Various social, financial, biological and technological systems can bemodeled by interdependent networks. It has been assumed that in order to remainfunctional, nodes in one network must receive the support from nodes belongingto different networks. So far these models have been limited to the case inwhich the failure propagates across networks only if the nodes lose all theirsupply nodes. In this paper we develop a more realistic model for twointerdependent networks in which each node has its own supply threshold, i.e.,they need the support of a minimum number of supply nodes to remain functional.In addition, we analyze different conditions of internal node failure due todisconnection from nodes within its own network. We show that several localinternal failure conditions lead to similar nontrivial results. When there areno internal failures the model is equivalent to a bipartite system, which canbe useful to model a financial market. We explore the rich behaviors of thesemodels that include discontinuous and continuous phase transitions. Using thegenerating functions formalism, we analytically solve all the models in thelimit of infinitely large networks and find an excellent agreement with thestochastic simulations.
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