Suppressing cascading failures on multiplex networks with flow by reinforcing critical edges

Understanding the robustness of networks is a key step to design stable networked systems and protect them from being collapsed under small initial failures. In a two-layer multiplex network with flow, where physical quantities such as traffic flow or electricity move, a global cascading failure may be triggered out when edges in one layer fail and their loads are redistributed within the layer and to the other layer. Here we propose an edge pressure index to identify the critical edges which are the most likely to initiate a cascading failure in the network with flow. By reinforcing the critical edges, the robustness of the multiplex networks can be significantly enhanced. Based on large simulations on both synthetic networks and real-world networks, we find that protecting critical edges identified by the edge pressure index outperforms protecting edges with the highest betweenness centrality and the randomly selected edges in enhancing the robustness of the network. Furthermore, preventing the initial cascading failures between layers is critical to reduce the cascading failure size, which is due to the accumulative effects of redistributed loads. When the layer under initial attack is a heterogeneous BA network, which is more robust under flow redistribution, the entire system is more robust because the edge failures in the BA network create less intra- and inter-layer load redistribution.