Robustness of edge-coupled interdependent networks with reinforced edges

Previous studies on cascade failures in interdependent networks have mainly focused on node coupling relationships. However, in realistic scenarios, interactions often occur at the edges connecting nodes rather than at the nodes themselves, giving rise to edge-coupled interdependent networks. In this article, we extend the model of partially edge-coupled interdependent networks by introducing reinforced edges with a ratio of rho. We analyse the formation of finite surviving components in edge-coupled networks, wherein the reinforced edges can function and support their neighbouring nodes to form functional components. To accomplish this, we develop a framework through a detailed mathematical derivation of the proposed model. We then investigate the critical value rho* of the reinforced edge ratio that can change the phase transition type of the network. Our model is verified by theoretical analysis, simulation experiments and real network systems. The results show that the introduction of a small proportion of reinforced edges in the edge-coupled interdependent network can avoid the sudden collapse of the network and significantly improve the robustness of the network.