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Establishment and structure properties of the scale-free Koch network

Liu Jia-Xue Kong Xiang-Mu

Establishment and structure properties of the scale-free Koch network

Liu Jia-Xue, Kong Xiang-Mu
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  • According the iterative algorithm of Koch curve, we generate a Koch network by considering tetrahedron as the basic unit of the iteration and investigate its structure properties, such as degree distribution, clustering coefficient, and average path length, degree correlation, analytically. The results show that the network is scale-free and the exponent of the degree distribution is γ≈332. The clustering coefficient tends to be 0870435 in the limit of large iteration and the study of the average path length proves that the network exhibits small world effect. We also find that the generated network is not degree uncorrelated because the function knn(k) is dependent on the degree of site.
    • Funds:
    [1]

    [1]Albert R, Barabási A L 2002 Rev. Mod. Phys. 74 47

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    [2]Dorogvtsev S N, Mendes J F F 2002 Adv. Phys. 51 1079

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    [3]Newman M E J 2003 SIAM Rev. 45 167

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    [4]Boccaletti S, Latora V, Moreno Y, Chavez M, Hwanga D U 2006 Phys. Rep. 424 175

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    [5]Wu J S, Di Z R 2004 Prog. Phys. 24 18 (in Chinese) [吴金闪、狄增如 2004 物理学进展 24 18]

    [6]

    [6]Ouyang M, Fei Q, Yu M H 2008 Acta Phys. Sin. 57 6763 (in Chinese) [欧阳敏、费奇、余明辉 2008 物理学报 57 6763]

    [7]

    [7]Costa L da F, Rodrigues F A, Travieso G, Boas P R V 2007 Adv. Phys. 56 167

    [8]

    [8]Barabási A L, Albert R 1999 Science 286 509

    [9]

    [9]Li J, Wang B H, Jiang P Q, Zhou T, Wang W X 2006 Acta Phys. Sin. 55 4051 (in Chinese) [李季、汪秉宏、蒋品群、周涛、王文旭 2006 物理学报 55 4051]

    [10]

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

    [11]

    ]Albert R, Jeong H, Barabási A L 2000 Nature 406 378

    [12]

    ]Callaway D S, Newman M E J, Strogatz S H, Watts D J 2000 Phys. Rev. Lett. 85 5468

    [13]

    ]Cohen R, Erez K, ben-Avraham D, Havlin S 2000 Phys. Rev. Lett. 85 4626

    [14]

    ]Cohen R, Erez K, ben-Avraham D, Havlin S 2001 Phys. Rev. Lett. 86 3682

    [15]

    ]Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200

    [16]

    ]Boguá M, Pastor-Satorras R, Vespignani A 2003 Phys. Rev. Lett. 90 028701

    [17]

    ]Song C, Havlin S, Makse H A 2006 Nat. Phys. 2 275

    [18]

    ]Zhang Z Z, Zhou S G, Zou T 2007 Eur. Phys. J. B 56 259

    [19]

    ]Chavez M, Hwang D U, Martinerie J, Boccaletti S 2006 Phys. Rev. E 74 066107

    [20]

    ]Molloy M, Reed B 1995 Random Struct. Algor. 6 161

    [21]

    ]Molloy M, Reed B 1998 Combinatorics, Probab. Comput. 7 295

    [22]

    ]Park J, Newman M E J 2003 Phys. Rev. E 68 026112

    [23]

    ]Maslov S, Sneppen K, Zaliznyak A 2004 Physica A 333 529

    [24]

    ]Catanzaro M, Boguá M, Pastor-Satorras R 2005 Phys. Rev. E 71 027103

    [25]

    ]Zhang Z Z, Zhou S G, Xie W L, Chen L C, Lin Y, Guan J H 2009 Phys. Rev. E 79 061113

    [26]

    ]Von Koch H 1906 Acta Math. 30 145

    [27]

    ]Lakhtakia A, Varadan V K, Messier R, Varadan V V 1987 J. Phys. A 20 3537

    [28]

    ]Watts D J, Strogatz H 1998 Nature 393 440

    [29]

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

    [30]

    ]Doye J P K, Massen C P 2005 Phys. Rev. E 71 016128

    [31]

    ]Zhang Z Z, Zhou S G, Zou T, Chen L C, Guan J H 2007 Eur. Phys. J. B 60 257

  • [1]

    [1]Albert R, Barabási A L 2002 Rev. Mod. Phys. 74 47

    [2]

    [2]Dorogvtsev S N, Mendes J F F 2002 Adv. Phys. 51 1079

    [3]

