Percolation and Loop Statistics in Complex Networks

Details Percolation and Loop Statistics in Complex Networks Percolation and Loop Statistics in Complex Networks

2007-07-04

arxiv.org/abs/0707.0560v2

Eur. Phys. J. B 66, 251 (2008)

Physics

Condensed Matter

Statistical Mechanics

4 pages and 2 figures; A major revision has been made

Scientific paper

10.1140/epjb/e2008-00401-9

Complex networks display various types of percolation transitions. We show that the degree distribution and the degree-degree correlation alone are not sufficient to describe diverse percolation critical phenomena. This suggests that a genuine structural correlation is an essential ingredient in characterizing networks. As a signature of the correlation we investigate a scaling behavior in $M_N(h)$, the number of finite loops of size $h$, with respect to a network size $N$. We find that networks, whose degree distributions are not too broad, fall into two classes exhibiting $M_N(h)\sim ({constant})$ and $M_N(h) \sim (\ln N)^\psi$, respectively. This classification coincides with the one according to the percolation critical phenomena.

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