  Percolation theory is extensively studied in statistical physics and
mathematics with applications in diverse fields. However, the research is
focused on systems with only one type of links, connectivity links. We review a
recently developed mathematical framework for analyzing percolation properties
of realistic scenarios of networks having links of two types, connectivity and
dependency links. This formalism was applied to study
Erd$\ddot{o}$s-R$\acute{e}$nyi (ER) networks that include also dependency
links. For an ER network with average degree $k$ that is composed of 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 randomly removed nodes. Here, we apply the
formalism to the study of random-regular (RR) networks and find a formula for
the size of the giant component in the percolation process:
$P_\infty=p^{s-1}(1-r^k)^s$ where $r$ is the solution of
$r=p^s(r^{k-1}-1)(1-r^k)+1$. These general results coincide, for $s=1$, with
the known equations for percolation in ER and RR networks respectively without
dependency links. In contrast to $s=1$, where the percolation transition is
second order, for $s>1$ it is of first order. Comparing the percolation
behavior of ER and RR networks we find a remarkable difference regarding their
resilience. We show, analytically and numerically, that in ER networks with low
connectivity degree or large dependency clusters, removal of even a finite
number (zero fraction) of the network nodes will trigger a cascade of failures
that fragments the whole network. This result is in contrast to RR networks
where such cascades and full fragmentation can be triggered only by removal of
a finite fraction of nodes in the network.
