Yuan Zhenhua, Peng Junhao, Gao Long, Shao Renxiang
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China.
Guangdong Provincial Key Laboratory, Co-sponsored by the Province and City of Information Security Technology, Guangzhou University, Guangzhou 510006, China.
Chaos. 2024 Mar 1;34(3). doi: 10.1063/5.0196934.
A class of self-similar networks, obtained by recursively replacing each edge of the current network with a well-designed structure (generator) and known as edge-iteration networks, has garnered considerable attention owing to its role in presenting rich network models to mimic real objects with self-similar structures. The generator dominates the structural and dynamic properties of edge-iteration networks. However, the general relationships between these networks' structural and dynamic properties and their generators remain unclear. We study the fractal and first-passage properties, such as the fractal dimension, walk dimension, resistance exponent, spectral dimension, and global mean first-passage time, which is the mean time for a walker, starting from a randomly selected node and reaching the fixed target node for the first time. We disclose the properties of the generators that dominate the fractal and first-passage properties of general edge-iteration networks. A clear relationship between the fractal and first-passage properties of the edge-iteration networks and the related properties of the generators are presented. The upper and lower bounds of these quantities are also discussed. Thus, networks can be customized to meet the requirements of fractal and dynamic properties by selecting an appropriate generator and tuning their structural parameters. The results obtained here shed light on the design and optimization of network structures.
一类自相似网络，通过用精心设计的结构（生成器）递归替换当前网络的每条边而获得，被称为边迭代网络，因其在呈现丰富的网络模型以模拟具有自相似结构的真实对象方面的作用而备受关注。生成器主导着边迭代网络的结构和动态特性。然而，这些网络的结构和动态特性与其生成器之间的一般关系仍不明确。我们研究分形和首次通过特性，如分形维数、游走维数、电阻指数、谱维数以及全局平均首次通过时间，全局平均首次通过时间是指一个漫步者从随机选择的节点出发首次到达固定目标节点的平均时间。我们揭示了主导一般边迭代网络分形和首次通过特性的生成器的性质。给出了边迭代网络的分形和首次通过特性与生成器相关性质之间的明确关系。还讨论了这些量的上下界。因此，通过选择合适的生成器并调整其结构参数，可以定制网络以满足分形和动态特性的要求。这里获得的结果为网络结构的设计和优化提供了启示。
