Journal
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
Volume 114, Issue 13, Pages 3311-3315Publisher
NATL ACAD SCIENCES
DOI: 10.1073/pnas.1621369114
Keywords
percolation; interdependent networks; phase transition; collapse
Categories
Funding
- Office of Naval Research [N00014-09-1-0380, N00014-12-1-0548, N62909-16-1-2170, N62909-14-1-N019]
- Defense Threat Reduction Agency [HDTRA-1-10-1-0014, HDTRA-1-09-1-0035]
- National Science Foundation [PHY-1505000, CHE-1213217, CMMI 1125290]
- Department of Energy [DE-AC07-05Id14517]
- US-Israel Binational Science Foundation-National Science Foundation [2015781]
- National Natural Science Foundation of China [61203156]
- Hundred-Talent Program of the Sun Yat-sen University
- Chinese Fundamental Research Funds for the Central Universities [16lgjc84]
- European Multiplex and Dynamics and Coevolution in Multilevel Strategic Interaction Games (CONGAS) Projects
- Israel Ministry of Science and Technology
- Italy Ministry of Foreign Affairs
- Next Generation Infrastructure (Bsik)
- Israel Science Foundation
- Forecasting Financial Crises (FOC) Program of the European Union
- Directorate For Engineering
- Div Of Civil, Mechanical, & Manufact Inn [1125290] Funding Source: National Science Foundation
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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 Erdos-Renyi(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.
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