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=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 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)∝k^-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 λ3 is expected to be valid also for strictly partial interdependent networks.