Eradicating Catastrophic Collapse in Interdependent Networks via   Reinforced Nodes
  In interdependent networks, it is usually assumed, based on percolation theory, that nodes become nonfunctional if they lose connection to the network giant component. However, in reality, some nodes, equipped with alternative resources, together with their connected neighbors can still be functioning once disconnected from the giant component. Here we propose and study a generalized percolation model that introduces a fraction of reinforced nodes in the interdependent networks that can function and support their neighborhood. We analyze, both analytically and via simulations, the order parameter$-$the functioning component$-$comprising both the giant component and smaller components that include at least one reinforced node. Remarkably, we find that for interdependent networks, we need to reinforce only a small fraction of nodes to prevent abrupt catastrophic collapses. Moreover, we find that the universal upper bound of this fraction is 0.1756 for two interdependent Erd\H{o}s-R\'{e}nyi (ER) networks, regular-random (RR) networks and scale-free (SF) networks with large average degrees. We also generalize our theory to interdependent networks of networks (NON). Our findings might yield insight for designing resilient interdependent infrastructure networks. 
Xin Yuan, Yanqing Hu, H. Eugene Stanley, Shlomo Havlin
physics.soc-ph