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.
|State||Published - 1 Mar 2016|
Bibliographical noteFunding Information:
We thank the DTRA, NSF (grants CMMI-1125290, CHE-1213217, PHY-1505000 and SES-1452061), Keck Foundation, European Commission FET Open Project (FOC 255987 and FOC-INCO 297149) and Office of Naval Research for financial support. S.H. acknowledges the European LINC and MULTIPLEX (EU-FET project 317532) projects, the Deutsche Forschungsgemeinschaft (DFG), the Israel Science Foundation, ONR and DTRA for financial support. L.A.B. thanks UNMdP and FONCyT, PICT 0429/13 for financial support. S.L.C. gratefully acknowledges the financial support of the Fulbright Program for visiting scholars. A.M. thanks Bijeli Zeko for useful discussions.