  The function of a real network depends not only on the reliability of its own
components, but is affected also by the simultaneous operation of other real
networks coupled with it. Robustness of systems composed of interdependent
network layers has been extensively studied in recent years. However, the
theoretical frameworks developed so far apply only to special models in the
limit of infinite sizes. These methods are therefore of little help in
practical contexts, given that real interconnected networks have finite size
and their structures are generally not compatible with those of graph toy
models. Here, we introduce a theoretical method that takes as inputs the
adjacency matrices of the layers to draw the entire phase diagram for the
interconnected network, without the need of actually simulating any percolation
process. We demonstrate that percolation transitions in arbitrary
interdependent networks can be understood by decomposing these system into
uncoupled graphs: the intersection among the layers, and the remainders of the
layers. When the intersection dominates the remainders, an interconnected
network undergoes a continuous percolation transition. Conversely, if the
intersection is dominated by the contribution of the remainders, the transition
becomes abrupt even in systems of finite size. We provide examples of real
systems that have developed interdependent networks sharing a core of "high
quality" edges to prevent catastrophic failures.
