escoboza, R.

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(16), 168702 (2012).
[CrossRef] [PubMed]

Showalter, K.

A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science323(5914), 614–617 (2009).
[CrossRef] [PubMed]

Sompolinsky, H.

D. J. Gross, I. Kanter, and H. Sompolinsky, “Mean-field theory of the Potts glass,” Phys. Rev. Lett.55(3), 304–307 (1985).
[CrossRef] [PubMed]

Sørensen, P. G.

S. De Monte, F. d’Ovidio, S. Danø, and P. G. Sørensen, “Dynamical quorum sensing: Population density encoded in cellular dynamics,” Proc. Natl. Acad. Sci. U.S.A.104(47), 18377–18381 (2007).
[CrossRef] [PubMed]

Strogatz, S. H.

J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, “Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing,” Proc. Natl. Acad. Sci. U.S.A.101(30), 10955–10960 (2004).
[CrossRef] [PubMed]

Taylor, A. F.

A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science323(5914), 614–617 (2009).
[CrossRef] [PubMed]

Thirumalai, D.

T. R. Kirkpatrick and D. Thirumalai, “Dynamics of the structural glass transition and the p-spin-interaction spin-glass model,” Phys. Rev. Lett.58(20), 2091–2094 (1987).
[CrossRef] [PubMed]

Tinsley, M. R.

A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science323(5914), 614–617 (2009).
[CrossRef] [PubMed]

Tsimring, L.

T. Danino, O. Mondragón-Palomino, L. Tsimring, and J. Hasty, “A synchronized quorum of genetic clocks,” Nature463(7279), 326–330 (2010).
[CrossRef] [PubMed]

Wang, F.

A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, and K. Showalter, “Dynamical quorum sensing and synchronization in large populations of chemical oscillators,” Science323(5914), 614–617 (2009).
[CrossRef] [PubMed]

Wiesenfeld, K.

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, “Observation of antiphase states in a multimode laser,” Phys. Rev. Lett.65(14), 1749–1752 (1990).
[CrossRef] [PubMed]

Willford, M.

P. Dallard, A. Fitzpatrick, A. Flint, S. Le Bourva, A. Low, R. Ridsdill Smith, and M. Willford, “The London millennium footbridge,” Structural Engineer79, 17–21 (2001).

Zamora-Munt, J.

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett.105(26), 264101 (2010).
[CrossRef] [PubMed]

Europhys. Lett.

M. Opper, “Learning in neural networks: Solvable dynamics,” Europhys. Lett.8(4), 389–392 (1989).
[CrossRef]

Nature

T. Danino, O. Mondragón-Palomino, L. Tsimring, and J. Hasty, “A synchronized quorum of genetic clocks,” Nature463(7279), 326–330 (2010).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

A. Priel, M. Blatt, T. Grossmann, E. Domany, and I. Kanter, “Computational capabilities of restricted two-layered perceptrons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics50(1), 577–595 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett.

T. R. Kirkpatrick and D. Thirumalai, “Dynamics of the structural glass transition and the p-spin-interaction spin-glass model,” Phys. Rev. Lett.58(20), 2091–2094 (1987).
[CrossRef] [PubMed]

D. J. Gross, I. Kanter, and H. Sompolinsky, “Mean-field theory of the Potts glass,” Phys. Rev. Lett.55(3), 304–307 (1985).
[CrossRef] [PubMed]

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett.105(26), 264101 (2010).
[CrossRef] [PubMed]

K. Wiesenfeld, C. Bracikowski, G. James, and R. Roy, “Observation of antiphase states in a multimode laser,” Phys. Rev. Lett.65(14), 1749–1752 (1990).
[CrossRef] [PubMed]

J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, “Explosive synchronization transitions in scale-free networks,” Phys. Rev. Lett.106(12), 128701 (2011).
[CrossRef] [PubMed]

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(16), 168702 (2012).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A.

J. Garcia-Ojalvo, M. B. Elowitz, and S. H. Strogatz, “Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing,” Proc. Natl. Acad. Sci. U.S.A.101(30), 10955–10960 (2004).
[CrossRef] [PubMed]

S. De Monte, F. d’Ovidio, S. Danø, and P. G. Sørensen, “Dynamical quorum sensing: Population density encoded in cellular dynamics,” Proc. Natl. Acad. Sci. U.S.A.104(47), 18377–18381 (2007).
[CrossRef] [PubMed]

Psychol. Rev.

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic of M non-identical elements interacting with each other via P hidden units (HUs), where σik stands for the coupling strength between element i and the kth HU. A dilution consists of setting a fraction of the couplings to vanishing values. (b) Schematic of an architecture with 2M elements and two non-identical HUs, with frequency detuning, wi between them, where only 2L out of the 2M elements have couplings to both HUs. Singly connected elements have coupling strengths σ1while elements coupled to two HUs have coupling strengths σ2.

Fig. 2
Fig. 2

Color chart of intensity correlation among all pair of lasers in the lower layer, ρ(i,j), for the architecture of Fig. 1(a) with M = 20 and P = 3 for three different coupling strengths. The lower layer lasers are sorted by increasing frequency. (a) σ = 24.55 where all pair correlations are below the threshold (below criticality). (b) σ = 24.67, correlation begins to form. (c) σ = 24.78, all pairs of lasers are correlated. The transition to crowd synchrony is identified at σc = 24.6398 as explained in the text.

Fig. 3
Fig. 3

(a) A power law behavior of synchronization time as a function of the deviation from σc. (b) Average correlation among all pairs of lower level lasers as a function of the normalized sigma (σ/σc) shows a hysteresis loop. (c) Decay of the correlation as a function of time, where σ is abruptly changed from a synchronized state, σ>σc, to σ = 0.995σc (see text for detail). Results indicate data collapse and are obtained for P = 3 and M in the range [20,100]. (d) Correlation decay exponent as a function of M obtained from the data of panel c.

Fig. 4
Fig. 4

(a) Critical coupling as a function of number of lower layer lasers, for different number of hidden units with the lack of dilution. (b) Critical coupling as a function of number of lower layer lasers, for different connectivity ratio, R, and with P = 4. The dashed lines are given by the middle expression of Eq. (4) with A~169.6.

Fig. 5
Fig. 5

Correlation among all lower layer lasers in Fig. 1(b) as a function of σ2, for different frequency detuning between the two HU lasers. As detuning is increased, the average correlation weakens. M = 40/80, L = 0.2M, σ1 = 1.3σc.

Equations (8)

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E ˙ i   = i w i E i +γ( 1+iα )( G i 1 ) E i + k=1 P σ ik E k ( tτ ) e i w 0 τ + D ζ i ( t )
E ˙ k  = i w k E k +γ( 1+iα )( G k 1 ) E k + i=1 M σ ik E i ( tτ ) e i w 0 τ + D ζ k ( t )
N ˙ i,k = γ e ( p i,k N i,k G i,k | E i,k |) 2
σ c M 0.463 P 0.492 R 0.993 ~ (MP) 0.5 R  
| E ˙ i |~ k=1 P σ ik E k ( tτ )~σRP E k ¯
| E ˙ k |~ i=1 M σ ik E i ( tτ )~ σRM E i ¯
| E ¨ i |=σRP| E ˙ k |= σ 2 R 2 PN E ¯ i 
a=σR PN

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