title: Multiple tipping points and optimal repairing in interacting networks.
abstract: Systems composed of many interacting dynamical networks-such as the human body with its biological networks or the global economic network consisting of regional clusters-often exhibit complicated collective dynamics. Three fundamental processes that are typically present are failure, damage spread and recovery. Here we develop a model for such systems and find a very rich phase diagram that becomes increasingly more complex as the number of interacting networks increases. In the simplest example of two interacting networks we find two critical points, four triple points, ten allowed transitions and two 'forbidden' transitions, as well as complex hysteresis loops. Remarkably, we find that triple points play the dominant role in constructing the optimal repairing strategy in damaged interacting systems. To test our model, we analyse an example of real interacting financial networks and find evidence of rapid dynamical transitions between well-defined states, in agreement with the predictions of our model.
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA.
authors.forename: Antonio
authors.initials: A
authors.lastname: Majdandzic
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA.|Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Physics Department, Universidad Nacional de Mar del Plata-CONICET, Funes 3350, 7600 Mar del Plata, Argentina.
authors.forename: Lidia A
authors.initials: LA
authors.lastname: Braunstein
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA.
authors.forename: Chester
authors.initials: C
authors.lastname: Curme
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA.|Administrative Sciences Department, Metropolitan College, Boston University, Boston, Massachusetts 02215 USA.
authors.forename: Irena
authors.initials: I
authors.lastname: Vodenska
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA.|Economics and Social Sciences Faculty, Central University of Venezuela, 1040 Caracas, Venezuela.
authors.forename: Sary
authors.initials: S
authors.lastname: Levy-Carciente
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA.
authors.forename: H Eugene
authors.initials: HE
authors.lastname: Stanley
authors.affiliation: Center for Polymer Studies and Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA.|Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel.
authors.forename: Shlomo
authors.initials: S
authors.lastname: Havlin