# Percolation of partially interdependent scale-free networks

Zhou, Di; Gao, Jianxi; Stanley, H. Eugene; Havlin, Shlomo
We study the percolation behavior of two interdependent scale-free (SF) networks under random failure of 1-p fraction of nodes. Our results are based on numerical solutions of analytical expressions and simulations. We find that as the coupling strength between the two networks q reduces from 1 (fully coupled) to 0 (no coupling), there exist two critical coupling strengths q1 and q2, which separate three different regions with different behavior of the giant component as a function of p. (i) For q≥q1, an abrupt collapse transition occurs at p=pc. (ii) For q2<q<q1, the giant component has a hybrid transition combined of both, abrupt decrease at a certain p=pcjump followed by a smooth decrease to zero for p<pcjump as p decreases to zero. (iii) For q≤q2, the giant component has a continuous second-order transition (at p=pc). We find that (a) for $\lambda$≤3, q1≡1; and for $\lambda$>3, q1 decreases with increasing $\lambda$. Here, $\lambda$ is the scaling exponent of the degree distribution, P(k)∝k−$\lambda$. (b) In the hybrid transition, at the q2<q<q1 region, the mutual giant component P∞ jumps discontinuously at p=pcjump to a very small but nonzero value, and when reducing p, P∞ continuously approaches to 0 at pc=0 for $\lambda$<3 and at pc>0 for $\lambda$>3. Thus, the known theoretical pc=0 for a single network with $\lambda$⩽3 is expected to be valid also for strictly partial interdependent networks.
Research areas:
Year:
2013
Type of Publication:
Article
Journal:
Physical Review E
Volume:
87
Number:
5
Pages:
052812
Month:
May
ISSN:
1539-3755
DOI:
10.1103/PhysRevE.87.052812
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