title: Recovery of Interdependent Networks.
abstract: Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse. An important question that has not been addressed is how to repair a failing system before it suffers total breakdown. Here we introduce a recovery strategy for nodes and develop an analytic and numerical framework for studying the concurrent failure and recovery of a system of interdependent networks based on an efficient and practically reasonable strategy. Our strategy consists of repairing a fraction of failed nodes, with probability of recovery γ, that are neighbors of the largest connected component of each constituent network. We find that, for a given initial failure of a fraction 1 - p of nodes, there is a critical probability of recovery above which the cascade is halted and the system fully restores to its initial state and below which the system abruptly collapses. As a consequence we find in the plane γ - p of the phase diagram three distinct phases. A phase in which the system never collapses without being restored, another phase in which the recovery strategy avoids the breakdown, and a phase in which even the repairing process cannot prevent system collapse.
authors.affiliation: Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)-Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata-CONICET, Funes 3350 (7600) Mar del Plata, Argentina.
authors.forename: M A
authors.initials: MA
authors.lastname: Di Muro
authors.affiliation: Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)-Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata-CONICET, Funes 3350 (7600) Mar del Plata, Argentina.
authors.forename: C E
authors.initials: CE
authors.lastname: La Rocca
authors.affiliation: Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA.
authors.forename: H E
authors.initials: HE
authors.lastname: Stanley
authors.affiliation: Department of Physics, Bar Ilan University, Ramat Gan, Israel.
authors.forename: S
authors.initials: S
authors.lastname: Havlin
authors.affiliation: Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR)-Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata-CONICET, Funes 3350 (7600) Mar del Plata, Argentina.|Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USA.
authors.forename: L A
authors.initials: LA
authors.lastname: Braunstein