ISSN 0253-2778

CN 34-1054/N

Open AccessOpen Access JUSTC Original Paper

Research on multi-fractals of weighted hypernetworks

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.03.016
  • Received Date: 15 July 2019
  • Accepted Date: 28 September 2019
  • Rev Recd Date: 28 September 2019
  • Publish Date: 31 March 2020
  • Hypernetwork is a kind of network that is more complex than ordinary complex networks, while hypernetwork dimension is a feasible tool for measuring it. For the case that both node weight and hyperedge weight in weighted hypernetworks can value among positive real number, negative real number, pure imaginary number and complex number, etc., fractal dimensions for various weighted hypernetworks is proposed and their multi-fractals are discussed. It is shown that among all types of weighted hypernetworks, except those with both node weight and hyperedge weight being positive or negative real numbers, other types of weighted hypernetworks share multi-fractals and can be divided into seven different categories that are all distributed in the entire complex plane. The analytic expressions of multi-fractal dimensions for all types of weighted hypernetworks are also presented. Finally, some important properties of the multi-fractal dimensions of these weighted hypernetworks are analyzed.
    Hypernetwork is a kind of network that is more complex than ordinary complex networks, while hypernetwork dimension is a feasible tool for measuring it. For the case that both node weight and hyperedge weight in weighted hypernetworks can value among positive real number, negative real number, pure imaginary number and complex number, etc., fractal dimensions for various weighted hypernetworks is proposed and their multi-fractals are discussed. It is shown that among all types of weighted hypernetworks, except those with both node weight and hyperedge weight being positive or negative real numbers, other types of weighted hypernetworks share multi-fractals and can be divided into seven different categories that are all distributed in the entire complex plane. The analytic expressions of multi-fractal dimensions for all types of weighted hypernetworks are also presented. Finally, some important properties of the multi-fractal dimensions of these weighted hypernetworks are analyzed.
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    [2]
    WATTS D J, STROGATZ S H. Collective dynamics of ‘small-world’ networks[J]. Nature, 1998, 393: 440-442.
    [3]
    NEWMAN M E J, WATTS D J. Renormalization group analysis of the small-world network model[J]. Physics Letter A, 1999, 293: 341-346.
    [4]
    BARABASI A L, ALBERT R. Emergence of scaling in random networks[J]. Science, 1999, 286: 509-512.
    [5]
    SONG C, JALVIN S, MAKSE H A. Self-similarity of complex networks[J]. Nature, 2005, 433: 392-395.
    [6]
    刘胜久, 李天瑞, 刘小伟. 网络维数: 一种度量复杂网络的新方法[J]. 计算机科学, 2019, 46(1): 51-56.
    LIU Shengjiu, LI Tianrui, LIU Xiaowei. Network dimension: A new measure for complex networks[J]. Computer Science, 2019, 46(1): 51-56.
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    HARTE D. Multifractals: Theory and Applications[M]. Chapman & Hall/CRC, 2001.
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    刘胜久, 李天瑞, 珠杰, 等. 带权图的多重分形研究[J]. 南京大学学报, 2020, 56(1): 85-97.
    LIU Shengjiu, LI Tianrui, ZHU Jie, et al. Research on multi-fractals of weighted graph[J]. Journal of Nanjing University, 2020, 56(1): 85-97.
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    刘胜久, 李天瑞, 杨宗霖, 等. 带权超网络的度量方法及其性质[J]. 计算机应用. 2019, 39(11): 3107-3113.
    LIU Shengjiu, LI Tianrui, YANG Zonglin, et al. Measure and properties of weighted hypernetwork[J]. Journal of Computer Applications. 2019, 39(11): 3107-3113.
    [13]
    黄汝激. 超网络的有向k超树分析法[J]. 电子科学学刊, 1987, 9(3): 244-255.
    HUANG Ruji. Directed k-hypertree method for hypernetwork analysis[J]. Journal of Electronics, 1987, 9(3): 244-255.
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    GALLO G, LONGO G, NGUYEN S. Directed hypergraph and applications[J]. Discrete Applied Mathematics, 1993, 42: 177-201.
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    FAHIR O E, DAVID A G. Directed Moore hypergraphs[J]. Discrete Applied Mathematics, 1995, 63: 117-127.
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    王建方. 超图的理论基础[M]. 北京: 高等教育出版社, 2006.
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    [17]
    FENG K Q, LI W W. Spectra of hypergraphs and applications[J]. Journal of Number Theory. 1996, 60(1): 1-22.
    [18]
    BALKA R, BUCZOLICH Z, ELEKEs M. A new fractal dimension: The topological Hausdorff dimension[J]. Advances in Mathematics, 2015, 274(1): 881-927.
    [19]
    SREENIVASAN K R, MENEVEAU C. The fractal facets of turbulence[J]. Journal of Fluid Mechanics, 1986, 173(173): 357-386.
    [20]
    梁俊杰, 李格升, 张尊华, 等. 球形火焰分形维数的计算方法[J]. 燃烧科学与技术, 2016, 22(1): 26-32.
    LIANG Junjie, LI Gesheng, ZHANG Zunhua, et al. Calculation method for fractal dimension of spherical flames[J]. Journal of Combustion Science and Technology, 2016, 22(1): 26-32.
    [21]
    陈彦光. 城市地理研究中的单分形、多分形和自仿射分形[J]. 地理科学进展, 2019, 38(1): 38-49.
    CHEN Yanguang. Monofractal, multifractals, and self-affine fractals in urban studies[J]. Progress in Geography, 2019, 38(1): 38-49.
    [22]
    赵静湉, 陈彦光, 李双成. 京津冀城市用地形态的双分形特征及其演化[J]. 地理科学进展, 2019, 38(1): 77-87.
    ZHAO Jingtian, CHEN Yanguang, LI Shuangcheng. Bi-fractal structure and evolution of the Beijing-Tianjin-Hebei region urban land-use patterns[J]. Progress in Geography, 2019, 38(1): 77-87.)
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Catalog

