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 q(1) and q(2), which separate three different regions with different behavior of the giant component as a function of p. (i) For q≥q(1), an abrupt collapse transition occurs at p=p(c). (ii) For q(2)3, q(1) decreases with increasing λ. Here, λ is the scaling exponent of the degree distribution, P(k)[proportionality]k(-λ). (b) In the hybrid transition, at the q(2)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.