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Disassortative networks generated by directed rewiring

Qu Jing Wang Sheng-Jun

Disassortative networks generated by directed rewiring

Qu Jing, Wang Sheng-Jun
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  • The degree correlation of nodes is known to considerably affect the network dynamics in systems with a complex network structure. Thus it is necessary to generate degree correlated networks for the study of network systems. The assortatively correlated networks can be generated effectively by rewiring connections in scale-free networks. However, disassortativity in scale-free networks due to rewiring has not been studied systematically.In this paper, we present the effectiveness of generating disassortative scale-free networks by rewiring the already formed structure of connections which are built using the evolving network model. In the rewiring, two randomly selected links are cut and the four ends are connected randomly by two new links. The rewiring will be reserved if the disassortativity changes to the direction we need, otherwise it will be aborted. However, if one or both of the new links already exist in the network or a node is connected to itself, the rewiring step is aborted and two new links are selected. Our result shows that the rewiring method can enhance the disassortativity of scale-free networks. However, it is notable that the disassortativity measured by the Pearson correlation coefficient cannot be tuned to-1 which is believed to be the complete disassortativity. We obtain that the minimum value of the Pearson correlation coefficient depends on the parameters of networks, and we study the effect of network parameters on the degree correlation of the rewired networks, including the network size, the connection density of the network, and the heterogeneity of node degrees in the network. The result suggests that the effect of rewiring process is poorer in networks with higher heterogeneity, large size and sparse density. Another measurement of degree correlation called Kendall-Gibbons' coefficient is also used here, which gives the value of degree correlation independent of the network size. We give the relation of Kendall-Gibbons' coefficient to network sizes in both original scale-free networks and rewired networks. Results show that there is no obvious variance in rewired networks when the network size changes. The Kendall-Gibbons' coefficient also shows that rewiring can effectively enhance the disassortativity of the scale-free network.We also study the effectiveness of rewiring by comparing it with two sets of data of real Internets. We use the evolving network model to generate networks which have the same parameters as the real Internet, including network sizes, connection density and degree distribution exponents. We obtain that the networks generated by rewiring procedure cannot reach the same degree correlation as the real networks. The degree distribution of real networks diverges from the model at the largest degree or the smallest degree, which provides a heuristic explanation for the special degree correlation of real networks. Therefore, the difference at the end of the distribution is not negligible.
      Corresponding author: Wang Sheng-Jun, wangshjun@snnu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11305098), the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2014JQ1028), the Fundamental Research Funds for the Central Universities, China (Grant No. GK201302008), and the Interdisciplinary Incubation Project of Shaanxi Normal University, China (Grant No. 5).
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    Liu G, Li Y S, Zhang X P 2013 Chin. Phys. B 22 068901

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    Contreras M G A, Fagiolo G 2014 Phys. Rev. E 90 062812

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    Wang Z, Szolnoki A, Perc M 2014 Phys. Rev. E 90 032813

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    Mastrandrea R, Squartini T, Fagiolo G, Garlaschelli D 2014 Phys. Rev. E 90 062804

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    Wang S J, Hilgetag C C, Zhou C 2011 Front. Comput. Neurosci. 5 30

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    Wang S J, Zhou C 2012 New J. Phys. 14 023005

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    Guez O C, Gozolchiani A, Havlin S 2014 Phys. Rev. E 90 062814

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    Xia H J, Li P P, Ke J H, Lin Z Q 2015 Chin. Phys. B 24 040203

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    Newman M E J 2003 SIAM Rev. 45 167

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    Hu M B, Jiang R, Wu Q S 2013 Chin. Phys. B 22 066301

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    Hu Y G, Wang S J, Jin T, Qu S X 2015 Acta Phys. Sin. 64 028901(in Chinese) [胡耀光, 王圣军, 金涛, 屈世显 2015 物理学报 64 028901]

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    Wang S J, Wu A C, Wu Z X, Xu X J, Wang Y H 2007 Phys. Rev. E 75 046113

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    Menche J, Valleriani A, Lipowsky R 2010 Phys. Rev. E 81 046103

    [21]

    Wu Y, Li P, Chen M, Xiao J, Kurths J 2009 Physica A 388 2987

    [22]

    Menche J, Valleriani A, Lipowsky R 2010 Europhys. Lett. 89 18002

    [23]

    Jin Y G, Zhong S M, An N 2015 Chin. Phys. B 24 049202

    [24]

    Li R Q, Tang M, Xu B M 2013 Acta Phys. Sin. 62 168903(in Chinese) [李睿琪, 唐明, 许伯铭 2013 物理学报 62 168903]

