Fractal dimensions of percolating networks

Reuven Cohen; Shlomo Havlin

doi:10.1016/j.physa.2004.01.005

Fractal dimensions of percolating networks

Reuven Cohen, Shlomo Havlin

Research output: Contribution to journal › Conference article › peer-review

17 Scopus citations

Abstract

We use the generating function formalism to calculate the fractal dimensions for the percolating cluster at criticality in Erdos-Rényi (ER) and random scale free (SF) networks, with degree distribution P(k)=ck ^-λ. We show that the chemical dimension is d_l=2 for ER and SF networks with λ>4, as in percolation in d≥d _c=6 dimensions. For 3<λ<4 we show that d _l=(λ-2)/(λ-3). The fractal dimension is d _f=4 (λ>4) and d_f=2(λ-2)/(λ-3) (3<λ<4), and the embedding dimension is d_c=6 (λ>4) and d_c=2(λ-1)/(λ-3) (3<λ<4). We discuss the meaning of these dimensions for networks.

Original language	English
Pages (from-to)	6-13
Number of pages	8
Journal	Physica A: Statistical Mechanics and its Applications
Volume	336
Issue number	1-2
DOIs	https://doi.org/10.1016/j.physa.2004.01.005
State	Published - 1 May 2004
Event	Proceedings of the XVIII Max Born Symposium at Statistical Physics - Ladek Zdroj, Poland Duration: 22 Sep 2003 → 25 Sep 2003

Bibliographical note

Funding Information:
We thank the Israel Science Foundation for financial support. We thank Daniel ben-Avraham for useful discussion.

Keywords

Fractal
Internet
Networks
Scale-free

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10.1016/j.physa.2004.01.005

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title = "Fractal dimensions of percolating networks",

abstract = "We use the generating function formalism to calculate the fractal dimensions for the percolating cluster at criticality in Erdos-R{\'e}nyi (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≥d c=6 dimensions. For 3<λ<4 we show that d l=(λ-2)/(λ-3). The fractal dimension is d f=4 (λ>4) and df=2(λ-2)/(λ-3) (3<λ<4), and the embedding dimension is dc=6 (λ>4) and dc=2(λ-1)/(λ-3) (3<λ<4). We discuss the meaning of these dimensions for networks.",

keywords = "Fractal, Internet, Networks, Scale-free",

author = "Reuven Cohen and Shlomo Havlin",

note = "Funding Information: We thank the Israel Science Foundation for financial support. We thank Daniel ben-Avraham for useful discussion. ; Proceedings of the XVIII Max Born Symposium at Statistical Physics ; Conference date: 22-09-2003 Through 25-09-2003",

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AU - Cohen, Reuven

AU - Havlin, Shlomo

N1 - Funding Information: We thank the Israel Science Foundation for financial support. We thank Daniel ben-Avraham for useful discussion.

PY - 2004/5/1

Y1 - 2004/5/1

N2 - We use the generating function formalism to calculate the fractal dimensions for the percolating cluster at criticality in Erdos-Rényi (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≥d c=6 dimensions. For 3<λ<4 we show that d l=(λ-2)/(λ-3). The fractal dimension is d f=4 (λ>4) and df=2(λ-2)/(λ-3) (3<λ<4), and the embedding dimension is dc=6 (λ>4) and dc=2(λ-1)/(λ-3) (3<λ<4). We discuss the meaning of these dimensions for networks.

AB - We use the generating function formalism to calculate the fractal dimensions for the percolating cluster at criticality in Erdos-Rényi (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≥d c=6 dimensions. For 3<λ<4 we show that d l=(λ-2)/(λ-3). The fractal dimension is d f=4 (λ>4) and df=2(λ-2)/(λ-3) (3<λ<4), and the embedding dimension is dc=6 (λ>4) and dc=2(λ-1)/(λ-3) (3<λ<4). We discuss the meaning of these dimensions for networks.

KW - Fractal

KW - Internet

KW - Networks

KW - Scale-free

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IS - 1-2

T2 - Proceedings of the XVIII Max Born Symposium at Statistical Physics

Y2 - 22 September 2003 through 25 September 2003

ER -

Fractal dimensions of percolating networks

Abstract

Bibliographical note

Keywords

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