  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.
