  Many systems, ranging from engineering to medical to societal, can only be
properly characterized by multiple interdependent networks whose normal
functioning depends on one another. Failure of a fraction of nodes in one
network may lead to a failure in another network. This in turn may cause
further malfunction of additional nodes in the first network and so on. Such a
cascade of failures, triggered by a failure of a small faction of nodes in only
one network, may lead to the complete fragmentation of all networks. We
introduce a model and an analytical framework for studying interdependent
networks. We obtain interesting and surprising results that should
significantly effect the design of robust real-world networks. For two
interdependent Erdos-Renyi (ER) networks, we find that the critical average
degree below which both networks collapse is <k_c>=2.445, compared to <k_c>=1
for a single ER network. Furthermore, while for a single network a broader
degree distribution of the network nodes results in higher robustness to random
failure, for interdependent networks, the broader the distribution is, the more
vulnerable the networks become to random failure.
