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 Erdos-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.
Original language | English |
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Pages (from-to) | 3311-3315 |
Number of pages | 5 |
Journal | Proceedings of the National Academy of Sciences of the United States of America |
Volume | 114 |
Issue number | 13 |
DOIs | |
State | Published - 28 Mar 2017 |
Bibliographical note
Funding Information:We thank the financial support of the Office of Naval Research Grants N00014-09-1-0380, N00014-12-1-0548, N62909-16-1-2170, and N62909-14-1-N019; Defense Threat Reduction Agency Grants HDTRA- 1-10-1-0014 and HDTRA-1-09-1-0035; National Science Foundation Grants PHY-1505000, CHE-1213217, and CMMI 1125290; Department of Energy Contract DE-AC07-05Id14517; and US-Israel Binational Science Foundation-National Science Foundation Grant 2015781. Y.H. is supported by National Natural Science Foundation of China Grant 61203156, the Hundred-Talent Program of the Sun Yat-sen University, and the Chinese Fundamental Research Funds for the Central Universities Grant 16lgjc84. Financial support was also provided by the European Multiplex and Dynamics and Coevolution in Multilevel Strategic Interaction Games (CONGAS) Projects; the Israel Ministry of Science and Technology with the Italy Ministry of Foreign Affairs; the Next Generation Infrastructure (Bsik); and the Israel Science Foundation. We also thank the Forecasting Financial Crises (FOC) Program of the European Union for support.
Funding
We thank the financial support of the Office of Naval Research Grants N00014-09-1-0380, N00014-12-1-0548, N62909-16-1-2170, and N62909-14-1-N019; Defense Threat Reduction Agency Grants HDTRA- 1-10-1-0014 and HDTRA-1-09-1-0035; National Science Foundation Grants PHY-1505000, CHE-1213217, and CMMI 1125290; Department of Energy Contract DE-AC07-05Id14517; and US-Israel Binational Science Foundation-National Science Foundation Grant 2015781. Y.H. is supported by National Natural Science Foundation of China Grant 61203156, the Hundred-Talent Program of the Sun Yat-sen University, and the Chinese Fundamental Research Funds for the Central Universities Grant 16lgjc84. Financial support was also provided by the European Multiplex and Dynamics and Coevolution in Multilevel Strategic Interaction Games (CONGAS) Projects; the Israel Ministry of Science and Technology with the Italy Ministry of Foreign Affairs; the Next Generation Infrastructure (Bsik); and the Israel Science Foundation. We also thank the Forecasting Financial Crises (FOC) Program of the European Union for support.
Funders | Funder number |
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Israel Ministry of Science and Technology | |
Next Generation Infrastructure | |
US-Israel Binational Science Foundation-National Science Foundation | 2015781 |
National Science Foundation | PHY-1505000, CMMI 1125290, CHE-1213217 |
Office of Naval Research | N00014-09-1-0380, N00014-12-1-0548, N62909-16-1-2170, N62909-14-1-N019 |
U.S. Department of Energy | DE-AC07-05Id14517 |
Directorate for Engineering | 1125290 |
Defense Threat Reduction Agency | HDTRA- 1-10-1-0014, HDTRA-1-09-1-0035 |
European Commission | |
National Natural Science Foundation of China | 61203156 |
Sun Yat-Sen University | |
Israel Science Foundation | |
Ministry for Foreign Affairs | |
Fundamental Research Funds for the Central Universities | 16lgjc84 |
Keywords
- Collapse
- Interdependent networks
- Percolation
- Phase transition