Percolation of partially interdependent scale-free networks

Di Zhou, Jianxi Gao, H. Eugene Stanley, Shlomo Havlin

Research output: Contribution to journalArticlepeer-review

104 Scopus citations


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 q 2<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≤q 2, the giant component has a continuous second-order transition (at p=pc). We find that (a) for λ≤3, q1≡1; and for λ>3, q1 decreases with increasing λ. Here, λ is the scaling exponent of the degree distribution, P(k)â̂k-λ. (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 λ<3 and at pc>0 for λ>3. Thus, the known theoretical pc=0 for a single network with λ≤3 is expected to be valid also for strictly partial interdependent networks.

Original languageEnglish
Article number052812
Pages (from-to)052812
JournalPhysical Review E
Issue number5
StatePublished - 29 May 2013


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