An algorithm for determining the fractal dimension of complex networks using a fixed number of boxes of flexible diameter

We present a novel box-covering algorithm for analyzing the fractal properties of complex networks. Unlike traditional algorithms that impose a predefined box size, our approach assigns nodes to boxes identified by the nearest local hubs without rigid distance constraints. This flexibility directly relates to the recently proposed scaling theory of fractal complex networks and is clearly consistent with the idea of hidden metric spaces in which network nodes are embedded. It also allows us to determine the box dimension of various real and model-based complex networks more accurately, including those previously unrecognized as fractal, such as the Internet at the level of autonomous systems. We show that our algorithm not only significantly reduces computational complexity compared to the classical greedy coloring method but also enables more precise determination of various scaling exponents describing the structure of fractal networks.本文提出一种用于分析复杂网络分形特性的新型盒覆盖算法（box-covering algorithm）。不同于传统算法预先设定固定盒尺寸的思路，本方法将节点分配至由最近局部枢纽（local hubs）标识的盒中，并未施加严格的距离约束。该特性与近期提出的分形复杂网络标度理论直接相关，且与网络节点嵌入其中的隐式度量空间（hidden metric spaces）理念高度一致。本算法可更为精准地测算各类真实网络与基于模型构建的复杂网络的盒维数（box dimension），其中涵盖此前未被识别为分形网络的对象，例如自治系统（autonomous systems）层级的互联网（Internet）。研究证实，相较于经典贪婪着色算法（greedy coloring method），本算法不仅大幅降低了计算复杂度，还能够更为精准地确定各类用于描述分形网络结构的标度指数（scaling exponents）。