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Contributed by H. Eugene Stanley, December 29, 2016 (sent for review April 21, 2016; reviewed by Antonio Coniglio and Michael F. Shlesinger)
Significance
Percolation theory assumes that only the largest connected component is functional. However, in reality, some components that are not connected to the largest component can also function. Here, we generalize the percolation theory by assuming a fraction of reinforced nodes that can function and support their components, although they are disconnected from the largest connected component. We find that the reinforced nodes reduce significantly the cascading failures in interdependent networks system. Moreover, including a small critical fraction of reinforced nodes can avoid abrupt catastrophic failures in such systems.
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.
Footnotes
- ↵1To whom correspondence may be addressed. Email: hes{at}bu.edu or yanqing.hu.sc{at}gmail.com.
Author contributions: X.Y., Y.H., and S.H. designed research; X.Y. and Y.H. performed research; S.H. contributed new reagents/analytic tools; X.Y., H.E.S., and S.H. analyzed data; and X.Y., Y.H., H.E.S., and S.H. wrote the paper.
Reviewers: A.C., University of Naples; and M.F.S., Office of Naval Research.
The authors declare no conflict of interest.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1621369114/-/DCSupplemental.
Catastrophic collapse in interdependent networks
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