Characteristics of phase transitions via intervention in random networks

doi:10.1088/1674-1056/23/7/076401

Abstract We present a percolation process in which the classical Erdös-Rényi (ER) random evolutionary network is intervened by the product rule (PR) from some moment t₀. The parameter t₀ is continuously tunable over the real interval [0, 1]. This model becomes the random network under the Achlioptas process at t₀= 0 and the ER network at t₀= 1. For the percolation process at t₀≤ 1, we introduce a relatively slow-growing point, after which the largest cluster begins growing faster than that in the ER model. A weakly discontinuous transition is generated in the percolation process at t₀ ≤ 0.5. We take the relatively slow-growing point as the lower pseudotransition point and the maximum gap point of the order parameter as the upper pseudotransition point. The critical point can be approximately predicted by each fitting function of the two points about t₀. This contributes to understanding the rapid mergence of the large clusters at the critical point. The numerical simulations indicate that the lower pseudotransition point and the upper pseudotransition point are equal in the thermodynamic limit. When t₀> 0.5, the percolation processes generate a continuous transition. The scaling analyses of several quantities are presented, including the relatively slow-growing point, the duration of the relatively slow-growing process, as well as the relatively maximum strength between the percolation percolation at t₀< 1 and the ER network about different t₀. The presented mechanism can be viewed as a two-stage percolation process that has many potential applications in the growth processes of real networks.

Keywords: percolation phase transitions networks

Received: 14 November 2013
Revised: 20 February 2014
Accepted manuscript online:

PACS:	64.60.ah	(Percolation)
	64.60.-i	(General studies of phase transitions)
	64.60.aq	(Networks)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 61172115 and 60872029), the High Technology Research and Development Program of China (Grant No. 2008AA01Z206), the Aeronautics Foundation of China (Grant No. 20100180003), the Fundamental Research Funds for the Central Universities, China (Grant No. ZYGX2009J037), and Project 9140A07030513DZ02098, China.

Corresponding Authors: Jia Xiao
E-mail: tsunamijia@163.com

About author: 64.60.ah; 64.60.-i; 64.60.aq

Cite this article:

Jia Xiao (贾啸), Hong Jin-Song (洪劲松), Yang Hong-Chun (杨宏春), Yang Chun (杨春), Shi Xiao-Hong (史晓红), Hu Jian-Quan (胡建全) Characteristics of phase transitions via intervention in random networks 2014 Chin. Phys. B 23 076401

[1]	Stauffer D and Aharony A 1994 Introduction to Percolation Theory (London: Taylor & Francis) pp. 18-60
[2]	Bollobás B 2011 Random Graphs (2nd edn.) (London: Cambridge University Press) pp. 22-113
[3]	Erdös P and Rényi A 1960 Publ. Math. Inst. Hungar. Acad. Sci. 5 17
[4]	Grimmett G 1999 Percolation (Berlin: Springer) pp. 60-126
[5]	Li Z W, Liu H J and Xu X 2013 Acta Phys. Sin. 62 096401 (in Chinese)
[6]	Wan B H, Zhang P, Zhang J, Di Z R and Fan Y 2012 Acta Phys. Sin. 61 166402 (in Chinese)
[7]	Da Costa R A, Dorogovtsev S N, Goltsev A V and Mendes J F F 2010 Phys. Rev. Lett. 105 255701
[8]	Nagler J, Tiessen T and Gutch H W 2012 Phys. Rev. X 2 031009
[9]	Nagler J, Levina A and Timme M 2011 Nat. Phys. 7 265
[10]	Araujo N A M and Herrmann H J 2010 Phys. Rev. Lett. 105 035701
[11]	Chen W and D'Souza R M 2011 Phys. Rev. Lett. 106 115701
[12]	D'Souza R M and Mitzenmacher M 2010 Phys. Rev. Lett. 104 195702
[13]	Manna S S and Chatterjee A 2011 Physica A 390 177
[14]	Araújo N A M, Andrade J S J, Ziff R M and Herrmann H J 2011 Phys. Rev. Lett. 106 095703
[15]	Fan J F, Liu M X, Li L S and Chen X S 2012 Phys. Rev. E 85 06110
[16]	Li Y, Tang G, Song L J, Xun Z P, Xia H and Hao D P 2013 Acta Phys. Sin. 62 046401 (in Chinese)
[17]	Achlioptas D, D'Souza R M and Spencer J 2009 Science 323 1453
[18]	Friedman E J and Landsberg A S 2009 Phys. Rev. Lett. 103 255701
[19]	Grassberger P, Christensen C, Bizhani G, Son S W and Paczuski M 2011 Phys. Rev. Lett. 106 225701
[20]	Lee H K, Kim B J and Park H 2011 Phys. Rev. E 84 020101
[21]	Cho Y S and Kahng B 2011 Phys. Rev. Lett. 107 275703
[22]	Sahimi M 1994 Applications of Percolation Theory (London: Taylor & Francis) pp. 26-88
[23]	Spencer J 2010 Not. Am. Math. Soc. 57 720
[24]	Parshani R, Buldyrev S V, Stanley H E and Havlin S 2010 Nature 464 1025
[25]	Liu Y, Slotine J J and Barabasi A L 2011 Nature 473 167
[26]	Radicchi F and Fortunato S 2010 Phys. Rev. E 81 036110

