The Power Function Hidden in the Vulnerability of Fractal Complex Networks

As a special kind of complex network, fractal complex networks have their unique characteristics such as fractal dimension, scale invariance, percolation threshold, etc. There have been some researches on the topology of fractal network, among which the calculation of fractal dimension, the construction of fractal model, the phase transition and the robustness of fractal network were the hotspots. But unlike the studies of the scale-free and small-world networks, there was little research on the relationship between the stability and the fractal feature, especially between the vulnerability and fractal dimension of the fractal networks. In this paper, the relationship between vulnerability and fractal dimension was discussed based on two kinds of fractal growth models and different vulnerability measurement methods. The same power functional relationships were obtained and a universal expression was proposed. Then, the effect of fractal feature on the vulnerability was exploring through the fitting error when fractal dimensions were calculated. Finally, the vulnerability of the real networks and their similarities to the fractal models in term of vulnerability showed the rationality of the fractal network models in establishing stable models. The results of this study have important reference for creating structurally stable artificial neural networks, transportation networks and information networks.