A new information dimension of complex networks

Abstract The fractal and self-similarity properties are revealed in many complex networks. The classical information dimension is an important method to study fractal and self-similarity properties of planar networks. However, it is not practical for real complex networks. In this Letter, a new information dimension of complex networks is proposed. The nodes number in each box is considered by using the box-covering algorithm of complex networks. The proposed method is applied to calculate the fractal dimensions of some real networks. Our results show that the proposed method is efficient when dealing with the fractal dimension problem of complex networks.

[1]  G. Calcagni Fractal universe and quantum gravity. , 2009, Physical review letters.

[2]  Zhongzhi Zhang,et al.  The Number of Spanning Trees of an Infinite Family of Outerplanar, Small-World and Self-Similar Graphs , 2012, ArXiv.

[3]  Zengru Di,et al.  Accuracy of the ball-covering approach for fractal dimensions of complex networks and a rank-driven algorithm. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Wenwu Yu,et al.  On pinning synchronization of complex dynamical networks , 2009, Autom..

[5]  강태원,et al.  [서평]「Chaos and Fractals : New Frontiers of Science」 , 1998 .

[6]  Igor I. Smolyaninov Metamaterial model of fractal time , 2012 .

[7]  郭龙,et al.  The Fractal Dimensions of Complex Networks , 2009 .

[8]  Araceli N. Proto,et al.  Empirical fractal geometry analysis of some speculative financial bubbles , 2012 .

[9]  Luciano da Fontoura Costa,et al.  Local Dimension of Complex Networks , 2012, ArXiv.

[10]  G. Bianconi,et al.  Shannon and von Neumann entropy of random networks with heterogeneous expected degree. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Mario Locci,et al.  Three algorithms for analyzing fractal software networks , 2010 .

[12]  A. Barabasi,et al.  Interactome Networks and Human Disease , 2011, Cell.

[13]  Bin Wu,et al.  Controlling the efficiency of trapping in treelike fractals. , 2013, The Journal of chemical physics.

[14]  Sankaran Mahadevan,et al.  Box-covering algorithm for fractal dimension of weighted networks , 2013, Scientific Reports.

[15]  J S Kim,et al.  Fractality in complex networks: critical and supercritical skeletons. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  M. Suzuki,et al.  Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations , 1990 .

[17]  Song Zheng,et al.  Adaptive projective synchronization in complex networks with time-varying coupling delay , 2009 .

[18]  Jean-Luc Thiffeault,et al.  Multiscale mixing efficiencies for steady sources. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Zhidong Teng,et al.  Synchronization of complex community networks with nonidentical nodes and adaptive coupling strength , 2011 .

[20]  B Kahng,et al.  Origin of the hub spectral dimension in scale-free networks. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Lucas Lacasa,et al.  Correlation dimension of complex networks , 2012, Physical review letters.

[22]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[23]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[24]  Shlomo Havlin,et al.  Origins of fractality in the growth of complex networks , 2005, cond-mat/0507216.

[25]  Nikos Nikiforakis,et al.  The growth of fractal dimension of an interface evolution from the interaction of a shock wave with a rectangular block of SF6 , 2011 .

[26]  Robert B. Ash,et al.  Information Theory , 2020, The SAGE International Encyclopedia of Mass Media and Society.

[27]  O. Shanker,et al.  DEFINING DIMENSION OF A COMPLEX NETWORK , 2007 .

[28]  Marcelo A. Moret,et al.  Classical and fractal analysis of vehicle demand on the ferry-boat system , 2012 .

[29]  Linyuan Lü,et al.  Similarity index based on local paths for link prediction of complex networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Michele Marchesi,et al.  Entropy of some CK Metrics to Assess Object-Oriented Software Quality , 2013, Int. J. Softw. Eng. Knowl. Eng..

[31]  Impulsive generalized function synchronization of complex dynamical networks , 2013 .

[32]  Yong Deng,et al.  SELF-SIMILARITY IN COMPLEX NETWORKS: FROM THE VIEW OF THE HUB REPULSION , 2013 .

[33]  P. H. Figueiredo,et al.  Multifractal behavior of wild-land and forest fire time series in Brazil , 2013 .

[34]  Guang H. Yue,et al.  Brain White Matter Shape Changes in Amyotrophic Lateral Sclerosis (ALS): A Fractal Dimension Study , 2013, PloS one.

[35]  O. Shanker,et al.  Graph zeta function and dimension of complex network , 2007 .

[36]  S. Mahadevan,et al.  Identifying influential nodes in weighted networks based on evidence theory , 2013 .

[37]  Hernán A. Makse,et al.  A review of fractality and self-similarity in complex networks , 2007 .

[38]  Giuseppe Vitiello,et al.  Fractals, coherent states and self-similarity induced noncommutative geometry , 2012, 1206.1854.

[39]  Wu Yi-man Fractal Theory and Its Applications in Pathology , 2004 .

[40]  Kun Zhao,et al.  Entropy of Dynamical Social Networks , 2011, PloS one.

[41]  Andrzej Bargiela,et al.  Fuzzy fractal dimensions and fuzzy modeling , 2003, Inf. Sci..

[42]  Przemysław Chełminiak,et al.  Emergence of fractal scale-free networks from stochastic evolution on the Cayley tree , 2013 .

[43]  S. Havlin,et al.  How to calculate the fractal dimension of a complex network: the box covering algorithm , 2007, cond-mat/0701216.

[44]  Hans J. Herrmann,et al.  Optimal box-covering algorithm for fractal dimension of complex networks , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  Jinde Cao,et al.  Synchronization of complex dynamical networks with nonidentical nodes , 2010 .

[46]  B. Mandelbrot,et al.  Fractal character of fracture surfaces of metals , 1984, Nature.

[47]  Santo Fortunato,et al.  Community detection in graphs , 2009, ArXiv.

[48]  Michele Marchesi,et al.  The fractal dimension of software networks as a global quality metric , 2013, Inf. Sci..

[49]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[50]  S. Havlin,et al.  Self-similarity of complex networks , 2005, Nature.