ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Belykh V. N., Belykh I. V., Hasler M. . Small-world networks: dynamical models and synchronization. Izvestiya VUZ. Applied Nonlinear Dynamics, 2003, vol. 11, iss. 3, pp. 67-76. DOI: 10.18500/0869-6632-2003-11-3-67-76

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon and_2003-3_belykh-etal.pdf
(downloads: 0)
Language: 
English
Heading: 
Deterministic Chaos
Article type: 
Article
UDC: 
537.86
DOI: 
10.18500/0869-6632-2003-11-3-67-76

Small-world networks: dynamical models and synchronization

Autors: 
Belykh Vladimir Nikolaevich, Volga State Academy of Water Transport (VGAVT)
Belykh Igor V., Lobachevsky State University of Nizhny Novgorod
Hasler Martin , Swiss Federal Institute of Technology Lausanne
Abstract: 

This paper provides а short review оf recent results оn synchronization in small-world dynamical networks of coupled oscillators. We also propose a new model of small-world networks of cells with a time-varying coupling and study its synchronization properties. It is shown that such а time-varying structure of the network can ensure more reliable synchronization than the conventional small-worlds. The term «small world» refers to a network of locally connected nodes having a few additional long-range shortcuts chosen at random. The addition оf thе shortcuts sharply reduces the average distance between the nodes and therefore provides the so-called small-world effect. Discovered first in social networks, the small-world effect appeared to be а characteristic оf many real-world structure both human-generated ог of biological origin. For social networks, this property implies that almost any pair of people in the world can be connected to one another by a short chain of intermediate acquaintances, of typical length about six. However, the structure оf social networks is not homogeneous, there are always key persons аn provide distant out-local world connections between people. This paper is written in honor оf the 60th birthday оf our friend and colleague, Wadim S. Anishchenko, who is one of such key persons in the Nonlinear Dynamics community.

Key words: 
-
Acknowledgments: 
L.B. and М.Н. acknowledge the financial support оf the Swiss National Science Foundation through Grant № 2100-065268. This work was also supported in part by INTAS (Grant № 01-2061) and RFFI (Grant № 02-01-00968).
Reference: 
  1. Strogatz SH. Exploring complex networks. Nature. 2001;410(6825):268–276. DOI: 10.1038/35065725.
  2. Watts DJ, Strogatz SH. Collective dynamics of «small-world» networks. Nature. 1998;393(6684):440–442. DOI: 10.1038/30918.
  3. Milgram S. The small-world problem. Psychol. Today. 1961;2:60.
  4. Eckmann J-P, Moses Е. Curvature оf co-lonks uncovers hidden thematic layers in the World Wide Web. Proc. Natl. Acad. Sci. U.S.A. 2002;99(9):5825–5829. DOI: 10.1073/pnas.032093399.
  5. Newman MEJ. Scientific collaboration networks. I. Network construction and fundamental results. Phys. Rev. Е. 2001;64(1):016131. DOI: 10.1103/PhysRevE.64.016131.
  6. Kuperman M, Abramson G. Small world effect in an epidemiological model. Phys. Rev. Lett. 2001;86(13):2909–2912. DOI: 10.1103/PhysRevLett.86.2909.
  7. Watts DJ. Small Worlds. Princeton: Princeton Univ. Press; 1999. 262 p.
  8. Cancho RF, Janssen C, Solé RV. Topology оf technology graphs: small world patterns in electronic circuits. Phys. Rev. Е. 2001;64(4):046119. DOI: 10.1103/PhysRevE.64.046119.
  9. Jeong H, Tombor B, Albert R, Olrvai ZN, Barabdsi A-L, Albert R. The large-scale organization of metabolic networks. Nature. 2000;407(6804):651–654. DOI: 10.1038/35036627.
  10. Lago-Ferndndez LF, Huerta R, Corbacho F, Sigienza JA. Fast response and temporal coherent oscillations in small-world networks. Phys. Rev. Lett. 2000;84(12):2758–2761. DOI: 10.1103/PhysRevLett.84.2758.
  11. Gade PM, Hu CK. Synchronous chaos in coupled map lattices with small-world interactions. Phys. Rev. Е. 2000;62(5):6409–6413. DOI: 10.1103/PhysRevE.62.6409.
  12. Wang X, Chen G. Synchronization in small-world dynamical networks. Int. J. Bifurc. Chaos. 2002;12(1):187-192. DOI: 10.1142/S0218127402004292.
  13. Fujisaka H, Yamada T. Stability theory of synchronized motions in coupled oscillatory systems. Prog. Theor. Phys. 1983;69(1):32–47. DOI: 10.1143/PTP.69.32.
  14. Afraimovich VS, Verichev NN, Rabinovich, MI. Stochastic synchronization of oscillation in dissipative systems. Radiophys. Quantum Electron. 1986;29(9):795–803. DOI: 10.1007/BF01034476.
  15. Ресога LM, Carroll TL. Synchronization in chaotic systems. Phys. Rev. Lett. 1990;64(8):821–824. DOI: 10.1103/PhysRevLett.64.821.
  16. Anishchenko VS, Vadivasova TE, Postnov DE, Safonova MA. Synchronization оf chaos. Int. J. Bifurc. Chaos. 1992;2(3):633–644. DOI: 10.1142/S0218127492000756.
  17. Wu CW, Chua LO. Synchronization in аn array оf linearly coupled dynamical systems. IЕЕЕ Trans. Circuits Syst., Т: Fundam. Theory Appl. 1996;43(2):161–165. DOI: 10.1109/81.486440.
  18. Pecora LM, Carroll TL. Master stability function for synchronized coupled systems. Phys. Rev. Lett. 1998;80(10):2109–2112. DOI: 10.1103/PhysRevLett.80.2109.
  19. Belykh VN, Belykh IV, Hasler M. Hierarchy and stability оf partially synchronous oscillations оf diffusively coupled dynamical systems. Phys. Rev. Е. 2000;62(5):6332–6345. DOI: 10.1103/PhysRevE.62.6332.
  20. Pikovsky А, Rosenblum M, Kurths J. Synchronization: А Universal Concept in Nonlinear Science. Cambridge: Cambridge University Press; 2001. 411 p. DOI: 10.1017/CBO9780511755743.
  21. Belykh IV, Belykh VN, Nevidin KV, Hasler M. Persistent clusters in lattices оf coupled nonidentical chaotic systems. CHAOS. 2003;13(1):165–178. DOI: 10.1063/1.1514202.
  22. Barahona M, Ресоrа LM. Synchronization in small world systems. Phys. Rev. Lett. 2002;89(5):054101. DOI: 10.1103/PhysRevLett.89.054101.
  23. Belykh IV, Belykh VN, Hasler M. Blinking model and synchronization in small-world networks with а time-varying coupling. Physica D. 2004;195(1–2):188–206. DOI: 10.1016/j.physd.2004.03.013.
  24. Belykh VN, Belykh IV, Hasler M. Connection graph stability method for synchronized coupled chaotic systems. Physica D. 2004;195(1–2):159–187. DOI: 10.1016/j.physd.2004.03.012.
  25. Bogoljubov NN, Mitropolski YA. Asymptotic Methods in Oscillation Theory. Moscow: Nauka; 1974. 448 p. (in Russian).
Received: 
19.09.2003
Accepted: 
02.11.2003
Available online: 
23.11.2023
Published: 
31.12.2003
  • 410 reads