Please wait a minute...
Chin. Phys. B, 2012, Vol. 21(8): 080504    DOI: 10.1088/1674-1056/21/8/080504
GENERAL Prev   Next  

Multifractal analysis of complex networks

Wang Dan-Linga b, Yu Zu-Guoa c, Anh Va
a School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Q 4001, Australia;
b School of Mathematics & Physics, University of Science & Technology Beijing, Beijing 10083, China;
c Hunan Key Laboratory for Computation & Simulation in Science & Engineering, and School of Mathematics & Computational Science, Xiangtan University, Xiangtan 411105, China
Abstract  Complex networks have recently attracted much attention in diverse areas of science and technology. Many networks such as the WWW and biological networks are known to display spatial heterogeneity which can be characterized by their fractal dimensions. Multifractal analysis is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. In this paper, we introduce a new box covering algorithm for multifractal analysis of complex networks. This algorithm is used to calculate the generalized fractal dimensions Dq of some theoretical networks, namely scale-free networks, small world networks, and random networks, and one kind of real networks, namely protein-protein interaction networks of different species. Our numerical results indicate the existence of multifractality in scale-free networks and protein-protein interaction networks, while the multifractal behavior is not clear-cut for small world networks and random networks. The possible variation of Dq due to changes in the parameters of the theoretical network models is also discussed.
Keywords:  complex networks      multifractality      box covering algorithm  
Received:  02 February 2012      Revised:  01 March 2012      Published:  01 July 2012
PACS:  05.45.Df (Fractals)  
  47.53.+n (Fractals in fluid dynamics)  
  89.75.Hc (Networks and genealogical trees)  
Fund: Project supported by the Australian Research Council (Grant No. DP0559807), the National Natural Science Foundation of China (Grant No. 11071282), the Science Fund for Changjiang Scholars and Innovative Research Team in University (PCSIRT) (Grant No. IRT1179), the Program for New Century Excellent Talents in University (Grant No. NCET-08-06867), the Research Foundation of the Education Department of Hunan Province of China (Grant No. 11A122), the Natural Science Foundation of Hunan Province of China (Grant No. 10JJ7001), the Science and Technology Planning Project of Hunan Province of China (Grant No. 2011FJ2011), the Lotus Scholars Program of Hunan Province of China, the Aid Program for Science and Technology Innovative Research Team in Higher Education Institutions of Hunan Province of China, and a China Scholarship Council-Queensland University of Technology Joint Scholarship.
Corresponding Authors:  Yu Zu-Guo     E-mail:

Cite this article: 

Wang Dan-Ling, Yu Zu-Guo, Anh V Multifractal analysis of complex networks 2012 Chin. Phys. B 21 080504

