Fractal dimensions of percolating networks

We use the generating function formalism to calculate the fractal dimensions for the percolating cluster at criticality in Erdos–Renyi (ER) and random scale free (SF) networks, with degree distribution P(k)=ck−λ. We show that the chemical dimension is dl=2 for ER and SF networks with λ>4, as in percolation in d⩾dc=6 dimensions. For 3 4) and df=2(λ−2)/(λ−3) (3 4) and dc=2(λ−1)/(λ−3) (3<λ<4). We discuss the meaning of these dimensions for networks.

[1]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[4]  Shlomo Havlin,et al.  Diffusion and Reactions in Fractals and Disordered Systems: Random walks and diffusion , 2000 .

[5]  S. Havlin,et al.  Diffusion and Reactions in Fractals and Disordered Systems , 2000 .

[6]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[7]  Sergey N. Dorogovtsev,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW (Physics) , 2003 .

[8]  Z. Burda,et al.  Statistical ensemble of scale-free random graphs. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  P. Erdos,et al.  On the strength of connectedness of a random graph , 1964 .

[10]  Béla Bollobás,et al.  Random Graphs , 1985 .

[11]  S. Havlin,et al.  Breakdown of the internet under intentional attack. , 2000, Physical review letters.

[12]  S. Havlin,et al.  Fractals and Disordered Systems , 1991 .

[13]  H. Stanley,et al.  Optimal paths in disordered complex networks. , 2003, Physical review letters.

[14]  S. N. Dorogovtsev,et al.  Metric structure of random networks , 2002, cond-mat/0210085.

[15]  D S Callaway,et al.  Network robustness and fragility: percolation on random graphs. , 2000, Physical review letters.

[16]  G. Weiss.  Aspects and Applications of the Random Walk , 1994 .

[17]  David Bawden,et al.  Book Review: Evolution and Structure of the Internet: A Statistical Physics Approach. , 2006 .

[18]  Fan Chung Graham,et al.  The Diameter of Sparse Random Graphs , 2001, Adv. Appl. Math..

[19]  Albert-László Barabási,et al.  Error and attack tolerance of complex networks , 2000, Nature.

[20]  Gesine Reinert,et al.  Small worlds , 2001, Random Struct. Algorithms.

[21]  S. Havlin,et al.  Scale-free networks are ultrasmall. , 2002, Physical review letters.

[22]  Albert-László Barabási,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW , 2004 .

[23]  Cohen,et al.  Resilience of the internet to random breakdowns , 2000, Physical review letters.

[24]  L. Sander,et al.  Geography in a scale-free network model. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Reuven Cohen,et al.  Percolation critical exponents in scale-free networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Reuven Cohen,et al.  Scale-free networks on lattices. , 2002, Physical review letters.

[27]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[28]  Carson C. Chow,et al.  Small Worlds , 2000 .

[29]  Bruce A. Reed,et al.  The Size of the Giant Component of a Random Graph with a Given Degree Sequence , 1998, Combinatorics, Probability and Computing.

[30]  S. N. Dorogovtsev,et al.  Anomalous percolation properties of growing networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.