title: Eradicating catastrophic collapse in interdependent networks via reinforced nodes.
abstract: In interdependent networks, it is usually assumed, based on percolation theory, that nodes become nonfunctional if they lose connection to the network giant component. However, in reality, some nodes, equipped with alternative resources, together with their connected neighbors can still be functioning after disconnected from the giant component. Here, we propose and study a generalized percolation model that introduces a fraction of reinforced nodes in the interdependent networks that can function and support their neighborhood. We analyze, both analytically and via simulations, the order parameter-the functioning component-comprising both the giant component and smaller components that include at least one reinforced node. Remarkably, it is found that, for interdependent networks, we need to reinforce only a small fraction of nodes to prevent abrupt catastrophic collapses. Moreover, we find that the universal upper bound of this fraction is 0.1756 for two interdependent Erdős-Rényi (ER) networks: regular random (RR) networks and scale-free (SF) networks with large average degrees. We also generalize our theory to interdependent networks of networks (NONs). These findings might yield insight for designing resilient interdependent infrastructure networks.
authors.affiliation: Center for Polymer Studies, Boston University, Boston, MA 02215.|Department of Physics, Boston University, Boston, MA 02215.
authors.forename: Xin
authors.initials: X
authors.lastname: Yuan
authors.affiliation: School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China; hes@bu.edu yanqing.hu.sc@gmail.com.|School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China.|Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China.
authors.forename: Yanqing
authors.initials: Y
authors.lastname: Hu
authors.affiliation: Center for Polymer Studies, Boston University, Boston, MA 02215; hes@bu.edu yanqing.hu.sc@gmail.com.|Department of Physics, Boston University, Boston, MA 02215.
authors.forename: H Eugene
authors.identifier: 0000-0003-2800-4495
authors.initials: HE
authors.lastname: Stanley
authors.affiliation: Minerva Center, Bar-Ilan University, Ramat-Gan 52900, Israel.|Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel.
authors.forename: Shlomo
authors.initials: S
authors.lastname: Havlin
keywords: collapse; interdependent networks; percolation; phase transition