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Small-world spectra in mean field theory

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Grabow, Carsten, Grosskinsky, Stefan and Timme, Marc. (2012) Small-world spectra in mean field theory. Physical Review Letters, Vol.108 (No.21). Article no. 218701. ISSN 1079-7114

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Abstract

Collective dynamics on small-world networks emerge in a broad range of systems with their spectra characterizing fundamental asymptotic features. Here we derive analytic mean field predictions for the spectra of small-world models that systematically interpolate between regular and random topologies by varying their randomness. These theoretical predictions agree well with the actual spectra (obtained by numerical diagonalization) for undirected and directed networks and from fully regular to strongly random topologies. These results may provide analytical insights to empirically found features of dynamics on small-world networks from various research fields, including biology, physics, engineering and social science.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Centre for Complexity Science
Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): System analysis, Topology
Journal or Publication Title: Physical Review Letters
Publisher: American Physical Society
ISSN: 1079-7114
Official Date: May 2012
Volume: Vol.108
Number: No.21
Number of Pages: 5
Page Range: Article no. 218701
Identification Number: 10.1103/PhysRevLett.108.218701
Status: Peer Reviewed
Publication Status: Published
Funder: Germany. Bundesministerium für Bildung und Forschung (BMBF), Universität Göttingen (GGNB), Max-Planck-Gesellschaft zur Förderung der Wissenschaften [Max Planck Society for the Advancement of Science], Engineering and Physical Sciences Research Council (EPSRC)
Grant number: 01GQ1005B (BMBF), GSC 226/1 (GGNB), EP/E501311/1 (EPSRC)
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References:

[1] A. Broder, R. Kumar, F. Maghoul, and P. Raghavan,
Comput. Netw. 32 (2000).
[2] D. Watts and S. Strogatz, Nature 393, 440 (1998).
[3] L. Amaral, A. Scala, M. Barthelemy, and H. Stanley,
Proc. Natl. Acad. Sci. U.S.A. 97, 11149 (2000).
[4] S. Jespersen, I. M. Sokolov, and A. Blumen, J. Chem.
Phys. 113 (2000).
[5] A. Wagner and D. A. Fell, Proc. R. Soc. London, Ser. B
268, 1803 (2001).
[6] T. Achacoso and W. Yamamoto, AY’s Neuroanatomy of
C. Elegans for Computation (CRC Press, Boca Raton,
FL, 1992).
[7] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza-
tion, A universal concept in nonlinear sciences, vol. 12
of Cambridge Nonlinear Science Series (Cambridge University
Press, Cambridge, UK, 2001).
[8] S. Strogatz, Nature 410, 268 (2001).
[9] A. Pluchino, V. Latora, and A. Rapisarda, Int. J. Mod.
Phys. C 16, 515 (2005).
[10] R. Olfati-Saber, Proc. Am. Control Conf. 4, 2371 (2005).
[11] C. Grabow, S. M. Hill, S. Grosskinsky, and M. Timme,
Europhys. Lett. 90, 48002 (2010).
[12] C. Grabow, S. Grosskinsky, and M. Timme, Eur. Phys.
J. B (2011).
[13] D. McMillen, N. Kopell, J. Hasty, and J. Collins, Proc.
Natl. Acad. Sci. U.S.A. 99, 679 (2002).
[14] T. S. Gardner, D. di Bernardo, D. Lorenz, and J. J.
Collins, Science 301, 102 (2003).
[15] M. E. J. Newman, C. Moore, and D.J. Watts, Phys. Rev.
Lett. 84, 3201 (2000).
[16] R. Monasson, Eur. Phys. J. B 12, 555 (1999).
[17] J. Jost and M. P. Joy, Phys. Rev. E 65 (2001).
[18] M. Barahona and L. M. Pecora, Phys. Rev. Lett. 89, 4
(2002).
[19] F. Mori and T. Odagaki, J. Phys. Soc. Jpn. 73, 3294
(2004).
[20] R. Kühn and J. van Mourik, J. Phys. A 44, 165205
(2011).
[21] G. Fagiolo, Phys. Rev. E 76, 026107 (2007).
[22] G. Golub and C. Van Loan, Matrix Computations (Johns
Hopkins Studies in Math. Sci.) (1996).
[23] E. P. Wigner, Proc. Cambridge Philos. Soc. 47, 790
(1951).
[24] M. Mehta, Random Matrices (Academic Press, New
York) (1991).
[25] H. J. Sommers, A. Crisanti, H. Sompolinsky, and
Y. Stein, Phys. Rev. Lett. 60, 1895 (1988).
[26] M. Timme, F. Wolf, and T. Geisel, Phys. Rev. Lett. 92,
074101 (2004).
[27] M. Timme, T. Geisel, and F. Wolf, Chaos 16, 015108
(2006).
[28] F. Götze and A. Tikhomirov, Ann. Probab. 38, 1444
(2010).
[29] I. J. Farkas, I. Derényi, A.-L. Barabási, and T. Vicsek,
Phys. Rev. E 64, 12 (2001).
[30] O. Sporns and E. Bullmore, Nat. Rev. Neurosci. 10, 186
(2009).
URI: http://wrap.warwick.ac.uk/id/eprint/44693

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