Numerical evaluation of the upper critical dimension of percolation in scale-free networks.
We propose numerical methods to evaluate the upper critical dimension d(c) of random percolation clusters in Erdös-Rényi networks and in scale-free networks with degree distribution P(k) approximately k(-lambda), where k is the degree of a node and lambda is the broadness of the degree distribution. Our results support the theoretical prediction, d(c) = 2(lambda - 1)(lambda - 3) for scale-free networks with 3 < lambda < 4 and d(c) = 6 for Erdös-Rényi networks and scale-free networks with lambda > 4 . When the removal of nodes is not random but targeted on removing the highest degree nodes we obtain d(c) = 6 for all lambda > 2 . Our method also yields a better numerical evaluation of the critical percolation threshold p(c) for scale-free networks. Our results suggest that the finite size effects increases when lambda approaches 3 from above.
Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA.
Zhenhua
Z
Wu
Cecilia
C
Lagorio
Lidia A
LA
Braunstein
Reuven
R
Cohen
Shlomo
S
Havlin
H Eugene
HE
Stanley