Percolation of partially interdependent scale-free networks

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=p_c. (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=p_c). 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 p_c=0 for λ<3 and at p_c>0 for λ>3. Thus, the known theoretical p_c=0 for a single network with λ≤3 is expected to be valid also for strictly partial interdependent networks.