  We study a problem of failure of two interdependent networks in the case of
correlated degrees of mutually dependent nodes. We assume that both networks (A
and B) have the same number of nodes $N$ connected by the bidirectional
dependency links establishing a one-to-one correspondence between the nodes of
the two networks in a such a way that the mutually dependent nodes have the
same number of connectivity links, i.e. their degrees coincide. This implies
that both networks have the same degree distribution $P(k)$. We call such
networks correspondently coupled networks (CCN). We assume that the nodes in
each network are randomly connected. We define the mutually connected clusters
and the mutual giant component as in earlier works on randomly coupled
interdependent networks and assume that only the nodes which belong to the
mutual giant component remain functional. We assume that initially a $1-p$
fraction of nodes are randomly removed due to an attack or failure and find
analytically, for an arbitrary $P(k)$, the fraction of nodes $\mu(p)$ which
belong to the mutual giant component. We find that the system undergoes a
percolation transition at certain fraction $p=p_c$ which is always smaller than
the $p_c$ for randomly coupled networks with the same $P(k)$. We also find that
the system undergoes a first order transition at $p_c>0$ if $P(k)$ has a finite
second moment. For the case of scale free networks with $2<\lambda \leq 3$, the
transition becomes a second order transition. Moreover, if $\lambda<3$ we find
$p_c=0$ as in percolation of a single network. For $\lambda=3$ we find an exact
analytical expression for $p_c>0$. Finally, we find that the robustness of CCN
increases with the broadness of their degree distribution.
