Cascading Failure in Fractal Scale-Free Networks

The fractal property widely exists in various complex self-organized systems, so a comprehensive analysis of the robustness of fractal networks is of great significance for the construction of resilient real-world systems. Previous studies have mainly focused on the structural robustness analysis of fractal networks, while this study considers the network functionality to investigate the robustness of fractal scale-free networks to catastrophic cascades and their relationship with the fractal dimension. The hierarchical multiplicative growth model with shortcuts is first created to generate scale-free networks capable of large-scale cascading failures. These networks have the same number of nodes, number of links, degree exponent, and clustering coefficient, but different fractal dimensions. Then, the cascades triggered by two targeted attack strategies within three types of cascade models are analyzed. The results on model networks indicate that the fractal topology can expand the range of load distribution and reduce the load disparity among nodes (or links). As a result of this effect, cascading failures in fractal scale-free networks are more predictable and controllable than their non-fractal counterparts. Notably, the smaller the fractal dimension, the more robust the network is to cascading failures. Finally, numerical results on four types of real-world networks confirm that the above conclusions are considered reliable. This study contributes to a deeper understanding of the evolutionary advantages of fractal topology from the perspective of network functionality, and may provide insight into the prevention, prediction, and control of cascading failures in real-world networks.