[1] Song C, Havlin S and Makse H A 2005 Nature 433 392
[2] Lee C Y and Jung S 2006 Phys. Rev. E 73 066102
[3] Guo L and Cai X 2009 Chin. Phys. Lett. 26 088901
[4] Erdös P and Rényi A 1960 Publ. Math. Inst. Hung. Acad. Sci. 5 17
[5] Milgram S 1967 Psychol. Today 2 60
[6] Albert R, Jeong H and Barabasi A L 1999 Nature 401 130
[7] Albert R and Barabasi A L 2002 Rev. Mod. Phys. 74 47
[8] Faloutsos M, Faloutsos P and Faloutsos C 1999 Comput. Commun. Rev. 29 251
[9] Mandelbrot B B 1983 The Fractal Geometry of Nature (New York: Academic Press)
[10] Feder J 1988 Fractals (New York: Plenum)
[11] Falconer K 1997 Techniques in Fractal Geometry (New York: Wiley)
[12] Eguiluz V M, Hernandez-Garcia E, Piro O and Klemm K 2003 Phys. Rev. E 68 055102(R)
[13] Song C, Gallos L K, Havlin S and Makse H A 2007 J. Stat. Mech.: Theor. Exp. 3 P03006
[14] Kim J S, Goh K I, Salvi G, Oh E, Kahng B and Kim D 2007 Phys. Rev. E 75 016110
[15] Zhou W X, Jiang Z Q and Sornette D 2007 Physica A 375 741
[16] Gao L, Hu Y and Di Z 2008 Phys. Rev. E 78 046109
[17] Shanker O 2007 Mod. Phys. Lett. B 21 321
[18] Rozenfeld H D and Makse H A 2009 Chem. Eng. Sci. 64 4572
[19] Rozenfeld H D, Song C and Makse H A 2010 Phys. Rev. Lett. 104 025701
[20] Liu J X and Kong X M 2010 Acta Phys. Sin. 59 2244 (in Chinese)
[21] Grassberger P and Procaccia I 1983 Phys. Rev. Lett. 50 346
[22] Halsey T C, Jensen M H, Kadanoff L P, Procaccia I and Shraiman B I 1986 Phys. Rev. A 33 1141
[23] Canessa E 2000 J. Phys. A: Math. Gen. 33 3637
[24] Anh V V, Tieng Q M and Tse Y K 2000 Int. Trans. Oper. Res. 7 349
[25] Yu Z G, Anh V and Lau K S 2001 Physica A 301 351
[26] Yu Z G, Anh V and Lau K S 2001 Phys. Rev. E 64 31903
[27] Yu Z G, Anh V and Wang B 2001 Phys. Rev. E 63 11903
[28] Anh V, Lau K S and Yu Z G 2002 Phys. Rev. E 66 031910
[29] Yu Z G, Anh V and Lau K S 2003 Phys. Rev. E 68 021913
[30] Yu Z G, Anh V and Lau K S 2004 J. Theor. Biol. 226 341
[31] Zhou L Q, Yu Z G, Deng J Q, Anh V and Long S C 2005 J. Theor. Biol. 232 559
[32] Yu Z G, Anh V, Lau K S and Zhou L Q 2006 Phys. Rev. E 73 031920
[33] Yu Z G, Xiao Q J, Shi L, Yu J W and Anh V 2010 Chin. Phys. B 19 068701
[34] Zhu S M, Yu Z G and Anh V 2011 Chin. Phys. B 20 010505
[35] Han J J and Fu W J 2010 Chin. Phys. B 19 010205
[36] Kantelhardt J W, Koscielny-Bunde E, Rybski D, Braun P, Bunde A and Havlin S 2006 J. Geophys. Res. 111 D01106
[37] Veneziano D, Langousis A and Furcolo P 2006 Water Resour. Res. 42 W06D15
[38] Venugopal V, Roux S G, Foufoula-Georgiou E and Arneodo A 2006 Water Resour. Res. 42 W06D14
[39] Yu Z G, Anh V, Wanliss J A and Watson S M 2007 Chaos, Solitons and Fractals 31 736
[40] Zang B J and Shang P J 2007 Chin. Phys. 16 565
[41] Yu Z G, Anh V and Eastes R 2009 J. Geophys. Res. 114 A 05214
[42] Yu Z G, Anh V, Wang Y, Mao D and Wanliss J 2010 J. Geophys. Res. 115 A 10219
[43] Polla G, Lovasz L and Vicsek T 2010 Proc. Natl. Acad. Sci. USA 107 7640
[44] Dijkstra E W 1959 Numerische Mathematik 1 269
[45] Newman M E J 2003 SIAM Rev. 45 167
[46] Barabsi A L and Albert R 1999 Science 286 509
[47] Newman M E J and Watts D J 1999 Phys. Lett. A 263 341
[48] Watts D J and Strogatz S H 1998 Nature 393 440
[49] Opheusden J H H, Bos M T A and van der Kaaden G 1996 Physica A 227 183
