Percolation and Loop Statistics in Complex Networks

Abstract

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.

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

References in corpus (2)

• Evolution of scale-free random graphs: Potts model formulation
• Finding long cycles in graphs

Cited by in corpus (1)

• Percolation transition and distribution of connected components in generalized random network ensembles