Percolation Transitions in Edge-Coupled Interdependent Networks with Reinforced Inter-Layer Links.

Prior research on cascading failures within interdependent networks has predominantly emphasized the coupling of nodes. Nevertheless, in practical networks, interactions often exist not just through the nodes themselves but also via the connections (edges) linking them, a configuration referred to as edge-coupled interdependent networks. Past research has shown that introducing a certain percentage of reinforced nodes or connecting edges can prevent catastrophic network collapses. However, the effect of reinforced inter-layer links in edge-coupled interdependent networks has yet to be addressed. Here, we develop a theoretical framework for studying percolation models in edge-coupled interdependent networks by introducing a proportion of reinforced inter-layer links and deriving detailed expressions for the giant and finite components and the percolation phase transition threshold. We find that there exists a required minimum proportion of the reinforced inter-layer links to prevent abrupt network collapse, which serves as a boundary to distinguish different phase transition types of a network. We provide both analytical and numerical solutions for random and scale-free networks, demonstrating that the proposed method exhibits superior reinforcement efficiency compared to intra-layer link reinforcement strategies. Theoretical analysis, simulation results, and real network systems validate our model and indicate that introducing a specific proportion of reinforced inter-layer links can prevent abrupt system failure and enhance network robustness in edge-coupled interdependent networks.