  Real data show that interdependent networks usually involve inter-similarity.
Intersimilarity means that a pair of interdependent nodes have neighbors in
both networks that are also interdependent (Parshani et al \cite{PAR10B}). For
example, the coupled world wide port network and the global airport network are
intersimilar since many pairs of linked nodes (neighboring cities), by direct
flights and direct shipping lines exist in both networks. Nodes in both
networks in the same city are regarded as interdependent. If two neighboring
nodes in one network depend on neighboring nodes in the another we call these
links common links. The fraction of common links in the system is a measure of
intersimilarity. Previous simulation results suggest that intersimilarity has
considerable effect on reducing the cascading failures, however, a theoretical
understanding on this effect on the cascading process is currently missing.
Here, we map the cascading process with inter-similarity to a percolation of
networks composed of components of common links and non common links. This
transforms the percolation of inter-similar system to a regular percolation on
a series of subnetworks, which can be solved analytically. We apply our
analysis to the case where the network of common links is an
Erd\H{o}s-R\'{e}nyi (ER) network with the average degree $K$, and the two
networks of non-common links are also ER networks. We show for a fully coupled
pair of ER networks, that for any $K\geq0$, although the cascade is reduced
with increasing $K$, the phase transition is still discontinuous. Our analysis
can be generalized to any kind of interdependent random networks system.
