Robustness of a Network of Networks
Abstract
Network research has been focused on studying the properties of a single isolated network, which rarely exists. We develop a general analytical framework for studying percolation of n interdependent networks. We illustrate our analytical solutions for three examples: (i) For any tree of n fully dependent ErdősRényi (ER) networks, each of average degree k¯, we find that the giant component is P_{∞}=p[1exp(k¯P_{∞})]^{n} where 1p is the initial fraction of removed nodes. This general result coincides for n=1 with the known secondorder phase transition for a single network. For any n>1 cascading failures occur and the percolation becomes an abrupt firstorder transition. (ii) For a starlike network of n partially interdependent ER networks, P_{∞} depends also on the topology—in contrast to case (i). (iii) For a looplike network formed by n partially dependent ER networks, P_{∞} is independent of n.
 Publication:

Physical Review Letters
 Pub Date:
 November 2011
 DOI:
 10.1103/PhysRevLett.107.195701
 arXiv:
 arXiv:1010.5829
 Bibcode:
 2011PhRvL.107s5701G
 Keywords:

 64.60.ah;
 05.10.a;
 05.40.a;
 89.75.Hc;
 Percolation;
 Computational methods in statistical physics and nonlinear dynamics;
 Fluctuation phenomena random processes noise and Brownian motion;
 Networks and genealogical trees;
 Physics  Data Analysis;
 Statistics and Probability;
 Computer Science  Social and Information Networks;
 Physics  Physics and Society
 EPrint:
 7 pages, 3 figures