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Based on the modified Newman and Ziff algorithm combined with the finite-size scaling theory, in this present work we analytically study the phase transition property of the explosive percolation model induced by Achlioptas process on the Erds Rnyi random network via numerical simulations for the basic percolation quantities including the order parameter, the average cluster size, the moments, the standard deviation and the cluster heterogeneity. It is explicitly shown that all these relevant quantities display a typical power-law scaling behavior, which is the characteristic of continuous phase transition at the percolation threshold despite the fact that the order parameter presents a certain feature of discontinuous transition at the same time. Strictly, the explosive percolation transition during the Erds Rnyi random network is a singular transition, which means that it is neither a standard discontinuous phase transition nor the continuous transition in the regular random percolation model.
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Keywords:
- Erd /
- s R /
- nyi random network /
- explosive percolation model
[1] Zalle R (translated by Huang Y) 1998 Amorphous Solid State Physics (Beijing: Peking University Press) (in Chinese) [Zalle著 (黄畇译) 1998 非晶态固体物理学 (北京:北京大学出版社)]
[2] de Arcangelis L, Redner L, Coniglio A 1985 Phys. Rev. B 31 4725
[3] Henley C L 1993 Phys. Rev. Lett. 71 2741
[4] Moore C, Newman M E J 2000 Phys. Rev. E 61 5678
[5] Jovanovi\’c B, Buldyrev S V, Havlin S, Stanley H E 1994 Phys. Rev. E 50 2403
[6] Solomon S, Weisbuch G, de Arcangelis L, Jan N, Stauffer D 2000 Physica (Amsterdam) 277A 239
[7] Stauffer D, Aharony A 1991 Introduction to Percolation Theory (2nd Ed.) (London: Taylor and Francis)
[8] Erdös P, Rényi A 1960 Publ. Math. Inst. Hungar. Acad. Sci. 5 17
[9] Callaway D, Hopcroft J, Kleinberg J, Newman M, Strogatz S 2001 Phys. Rev. E 64 041902
[10] Kim J, Krapivsky P, Kahng B, Redner S 2002 Phys. Rev. E 66 055101
[11] Achlioptas D, Souza R M D, Spencer J 2009 Science 323 1453
[12] Ziff R M 2009 Phys. Rev. Lett. 103 045701
[13] Ziff R M 2010 Phys. Rev. E 82 051105
[14] Radicchi F, Fortunato S 2009 Phys. Rev. Lett. 103 168701
[15] Cho Y S, Kim J S, Park J, Kahng B, Kim D 2009 Phys. Rev. Lett. 103 135702
[16] Araújo N A M, Herrmann H J 2010 Phys. Rev. Lett. 105 035701
[17] Moreira A A, Oliveira E A, Reis S D S, Herrmann H J, Andrade Jr J S 2010 Phys. Rev. E 81 040101(R)
[18] Friedman E J, Landsberg A S 2009 Phys. Rev. Lett. 103 255701
[19] Souza R M D, Mitzenmacher M 2010 Phys. Rev. Lett. 104 195702
[20] Araújo N A M, Andrade Jr J S, Ziff R M, Herrmann Jr H 2011 Phys. Rev. Lett. 106 095703
[21] Manna S S, Chatterjee A 2011 Physica (Amsterdam) 390A 177
[22] Pan R K, Kivela M, Saramaki J, Kaski K, Kertesz K 2011 Phys. Rev. Lett. 83 046112
[23] Grassberger P, Christensen C, Bizhani G, Son S W, Paczuski M 2011 Phys. Rev. Lett. 106 255701
[24] Riordan O, Warnke L 2011 Science 333 322
[25] Lee H K, Kim B J, Park H 2011 Phys. Rev. E 84 020101
