Fractal characteristics of complex networks with cascading failures

Due to fault propagation, a small initial shock can trigger large cascades in complex networks, which reduces network reliability. Studying the characteristics of failing nodes of networks under fault propagation process can help to design a more reliable network. Previous researches have shown that the structure consisting of adjacent failing nodes caused by fault propagation on grid networks has fractal characteristics. However, the topologies of many real networks appear more complicated and the failing nodes are not often connected. Therefore, in this paper, we propose a new shortest-path-based method to construct a fault network, and then investigate fractal characteristic of the fault network during fault propagation process. First, failing nodes are obtained by performing cascading failure process on complex networks, and a fault network is constructed by linking the failing nodes, in which the edge weight is the length of shortest path between failing nodes. Then, the fractal dimension of fault networks, an effective criterion for fractal characteristics, is calculated based on boxing-covering algorithm for weighted networks (BCANw). Numerical simulations performing cascading failure process on random networks show that the fractal dimension of fault networks during fault propagation process is constant when the critical condition for large-scale collapse has not been reached. Invariance of fractal dimension of fault networks, denoting the spatial density of failures, indicates the impact of fault propagation on failing nodes.