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 $q \geq q_1$, 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 $p=p^{jump}_c$ followed by a smooth decrease to zero for $p < p^{jump}_c$ as $p$ decreases to zero. (iii) For $q \leq q_2$, the giant component has a continuous second-order transition (at $p=p_c$). We find that $(a)$ for $λ\leq 3$, $q_1 \equiv 1$; and for $λ> 3$, $q_1$ decreases with increasing $λ$. $(b)$ In the hybrid transition, at the $q_2 < q < q_1$ region, the mutual giant component $P_{\infty}$ jumps discontinuously at $p=p^{jump}_c$ to a very small but non-zero value, and when reducing $p$, $P_{\infty}$ 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 $λ\leqslant 3$ is expected to be valid also for strictly partial interdependent networks.