Percolation in networks composed of connectivity and dependency links

Networks composed from both connectivity and dependency links were found to be more vulnerable compared to classical networks with only connectivity links. Their percolation transition is usually of a first order compared to the second order transition found in classical networks. We analytically analyze the effect of different distributions of dependencies links on the robustness of networks. For a random Erd$\ddot{o}$s-R$\acute{e}$nyi (ER) network with average degree $k$ that is divided into dependency clusters of size $s$, the fraction of nodes that belong to the giant component, $P_\infty$, is given by $P_\infty=p^{s-1} [1-\exp{(-kpP_\infty)}]^s$ where $1-p$ is the initial fraction of removed nodes. Our general result coincides with the known Erd$\ddot{o}$s-R$\acute{e}$nyi equation for random networks for $s=1$ and with the result of Parshani et al (PNAS, in press, 2011) for $s=2$. For networks with Poissonian distribution of dependency links we find that $P_\infty$ is given by $P_\infty = f_{k,p}(P_\infty) e^{(<s>-1)(pf_{k,p}(P_\infty)-1)}$ where $f_{k,p}(P_\infty) \equiv 1-\exp{(-kpP_\infty)}$ and $<s>$ is the mean value of the size of dependency clusters. For networks with Gaussian distribution of dependency links we show how the average and width of the distribution affect the robustness of the networks.