Skip to Main Content

Access Provided by Xinjiang University

Physical Review E

Access by Xinjiang University

Abrupt percolation in small equilibrated networks

Themis Matsoukas^*

Department of Chemical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA

^*matsoukas@engr.psu.edu

Phys. Rev. E 91, 052105 – Published 5 May, 2015

DOI: https://doi.org/10.1103/PhysRevE.91.052105

Abstract

Networks can exhibit an abrupt transition in the form of a spontaneous self-organization of a sizable fraction of the population into a giant component of connected members. This behavior has been demonstrated in random graphs under suppressive rules that passively or actively attempt to delay the formation of the giant cluster. We show that suppressive rules are not a necessary condition for a sharp transition at the percolation threshold. Rather, a finite system with aggressive tendency to form a giant cluster may exhibit an instability at the percolation threshold that is relieved through an abrupt and discontinuous transition to the stable branch. We develop the theory for a class of equilibrated networks that produce this behavior and find that the discontinuous jump is especially pronounced in small networks but disappears when the size of the system is infinite.

Article Text

References (25)

P. J. Flory, J. Am. Chem. Soc. 63, 3083 (1941).
P. Erdős and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci 5, 17 (1960).
C. Moore and M. E. J. Newman, Phys. Rev. E 61, 5678 (2000).
R. Axelrod, J. Conflict Res. 41, 203 (1997).
R. M. Ziff, J. Stat. Phys. 23, 241 (1980).
T. Matsoukas, Sci. Rep. 5, 8855 (2015).
D. Achlioptas, R. M. D'Souza, and J. Spencer, Science 323, 1453 (2009).
O. Riordan and L. Warnke, Science 333, 322 (2011).
J. Nagler, A. Levina, and M. Timme, Nat. Phys. 7, 265 (2011).
Y. S. Cho, S. Hwang, H. J. Herrmann, and B. Kahng, Science 339, 1185 (2013).
F. Radicchi and S. Fortunato, Phys. Rev. Lett. 103, 168701 (2009).
R. M. Ziff, Phys. Rev. Lett. 103, 045701 (2009).
E. J. Friedman and A. S. Landsberg, Phys. Rev. Lett. 103, 255701 (2009).
R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Phys. Rev. Lett. 105, 255701 (2010).
R. M. D'Souza and M. Mitzenmacher, Phys. Rev. Lett. 104, 195702 (2010).
S. Manna and A. Chatterjee, Physica A: Stat. Mech. Appl. 390, 177 (2011).
J. Gao, S. V. Buldyrev, H. E. Stanley, and S. Havlin, Nat. Phys. 8, 40 (2012).
G. J. Baxter, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Phys. Rev. Lett. 109, 248701 (2012).
P. Echenique, J. Gómez-Gardeñes, and Y. Moreno, Europhys. Lett. 71, 325 (2005).
T. Gross, Carlos J. Dommar D'Lima, and B. Blasius, Phys. Rev. Lett. 96, 208701 (2006).
S. Boettcher, V. Singh, and R. M. Ziff, Nat. Commun. 3, 787 (2012).
S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, Nucl. Phys. B 666, 396 (2003).
T. Matsoukas, Phys. Rev. E 90, 022113 (2014).
E. Ben-Naim and P. L. Krapivsky, Phys. Rev. E 68, 031104 (2003).
F. Leyvraz, Phys. Rep. 383, 95 (2003).