    [3]Newman M E J 2003 SIAM Rev. 45 167

    [4]

    [4]Boccaletti S, Latora V, Moreno Y, Chavez M, Hwanga D U 2006 Phys. Rep. 424 175

    [5]

    [5]Wu J S, Di Z R 2004 Prog. Phys. 24 18 (in Chinese) [吴金闪、狄增如 2004 物理学进展 24 18]

    [6]

    [6]Ouyang M, Fei Q, Yu M H 2008 Acta Phys. Sin. 57 6763 (in Chinese) [欧阳敏、费奇、余明辉 2008 物理学报 57 6763]

    [7]

    [7]Costa L da F, Rodrigues F A, Travieso G, Boas P R V 2007 Adv. Phys. 56 167

    [8]

    [8]Barabási A L, Albert R 1999 Science 286 509

    [9]

    [9]Li J, Wang B H, Jiang P Q, Zhou T, Wang W X 2006 Acta Phys. Sin. 55 4051 (in Chinese) [李季、汪秉宏、蒋品群、周涛、王文旭 2006 物理学报 55 4051]

    [10]

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

    [11]

    ]Albert R, Jeong H, Barabási A L 2000 Nature 406 378

    [12]

    ]Callaway D S, Newman M E J, Strogatz S H, Watts D J 2000 Phys. Rev. Lett. 85 5468

    [13]

    ]Cohen R, Erez K, ben-Avraham D, Havlin S 2000 Phys. Rev. Lett. 85 4626

    [14]

    ]Cohen R, Erez K, ben-Avraham D, Havlin S 2001 Phys. Rev. Lett. 86 3682

    [15]

    ]Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200

    [16]

    ]Boguá M, Pastor-Satorras R, Vespignani A 2003 Phys. Rev. Lett. 90 028701

    [17]

    ]Song C, Havlin S, Makse H A 2006 Nat. Phys. 2 275

    [18]

    ]Zhang Z Z, Zhou S G, Zou T 2007 Eur. Phys. J. B 56 259

    [19]

    ]Chavez M, Hwang D U, Martinerie J, Boccaletti S 2006 Phys. Rev. E 74 066107

    [20]

    ]Molloy M, Reed B 1995 Random Struct. Algor. 6 161

    [21]

    ]Molloy M, Reed B 1998 Combinatorics, Probab. Comput. 7 295

    [22]

    ]Park J, Newman M E J 2003 Phys. Rev. E 68 026112

    [23]

    ]Maslov S, Sneppen K, Zaliznyak A 2004 Physica A 333 529

    [24]

    ]Catanzaro M, Boguá M, Pastor-Satorras R 2005 Phys. Rev. E 71 027103

    [25]

    ]Zhang Z Z, Zhou S G, Xie W L, Chen L C, Lin Y, Guan J H 2009 Phys. Rev. E 79 061113

    [26]

    ]Von Koch H 1906 Acta Math. 30 145

    [27]

    ]Lakhtakia A, Varadan V K, Messier R, Varadan V V 1987 J. Phys. A 20 3537

    [28]

    ]Watts D J, Strogatz H 1998 Nature 393 440

    [29]

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

    [30]

    ]Doye J P K, Massen C P 2005 Phys. Rev. E 71 016128

    [31]

    ]Zhang Z Z, Zhou S G, Zou T, Chen L C, Guan J H 2007 Eur. Phys. J. B 60 257

  • [1] Simulation of the nonlinear cahn-hilliard equation based onthe local refinement pure meshless method. Acta Physica Sinica, 2020, (): . doi: 10.7498/aps.69.20191829
    [2] Preparing GaN nanowires on Al2O3 substrate without catalyst and its optical property research. Acta Physica Sinica, 2020, (): . doi: 10.7498/aps.69.20191923
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  • Received Date:  06 June 2009
  • Accepted Date:  22 August 2009
  • Published Online:  15 April 2010

Establishment and structure properties of the scale-free Koch network

  • 1. 曲阜师范大学物理工程学院,曲阜 273165

Abstract: According the iterative algorithm of Koch curve, we generate a Koch network by considering tetrahedron as the basic unit of the iteration and investigate its structure properties, such as degree distribution, clustering coefficient, and average path length, degree correlation, analytically. The results show that the network is scale-free and the exponent of the degree distribution is γ≈332. The clustering coefficient tends to be 0870435 in the limit of large iteration and the study of the average path length proves that the network exhibits small world effect. We also find that the generated network is not degree uncorrelated because the function knn(k) is dependent on the degree of site.

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