    [1]
    ERDSP, RéNYI A R. On random graphs Ⅰ[J]. Publicationes Mathematicae, 1959, 6: 290-297.
    [2]
    WATTS D J, STROGATZ S H. Collective dynamics of ‘small-world’ networks[J]. Nature, 1998, 393: 440-442.
    [3]
    NEWMAN M E J, WATTS D J. Renormalization group analysis of the small-world network model[J]. Physics Letter A, 1999, 293: 341-346.
    [4]
    BARABASI A L, ALBERT R. Emergence of scaling in random networks[J]. Science, 1999, 286: 509-512.
    [5]
    SONG C, JALVIN S, MAKSE H A. Self-similarity of complex networks[J]. Nature, 2005, 433: 392-395.
    [6]
    刘胜久, 李天瑞, 刘小伟. 网络维数: 一种度量复杂网络的新方法[J]. 计算机科学, 2019, 46(1): 51-56.
    LIU Shengjiu, LI Tianrui, LIU Xiaowei. Network dimension: A new measure for complex networks[J]. Computer Science, 2019, 46(1): 51-56.
    [7]
    HARTE D. Multifractals: Theory and Applications[M]. Chapman & Hall/CRC, 2001.
    [8]
    刘胜久, 李天瑞, 珠杰, 等. 带权图的多重分形研究[J]. 南京大学学报, 2020, 56(1): 85-97.
    LIU Shengjiu, LI Tianrui, ZHU Jie, et al. Research on multi-fractals of weighted graph[J]. Journal of Nanjing University, 2020, 56(1): 85-97.
    [9]
    ERDS P, HAJNAL A. On the chromatic number of graphs and set systems[J]. Acta Mathematica Academiae Scientiarum Hungarica. 1966, 17: 61-99.
    [10]
    BERGE C. Graphs and Hypergraphs[M]. Amsterdam: North-Holland Publishing Company, 1973.
    [11]
    ESTRADA E, RODRIGUES V R. Subgraphcentrality in complex networks[J]. Physical Review E, 2005, 71(5): 1-9.
    [12]
    刘胜久, 李天瑞, 杨宗霖, 等. 带权超网络的度量方法及其性质[J]. 计算机应用. 2019, 39(11): 3107-3113.
    LIU Shengjiu, LI Tianrui, YANG Zonglin, et al. Measure and properties of weighted hypernetwork[J]. Journal of Computer Applications. 2019, 39(11): 3107-3113.
    [13]
    黄汝激. 超网络的有向k超树分析法[J]. 电子科学学刊, 1987, 9(3): 244-255.
    HUANG Ruji. Directed k-hypertree method for hypernetwork analysis[J]. Journal of Electronics, 1987, 9(3): 244-255.
    [14]
    GALLO G, LONGO G, NGUYEN S. Directed hypergraph and applications[J]. Discrete Applied Mathematics, 1993, 42: 177-201.
    [15]
    FAHIR O E, DAVID A G. Directed Moore hypergraphs[J]. Discrete Applied Mathematics, 1995, 63: 117-127.
    [16]
    王建方. 超图的理论基础[M]. 北京: 高等教育出版社, 2006.
    WANG Jianfang. Theoretical Principle of Hypergraph[M]. Beijing: Higher Education Press, 2006.
    [17]
    FENG K Q, LI W W. Spectra of hypergraphs and applications[J]. Journal of Number Theory. 1996, 60(1): 1-22.
    [18]
    BALKA R, BUCZOLICH Z, ELEKEs M. A new fractal dimension: The topological Hausdorff dimension[J]. Advances in Mathematics, 2015, 274(1): 881-927.
    [19]
    SREENIVASAN K R, MENEVEAU C. The fractal facets of turbulence[J]. Journal of Fluid Mechanics, 1986, 173(173): 357-386.
    [20]
    梁俊杰, 李格升, 张尊华, 等. 球形火焰分形维数的计算方法[J]. 燃烧科学与技术, 2016, 22(1): 26-32.
    LIANG Junjie, LI Gesheng, ZHANG Zunhua, et al. Calculation method for fractal dimension of spherical flames[J]. Journal of Combustion Science and Technology, 2016, 22(1): 26-32.
    [21]
    陈彦光. 城市地理研究中的单分形、多分形和自仿射分形[J]. 地理科学进展, 2019, 38(1): 38-49.
    CHEN Yanguang. Monofractal, multifractals, and self-affine fractals in urban studies[J]. Progress in Geography, 2019, 38(1): 38-49.
    [22]
    赵静湉, 陈彦光, 李双成. 京津冀城市用地形态的双分形特征及其演化[J]. 地理科学进展, 2019, 38(1): 77-87.
    ZHAO Jingtian, CHEN Yanguang, LI Shuangcheng. Bi-fractal structure and evolution of the Beijing-Tianjin-Hebei region urban land-use patterns[J]. Progress in Geography, 2019, 38(1): 77-87.)

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