Article,
Robustness of a network formed of spatially embedded networks
Physical Review E, (Jul 17, 2014)
DOI: 10.1103/PhysRevE.90.012809
Abstract
We present analytic and numeric results for percolation in a network formed
of interdependent spatially embedded networks. We show results for a treelike
and a random regular network of networks each with \$(i)\$ unconstrained
interdependent links and \$(ii)\$ interdependent links restricted to a maximum
length, \$r\$. Analytic results are given for each network of networks with
unconstrained dependency links and compared with simulations. For the case of
two spatially embedded networks it was found that only for \$r>r\_c\approx8\$ does
the system undergo a first order phase transition. We find that for treelike
networks of networks \$r\_c\$ significantly decreases as \$n\$ increases and rapidly
reaches its limiting value, \$r=1\$. For cases where the dependencies form loops,
such as in random regular networks, we show analytically and confirm through
simulations, that there is a certain fraction of dependent nodes, \$q\_max\$,
above which the entire network structure collapses even if a single node is
removed. This \$q\_max\$ decreases quickly with \$m\$, the degree of the random
regular network of networks. Our results show the extreme sensitivity of
coupled spatial networks and emphasize the susceptibility of these networks to
sudden collapse. The theory derived here can be used to find the robustness of
any network of networks where the profile of percolation of a single network is
known.