[1]	Modeling of thermal conductivity for disordered carbon nanotube networks Hao Yin(殷浩), Zhiguo Liu(刘治国), and Juekuan Yang(杨决宽). Chin. Phys. B, 2023, 32(4): 044401.
[2]	Meshfree-based physics-informed neural networks for the unsteady Oseen equations Keyi Peng(彭珂依), Jing Yue(岳靖), Wen Zhang(张文), and Jian Li(李剑). Chin. Phys. B, 2023, 32(4): 040208.
[3]	Analysis of cut vertex in the control of complex networks Jie Zhou(周洁), Cheng Yuan(袁诚), Zu-Yu Qian(钱祖燏), Bing-Hong Wang(汪秉宏), and Sen Nie(聂森). Chin. Phys. B, 2023, 32(2): 028902.
[4]	Vertex centrality of complex networks based on joint nonnegative matrix factorization and graph embedding Pengli Lu(卢鹏丽) and Wei Chen(陈玮). Chin. Phys. B, 2023, 32(1): 018903.
[5]	Exploring fundamental laws of classical mechanics via predicting the orbits of planets based on neural networks Jian Zhang(张健), Yiming Liu(刘一鸣), and Zhanchun Tu(涂展春). Chin. Phys. B, 2022, 31(9): 094502.
[6]	Power-law statistics of synchronous transition in inhibitory neuronal networks Lei Tao(陶蕾) and Sheng-Jun Wang(王圣军). Chin. Phys. B, 2022, 31(8): 080505.
[7]	Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model Yan-Wei Dai(代艳伟), Sheng-Hao Li(李生好), and Xi-Hao Chen(陈西浩). Chin. Phys. B, 2022, 31(7): 070502.
[8]	A novel similarity measure for mining missing links in long-path networks Yijun Ran(冉义军), Tianyu Liu(刘天宇), Tao Jia(贾韬), and Xiao-Ke Xu(许小可). Chin. Phys. B, 2022, 31(6): 068902.
[9]	Influence fast or later: Two types of influencers in social networks Fang Zhou(周方), Chang Su(苏畅), Shuqi Xu(徐舒琪), and Linyuan Lü(吕琳媛). Chin. Phys. B, 2022, 31(6): 068901.
[10]	Synchronization in multilayer networks through different coupling mechanisms Xiang Ling(凌翔), Bo Hua(华博), Ning Guo(郭宁), Kong-Jin Zhu(朱孔金), Jia-Jia Chen(陈佳佳), Chao-Yun Wu(吴超云), and Qing-Yi Hao(郝庆一). Chin. Phys. B, 2022, 31(4): 048901.
[11]	Characteristics of vapor based on complex networks in China Ai-Xia Feng(冯爱霞), Qi-Guang Wang(王启光), Shi-Xuan Zhang(张世轩), Takeshi Enomoto(榎本刚), Zhi-Qiang Gong(龚志强), Ying-Ying Hu(胡莹莹), and Guo-Lin Feng(封国林). Chin. Phys. B, 2022, 31(4): 049201.
[12]	Robust H_∞ state estimation for a class of complex networks with dynamic event-triggered scheme against hybrid attacks Yahan Deng(邓雅瀚), Zhongkai Mo(莫中凯), and Hongqian Lu(陆宏谦). Chin. Phys. B, 2022, 31(2): 020503.
[13]	Explosive synchronization: From synthetic to real-world networks Atiyeh Bayani, Sajad Jafari, and Hamed Azarnoush. Chin. Phys. B, 2022, 31(2): 020504.
[14]	Parameter estimation of continuous variable quantum key distribution system via artificial neural networks Hao Luo(罗浩), Yi-Jun Wang(王一军), Wei Ye(叶炜), Hai Zhong(钟海), Yi-Yu Mao(毛宜钰), and Ying Guo(郭迎). Chin. Phys. B, 2022, 31(2): 020306.
[15]	Learnable three-dimensional Gabor convolutional network with global affinity attention for hyperspectral image classification Hai-Zhu Pan(潘海珠), Mo-Qi Liu(刘沫岐), Hai-Miao Ge(葛海淼), and Qi Yuan(袁琪). Chin. Phys. B, 2022, 31(12): 120701.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Characteristics of phase transitions via intervention in random networks

Cite this article:

Online attention