An in-depth analysis of the attack vulnerability of fractal scale-free networks is of great significance for designing robust networks. Previous studies have mainly focused on the impact of fractal property on attack vulnerability of scale-free networks under static node attacks, while we extend the study to the cases of various types of targeted attacks, and explore the relationship between the attack vulnerability of fractal scale-free networks and the fractal dimension. A hierarchical multiplicative growth model is first proposed to generate scale-free networks with the same structural properties except for the fractal dimension. Furthermore, the fractal dimension of the network is calculated using two methods, namely, the box-covering method and the cluster-growing method, to exclude the possibility of differences in conclusions caused by the methods of calculating the fractal dimension for the subsequent relationship analysis. Finally, four attack strategies are used to attack the network, and the network performance is quantitatively measured by three structural indicators. Results on model networks show that compared to non-fractal modular networks, fractal scale-free networks are more robust to both static and dynamic targeted attacks on nodes and links, and the robustness of the network increases as the fractal dimension decreases. However, there is a cost in that as the fractal dimension decreases, the network becomes less efficient and more vulnerable to random failures on links. These findings contribute to a deeper understanding of the impact of fractal property on scale-free network performance and may be useful for designing resilient infrastructures.

深入分析分形无标度网络的攻击脆弱性对于设计鲁棒网络具有重要意义。以往的研究主要集中在静态节点攻击下分形特性对无标度网络攻击脆弱性的影响，而我们将研究扩展到各类针对性攻击的案例，探讨分形尺度攻击脆弱性之间的关系-自由网络和分形维数。首先提出了分层乘性增长模型来生成除分形维数之外具有相同结构特性的无标度网络。此外，采用盒覆盖法和簇生长法两种方法计算网络的分形维数，以排除分形维数计算方法造成结论差异的可能性，以便后续的关系分析。 。最后采用四种攻击策略对网络进行攻击，并通过三个结构指标定量衡量网络性能。模型网络的结果表明，与非分形模块化网络相比，分形无标度网络对于节点和链路的静态和动态针对性攻击都更加鲁棒，并且网络的鲁棒性随着分形维数的减小而增强。然而，随着分形维数的减小，网络的效率会降低，并且更容易受到链路上随机故障的影响，这是有代价的。这些发现有助于更深入地了解分形属性对无标度网络性能的影响，并且可能有助于设计弹性基础设施。