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In interdependent networks, it is usually assumed, based on percolationtheory, that nodes become nonfunctional if they lose connection to the networkgiant component. However, in reality, some nodes, equipped with alternativeresources, together with their connected neighbors can still be functioningonce disconnected from the giant component. Here we propose and study ageneralized percolation model that introduces a fraction of reinforced nodes inthe interdependent networks that can function and support their neighborhood.We analyze, both analytically and via simulations, the order parameter$-$thefunctioning component$-$comprising both the giant component and smallercomponents that include at least one reinforced node. Remarkably, we find thatfor interdependent networks, we need to reinforce only a small fraction ofnodes to prevent abrupt catastrophic collapses. Moreover, we find that theuniversal upper bound of this fraction is 0.1756 for two interdependentErd\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 tointerdependent networks of networks (NON). Our findings might yield insight fordesigning resilient interdependent infrastructure networks.

Eradicating Catastrophic Collapse in Interdependent Networks via Reinforced Nodes