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We study a problem of failure of two interdependent networks in the case ofcorrelated degrees of mutually dependent nodes. We assume that both networks (Aand B) have the same number of nodes $N$ connected by the bidirectionaldependency links establishing a one-to-one correspondence between the nodes ofthe two networks in a such a way that the mutually dependent nodes have thesame number of connectivity links, i.e. their degrees coincide. This impliesthat both networks have the same degree distribution $P(k)$. We call suchnetworks correspondently coupled networks (CCN). We assume that the nodes ineach network are randomly connected. We define the mutually connected clustersand the mutual giant component as in earlier works on randomly coupledinterdependent networks and assume that only the nodes which belong to themutual giant component remain functional. We assume that initially a $1-p$fraction of nodes are randomly removed due to an attack or failure and findanalytically, for an arbitrary $P(k)$, the fraction of nodes $\mu(p)$ whichbelong to the mutual giant component. We find that the system undergoes apercolation transition at certain fraction $p=p_c$ which is always smaller thanthe $p_c$ for randomly coupled networks with the same $P(k)$. We also find thatthe system undergoes a first order transition at $p_c>0$ if $P(k)$ has a finitesecond moment. For the case of scale free networks with $2<\lambda \leq 3$, thetransition 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 exactanalytical expression for $p_c>0$. Finally, we find that the robustness of CCNincreases with the broadness of their degree distribution.

Interdependent networks with correlated degrees of mutually dependent nodes