Abstract: | Network science have been focused on the properties of a single isolated network that does not interact or depends on other networks. In reality, many real-networks, such as power grid, protein networks, transportation and communication infrastructures interact and depend on other networks. I will present a framework for studying the vulnerability of networks of interdependent networks. In interdependent networks, when nodes in one network fail, they cause dependent nodes in other networks to also fail. This may happen recursively and can lead to a cascade of failures and to a sudden fragmentation of the system. I will present analytical solutions for the critical threshold and the giant component of a network of n interdependent networks. I will show, that the general theory has many novel features that are not present in the classical network theory. Our results for network of n networks suggest that the classical percolation theory extensively studied in physics and mathematics is a limiting case of n=1 of the general theory of percolation in network of networks. I will also show that interdependent networks embedded in space are significantly more vulnerable compared to non embedded networks. References: [1] S. Buldyrev, R. Parshani, G. Paul, H.E. Stanley, S. Havlin, Nature, 465, 0893 (2010) [2] R. Parshani, S. Buldyrev, S. Havlin, PRL, 105, 048701 (2010) [3] R. Parshani, S.V. Buldyrev, S. Havlin, PNAS 108, 1007 (2011) [4] J. Gao, S. Buldyrev, H. E. Stanley, S. Havlin, Nature Physics, 8, 40 (2012). [5] W. Li, A. Bashan, S. V. Buldyrev, H. E. Stanley, and S. Havlin Phys. Rev. Lett. 108, 228702 (2012) [6] A. Bashan et al, Nature Physics, 9, 667 (2013) |