A High Robustness and Low Cost Model for Cascading Failures

We study numerically the cascading failure problem by using artificially created scale-free networks and the real network structure of the power grid. The capacity for a vertex is assigned as a monotonically increasing function of the load (or the betweenness centrality). Through the use of a simple functional form with two free parameters, revealed is that it is indeed possible to make networks more robust while spending less cost. We suggest that our method to prevent cascade by protecting less vertices is particularly important for the design of more robust real-world networks to cascading failures. The network robustness has been one of the most central topics in the complex network research [1]. In scale-free networks, the existence of hub vertices with high degrees has been shown to yield fragility to intentional attacks, while at the same time the network becomes robust to random failures due to the heterogeneous degree distribution [2–5]. On the other hand, for the description of dynamic processes on top of networks, it has been suggested that the information flow across the network is one of the key issues, which can be captured well by the betweenness centrality or the load [6]. Cascading failures can happen in many infrastructure networks, including the electrical power grid, Internet, road systems, and so on. At each vertex of the power grid, the electric power is either produced or transferred to other vertices, and it is possible that from some reasons a vertex is overloaded beyond the given capacity, which is the maximum electric power the vertex can handle. The breakdown of the heavily loaded single vertex will cause the redistribution of loads over the remaining vertices, which can trigger breakdowns of newly overloaded vertices. This process will go on until all the loads of the remaining vertices are below their capacities. For some real networks, the breakdown of a single vertex is sufficient to collapse the entire system, which is exactly what happened on August 14, 2003 when an initial minor disturbance in Ohio triggered the largest blackout in the history of United States in which millions of people suffered without electricity for as long as 15 hours [7]. A number of aspects of cascading failures in complex networks have been discussed in the literature [8–16], including the model for describing cascade phenomena [8], the control and defense strategy against cascading failures [9, 10], the analytical calculation of capacity parameter [11], and the modelling of the real-world data [12]. In a recent paper [16], the cascade process in scale-free networks with community structure has been investigated, and it has been found that a smaller modularity is easier to trigger cascade, which implies the importance of the modularity and community structure in cascading failures. In the research of the cascading failures, the following two issues are closely related to each other and of significant interests: One is how to improve the network robustness to cascading failures, and the other particularly important issue is how to design manmade networks with a less cost. In most circumstances, a high robustness and a low cost are difficult to achieve simultaneously. For example, while a network with more edges are more robust to failures, in practice, the number of edges is often limited by the cost to construct them. In brevity, it costs much to build a robust network. Very recently, Schäfer et. al. proposed a new proactive measure to increase the robustness of heterogeneous loaded networks to cascades. By defining the load dependent weights, the network turns to be more homogeneous and the total load is decreased, which means the investment cost is also reduced [15]. In the present Letter, for simplicity, we try to find a possible way of protecting networks based on the flow along shortest-