Percolation of Partially Interdependent Scale-free Networks

Abstract

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 $q\ge q_1$, 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 $q\le q_2$, the giant component has a continuous second-order transition (at $p=p_c$). We find that $(a)$ for $\lambda \le 3$, $q_1\equiv 1$; and for $\lambda >3$, $q_1$ decreases with increasing $\lambda$. $(b)$ In the hybrid transition, at the $q_2<q<q_1$ region, the mutual giant component $P_{\infty }$ jumps discontinuously at $p=p_c^{jump}$ to a very small but non-zero value, and when reducing $p$, $P_{\infty }$ continuously approaches to 0 at $p_c=0$ for $\lambda <3$ and at $p_c>0$ for $\lambda >3$. Thus, the known theoretical $p_c=0$ for a single network with $\lambda ⩽3$ is expected to be valid also for strictly partial interdependent networks.

20 pages, 17 figures

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