    [25]

    Ren Z M, Liu J G, Shao F, Hu Z L, Guo Q 2013 Acta Phys. Sin. 62 108902(in Chinese) [任卓明, 刘建国, 邵凤, 胡兆龙, 郭强 2013 物理学报 62 108902]

    [26]

    Rong Z, Li X, Wang X 2007 Phys. Rev. E 76 027101

    [27]

    Rong Z, Wu Z X 2009 Europhys. Lett. 87 30001

    [28]

    Rong Z, Wu Z X, Chen G 2013 Europhys. Lett. 102 68005

    [29]

    Maslov S, Sneppen K 2002 Science 296 910

    [30]

    Watts D J, Strogatz S H 1998 Nature 393 440

    [31]

    Xulvi-Brunet R, Sokolov I M 2004 Phys. Rev. E 70 066102

    [32]

    Barabsi A L, Albert R 1999 Science 286 509

    [33]

    Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D U 2006 Phys. Rep. 424 175

    [34]

    Larremore D B, Shew W L, Restrepo J G 2011 Phys. Rev. Lett. 106 058101

    [35]

    Raschke M, Schlpfer M, Nibali R 2010 Phys. Rev. E 82 037102

    [36]

    Dorogovtsev S N, Ferreira A L, Goltsev A V, Mendes J F F 2010 Phys. Rev. E 81 031135

    [37]

    Zhou S, Mondragn R J 2007 New J. Phys. 9 173

    [38]

    Zhang G Q, Zhang G Q, Yang Q F, Cheng S Q, Zhou T 2008 New J. Phys. 10 123027

  • [1]

    Goh K-I, Oh E, Kahng B, Kim D 2003 Phys. Rev. E 67 017101

    [2]

    Liu G, Li Y S, Zhang X P 2013 Chin. Phys. B 22 068901

    [3]

    Contreras M G A, Fagiolo G 2014 Phys. Rev. E 90 062812

    [4]

    Wang Z, Szolnoki A, Perc M 2014 Phys. Rev. E 90 032813

    [5]

    Mastrandrea R, Squartini T, Fagiolo G, Garlaschelli D 2014 Phys. Rev. E 90 062804

    [6]

    Pastor-Satorras R, Vzquez A, Vespignani A 2001 Phys. Rev. Lett. 87 258701

    [7]

    Kenmogne F, Yeml D, Kengne J, Ndjanfang D 2014 Phys. Rev. E 90 052921

    [8]

    Wang S J, Hilgetag C C, Zhou C 2011 Front. Comput. Neurosci. 5 30

    [9]

    Wang S J, Zhou C 2012 New J. Phys. 14 023005

    [10]

    Guez O C, Gozolchiani A, Havlin S 2014 Phys. Rev. E 90 062814

    [11]

    Xia H J, Li P P, Ke J H, Lin Z Q 2015 Chin. Phys. B 24 040203

    [12]

    Zhou T, Bai W J, Wang B H, Liu Z J, Yan G 2005 Physics 34 31 (in Chinese) [周涛, 柏文洁, 汪秉宏, 刘之景, 严钢 2005 物理 34 31]

    [13]

    Chen G R 2008 Advances in Mechanics 38 653 (in Chinese) [陈关荣 2008 力学进展 38 653]

    [14]

    Wang X F, Li X, Chen G R 2006 Complex Networks: Theory and It's Applications (Beijing: Tsinghua University Press) p49 (in Chinese) [汪小帆, 李翔, 陈关荣 2006 复杂网络理论及其应用 (北京: 清华大学出版社) 第49页]

    [15]

    Newman M E J 2002 Phys. Rev. Lett. 89 208701

    [16]

    Newman M E J 2003 SIAM Rev. 45 167

    [17]

    Hu M B, Jiang R, Wu Q S 2013 Chin. Phys. B 22 066301

    [18]

    Hu Y G, Wang S J, Jin T, Qu S X 2015 Acta Phys. Sin. 64 028901(in Chinese) [胡耀光, 王圣军, 金涛, 屈世显 2015 物理学报 64 028901]

    [19]

    Wang S J, Wu A C, Wu Z X, Xu X J, Wang Y H 2007 Phys. Rev. E 75 046113

    [20]

    Menche J, Valleriani A, Lipowsky R 2010 Phys. Rev. E 81 046103

    [21]

    Wu Y, Li P, Chen M, Xiao J, Kurths J 2009 Physica A 388 2987

    [22]

    Menche J, Valleriani A, Lipowsky R 2010 Europhys. Lett. 89 18002

    [23]