[26] da Costa R A, Dorogovtsev S N, Goltsev A V, Mendes J F F 2010 Phys. Rev. Lett. 105 255701
[27] Newman M E J, Ziff R M 2000 Phys. Rev. Lett. 85 4104
[28] Newman M E J, Ziff R M 2001 Phys. Rev. E 64 016706
[29] Radicchi F, Fortunato S 2010 Phys. Rev. E 81 036110
[30] Landau D P, Binder K 2000 A Guide to Monte Carlo Simulations in Statistical Physics (England: Cambridge University Press)
[31] Ódor G 2004 Rev. Mod. Phys. 76 663
[32] Cohen R, ben-Avraham D, Havlin S 2002 Phys. Rev. E 66 036113
[33] Noh J D, Lee H E, Park H 2011 Phys. Rev. E 84 010101
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[1] Zalle R (translated by Huang Y) 1998 Amorphous Solid State Physics (Beijing: Peking University Press) (in Chinese) [Zalle著 (黄畇译) 1998 非晶态固体物理学 (北京:北京大学出版社)]
[2] de Arcangelis L, Redner L, Coniglio A 1985 Phys. Rev. B 31 4725
[3] Henley C L 1993 Phys. Rev. Lett. 71 2741
[4] Moore C, Newman M E J 2000 Phys. Rev. E 61 5678
[5] Jovanovi\’c B, Buldyrev S V, Havlin S, Stanley H E 1994 Phys. Rev. E 50 2403
[6] Solomon S, Weisbuch G, de Arcangelis L, Jan N, Stauffer D 2000 Physica (Amsterdam) 277A 239
[7] Stauffer D, Aharony A 1991 Introduction to Percolation Theory (2nd Ed.) (London: Taylor and Francis)
[8] Erdös P, Rényi A 1960 Publ. Math. Inst. Hungar. Acad. Sci. 5 17
[9] Callaway D, Hopcroft J, Kleinberg J, Newman M, Strogatz S 2001 Phys. Rev. E 64 041902
[10] Kim J, Krapivsky P, Kahng B, Redner S 2002 Phys. Rev. E 66 055101
[11] Achlioptas D, Souza R M D, Spencer J 2009 Science 323 1453
[12] Ziff R M 2009 Phys. Rev. Lett. 103 045701
[13] Ziff R M 2010 Phys. Rev. E 82 051105
[14] Radicchi F, Fortunato S 2009 Phys. Rev. Lett. 103 168701
[15] Cho Y S, Kim J S, Park J, Kahng B, Kim D 2009 Phys. Rev. Lett. 103 135702
[16] Araújo N A M, Herrmann H J 2010 Phys. Rev. Lett. 105 035701
[17] Moreira A A, Oliveira E A, Reis S D S, Herrmann H J, Andrade Jr J S 2010 Phys. Rev. E 81 040101(R)
[18] Friedman E J, Landsberg A S 2009 Phys. Rev. Lett. 103 255701
[19] Souza R M D, Mitzenmacher M 2010 Phys. Rev. Lett. 104 195702
[20] Araújo N A M, Andrade Jr J S, Ziff R M, Herrmann Jr H 2011 Phys. Rev. Lett. 106 095703
[21] Manna S S, Chatterjee A 2011 Physica (Amsterdam) 390A 177
[22] Pan R K, Kivela M, Saramaki J, Kaski K, Kertesz K 2011 Phys. Rev. Lett. 83 046112
[23] Grassberger P, Christensen C, Bizhani G, Son S W, Paczuski M 2011 Phys. Rev. Lett. 106 255701
[24] Riordan O, Warnke L 2011 Science 333 322
[25] Lee H K, Kim B J, Park H 2011 Phys. Rev. E 84 020101
[26] da Costa R A, Dorogovtsev S N, Goltsev A V, Mendes J F F 2010 Phys. Rev. Lett. 105 255701
[27] Newman M E J, Ziff R M 2000 Phys. Rev. Lett. 85 4104
[28] Newman M E J, Ziff R M 2001 Phys. Rev. E 64 016706
[29] Radicchi F, Fortunato S 2010 Phys. Rev. E 81 036110
[30] Landau D P, Binder K 2000 A Guide to Monte Carlo Simulations in Statistical Physics (England: Cambridge University Press)
[31] Ódor G 2004 Rev. Mod. Phys. 76 663
[32] Cohen R, ben-Avraham D, Havlin S 2002 Phys. Rev. E 66 036113
[33] Noh J D, Lee H E, Park H 2011 Phys. Rev. E 84 010101
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