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)<q<q(1), the giant component has a hybrid transition combined of both, abrupt decrease at a certain p=p(c)(jump) followed by a smooth decrease to zero for p<p(c)(jump) 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, q(1)≡1; and for λ>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)<q<q(1) region, the mutual giant component P(∞) jumps discontinuously at p=p(c)(jump) 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.