A theoretical framework for the percolation of interdependent networks will be presented. In interdependent networks, such as infrastructures, 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 a sudden fragmentation of the system. This is in contrast to a single network where the percolation transition due to failures is continuous.  I will present analytical solutions based on the percolation theory, of the functional network and cascading failures, for a network of n interdependent networks. Our analytical results show that the percolation theory of a single network studied for over 90 years is just a limited case, n=1, of the general and a significantly richer case of n>1. I will also show that interdependent networks embedded in space are extremely vulnerable and have significantly richer behavior compared to non-embedded networks. In particular, it will be shown that localized attacks of a microscopic critical size lead to cascading failures that dynamically propagate like nucleation and yield an abrupt macroscopic phase transition.  I will finally show that the abstract interdependent percolation theory and its novel behavior in networks of networks can be realized and proven in controlled experiments performed by Aviad Frydman on real physical systems.  I will present recent experiments that support the interdependent network theory in measurements of interdependent superconducting networks. Here, a novel abrupt phase transition is observed due to microscopic interactions between the macroscopic systems. This is in contrast to an isolated system that shows a continuous phase transition.

References
 S. Buldyrev, G. Paul, H.E. Stanley, S. Havlin, Nature, 464, 08932 (2010)
 J. Gao, S. Buldyrev, H. E. Stanley, S. Havlin, Nature Physics, 8, 40 (2012)

 A. Bashan et al, Nature Physics, 9, 667 (2013)
A. Majdandzic et al, Nature Physics 10 (1), 34 (2014); Nature Comm. 7, 10850 (2016)
 M. Danziger et al, Nature Physics  15(2), 178 (2019)
 B. Gross et al, PRL  129, 268301 (2022)
 I. Bonamassa et al, Interdependent superconducting networks, Nature Physics 19, 1163 (2023)