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Gao J, Buldyrev SV, Havlin S, et al. Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes. Phys Rev E Stat Nonlin Soft Matter Phys. 2012;85(6):066134doi: 10.1103/PhysRevE.85.066134.
Gao, J., Buldyrev, S. V., Havlin, S., & Stanley, H. E. (2012). Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes. Physical review. E, Statistical, nonlinear, and soft matter physics, 85(6), 066134. https://doi.org/10.1103/PhysRevE.85.066134
Gao, Jianxi, et al. "Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes." Physical review. E, Statistical, nonlinear, and soft matter physics vol. 85,6 (2012): 066134. doi: https://doi.org/10.1103/PhysRevE.85.066134
Gao J, Buldyrev SV, Havlin S, Stanley HE. Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes. Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jun;85(6):066134. doi: 10.1103/PhysRevE.85.066134. Epub 2012 Jun 29. PMID: 23005189.
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Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jun;85(6):066134. doi: 10.1103/PhysRevE.85.066134. Epub 2012 Jun 29.

Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes.

Physical review. E, Statistical, nonlinear, and soft matter physics

Jianxi Gao, S V Buldyrev, S Havlin, H E Stanley

Affiliations

  1. Department of Automation, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, PR China.

PMID: 23005189 DOI: 10.1103/PhysRevE.85.066134

Abstract

Many real-world networks interact with and depend upon other networks. We develop an analytical framework for studying a network formed by n fully interdependent randomly connected networks, each composed of the same number of nodes N. The dependency links connecting nodes from different networks establish a unique one-to-one correspondence between the nodes of one network and the nodes of the other network. We study the dynamics of the cascades of failures in such a network of networks (NON) caused by a random initial attack on one of the networks, after which a fraction p of its nodes survives. We find for the fully interdependent loopless NON that the final state of the NON does not depend on the dynamics of the cascades but is determined by a uniquely defined mutual giant component of the NON, which generalizes both the giant component of regular percolation of a single network (n=1) and the recently studied case of the mutual giant component of two interdependent networks (n=2). We also find that the mutual giant component does not depend on the topology of the NON and express it in terms of generating functions of the degree distributions of the network. Our results show that, for any n≥2 there exists a critical p=p(c)>0 below which the mutual giant component abruptly collapses from a finite nonzero value for p≥p(c) to zero for p2, a RR NON is stable for any n with p(c)<1). This results arises from the critical role played by singly connected nodes which exist in an ER NON and enhance the cascading failures, but do not exist in a RR NON.

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