    Jin Y G, Zhong S M, An N 2015 Chin. Phys. B 24 049202

    [24]

    Li R Q, Tang M, Xu B M 2013 Acta Phys. Sin. 62 168903(in Chinese) [李睿琪, 唐明, 许伯铭 2013 物理学报 62 168903]

    [25]

    Ren Z M, Liu J G, Shao F, Hu Z L, Guo Q 2013 Acta Phys. Sin. 62 108902(in Chinese) [任卓明, 刘建国, 邵凤, 胡兆龙, 郭强 2013 物理学报 62 108902]

    [26]

    Rong Z, Li X, Wang X 2007 Phys. Rev. E 76 027101

    [27]

    Rong Z, Wu Z X 2009 Europhys. Lett. 87 30001

    [28]

    Rong Z, Wu Z X, Chen G 2013 Europhys. Lett. 102 68005

    [29]

    Maslov S, Sneppen K 2002 Science 296 910

    [30]

    Watts D J, Strogatz S H 1998 Nature 393 440

    [31]

    Xulvi-Brunet R, Sokolov I M 2004 Phys. Rev. E 70 066102

    [32]

    Barabsi A L, Albert R 1999 Science 286 509

    [33]

    Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D U 2006 Phys. Rep. 424 175

    [34]

    Larremore D B, Shew W L, Restrepo J G 2011 Phys. Rev. Lett. 106 058101

    [35]

    Raschke M, Schlpfer M, Nibali R 2010 Phys. Rev. E 82 037102

    [36]

    Dorogovtsev S N, Ferreira A L, Goltsev A V, Mendes J F F 2010 Phys. Rev. E 81 031135

    [37]

    Zhou S, Mondragn R J 2007 New J. Phys. 9 173

    [38]

    Zhang G Q, Zhang G Q, Yang Q F, Cheng S Q, Zhou T 2008 New J. Phys. 10 123027

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  • Received Date:  16 March 2015
  • Accepted Date:  10 May 2015
  • Published Online:  05 October 2015

Disassortative networks generated by directed rewiring

    Corresponding author: Wang Sheng-Jun, wangshjun@snnu.edu.cn
  • 1. School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710119, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant No. 11305098), the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2014JQ1028), the Fundamental Research Funds for the Central Universities, China (Grant No. GK201302008), and the Interdisciplinary Incubation Project of Shaanxi Normal University, China (Grant No. 5).

Abstract: The degree correlation of nodes is known to considerably affect the network dynamics in systems with a complex network structure. Thus it is necessary to generate degree correlated networks for the study of network systems. The assortatively correlated networks can be generated effectively by rewiring connections in scale-free networks. However, disassortativity in scale-free networks due to rewiring has not been studied systematically.In this paper, we present the effectiveness of generating disassortative scale-free networks by rewiring the already formed structure of connections which are built using the evolving network model. In the rewiring, two randomly selected links are cut and the four ends are connected randomly by two new links. The rewiring will be reserved if the disassortativity changes to the direction we need, otherwise it will be aborted. However, if one or both of the new links already exist in the network or a node is connected to itself, the rewiring step is aborted and two new links are selected. Our result shows that the rewiring method can enhance the disassortativity of scale-free networks. However, it is notable that the disassortativity measured by the Pearson correlation coefficient cannot be tuned to-1 which is believed to be the complete disassortativity. We obtain that the minimum value of the Pearson correlation coefficient depends on the parameters of networks, and we study the effect of network parameters on the degree correlation of the rewired networks, including the network size, the connection density of the network, and the heterogeneity of node degrees in the network. The result suggests that the effect of rewiring process is poorer in networks with higher heterogeneity, large size and sparse density. Another measurement of degree correlation called Kendall-Gibbons' coefficient is also used here, which gives the value of degree correlation independent of the network size. We give the relation of Kendall-Gibbons' coefficient to network sizes in both original scale-free networks and rewired networks. Results show that there is no obvious variance in rewired networks when the network size changes. The Kendall-Gibbons' coefficient also shows that rewiring can effectively enhance the disassortativity of the scale-free network.We also study the effectiveness of rewiring by comparing it with two sets of data of real Internets. We use the evolving network model to generate networks which have the same parameters as the real Internet, including network sizes, connection density and degree distribution exponents. We obtain that the networks generated by rewiring procedure cannot reach the same degree correlation as the real networks. The degree distribution of real networks diverges from the model at the largest degree or the smallest degree, which provides a heuristic explanation for the special degree correlation of real networks. Therefore, the difference at the end of the distribution is not negligible.

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