[50] Smith T G and Lange G D 1998 Fractal in Biology and Medicine eds. Nonnenmacher T F, Losa G A and Weibel E R (Basel: Birkhäuser)
[51] Fernández E, Bolea J A, Ortega G and Louis E 1999 J. Neurosci. Methods 89 151
[52] Cytoscape software:
[53] BioGRID:
[54] DIP:
[55] Lee E, Jung H, Radivojac P, Kim J W and Lee D 2009 BMC Bioinformatics 10 (Suppl. 2) S2
[1] Influential nodes identification in complex networks based on global and local information
Yuan-Zhi Yang(杨远志), Min Hu(胡敏), Tai-Yu Huang(黄泰愚). Chin. Phys. B, 2020, 29(8): 088903.
[2] Identifying influential spreaders in complex networks based on entropy weight method and gravity law
Xiao-Li Yan(闫小丽), Ya-Peng Cui(崔亚鹏), Shun-Jiang Ni(倪顺江). Chin. Phys. B, 2020, 29(4): 048902.
[3] Modeling and analysis of the ocean dynamic with Gaussian complex network
Xin Sun(孙鑫), Yongbo Yu(于勇波), Yuting Yang(杨玉婷), Junyu Dong(董军宇), Christian Böhm(陈学恩), Xueen Chen. Chin. Phys. B, 2020, 29(10): 108901.
[4] Pyramid scheme model for consumption rebate frauds
Yong Shi(石勇), Bo Li(李博), Wen Long(龙文). Chin. Phys. B, 2019, 28(7): 078901.
[5] Theoretical analyses of stock correlations affected by subprime crisis and total assets: Network properties and corresponding physical mechanisms
Shi-Zhao Zhu(朱世钊), Yu-Qing Wang(王玉青), Bing-Hong Wang(汪秉宏). Chin. Phys. B, 2019, 28(10): 108901.
[6] Coordinated chaos control of urban expressway based on synchronization of complex networks
Ming-bao Pang(庞明宝), Yu-man Huang(黄玉满). Chin. Phys. B, 2018, 27(11): 118902.
[7] Detecting overlapping communities based on vital nodes in complex networks
Xingyuan Wang(王兴元), Yu Wang(王宇), Xiaomeng Qin(秦小蒙), Rui Li(李睿), Justine Eustace. Chin. Phys. B, 2018, 27(10): 100504.
[8] Dominant phase-advanced driving analysis of self-sustained oscillations in biological networks
Zhi-gang Zheng(郑志刚), Yu Qian(钱郁). Chin. Phys. B, 2018, 27(1): 018901.
[9] Ranking important nodes in complex networks by simulated annealing
Yu Sun(孙昱), Pei-Yang Yao(姚佩阳), Lu-Jun Wan(万路军), Jian Shen(申健), Yun Zhong(钟赟). Chin. Phys. B, 2017, 26(2): 020201.
[10] Empirical topological investigation of practical supply chains based on complex networks
Hao Liao(廖好), Jing Shen(沈婧), Xing-Tong Wu(吴兴桐), Bo-Kui Chen(陈博奎), Mingyang Zhou(周明洋). Chin. Phys. B, 2017, 26(11): 110505.
[11] An improved genetic algorithm with dynamic topology
Kai-Quan Cai(蔡开泉), Yan-Wu Tang(唐焱武), Xue-Jun Zhang(张学军), Xiang-Min Guan(管祥民). Chin. Phys. B, 2016, 25(12): 128904.
[12] Subtle role of latency for information diffusion in online social networks
Fei Xiong(熊菲), Xi-Meng Wang(王夕萌), Jun-Jun Cheng(程军军). Chin. Phys. B, 2016, 25(10): 108904.
[13] Synchronization of Markovian jumping complex networks with event-triggered control
Shao Hao-Yu, Hu Ai-Hua, Liu Dan. Chin. Phys. B, 2015, 24(9): 098902.
[14] Load-redistribution strategy based on time-varying load against cascading failure of complex network
Liu Jun, Xiong Qing-Yu, Shi Xin, Wang Kai, Shi Wei-Ren. Chin. Phys. B, 2015, 24(7): 076401.
[15] Degree distribution and robustness of cooperativecommunication network with scale-free model
Wang Jian-Rong, Wang Jian-Ping, He Zhen, Xu Hai-Tao. Chin. Phys. B, 2015, 24(6): 060101.
No Suggested Reading articles found!