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=== 基于最短路的方法 ===

=== 基于最短路的方法 ===

基于最短路长度定义节点间距，借助分形几何的盒覆盖思想[17,18]实现迭代重整化群变换。该方法以最短路距离为度量，将指定距离内的节点聚合成粗粒化节点，生成重整化群序列[14]，可揭示网络的自相似性与分形特征[14,19–21]，并用于新生长机制[22]和网络普适类分类[5,23]。但受小世界特性限制，该方法难以有效解耦和揭示标度行为，度分布自相似性不伴随关联函数标度律，尤其在网络集聚方面。

基于最短路长度定义节点间距，借助分形几何的盒覆盖思想<ref name="Mandelbrot_Fractal_Geometry">{{cite journal|author1=Mandelbrot B. B.|title=The Fractal Geometry of Nature Vol. 2|journal=Freeman|year=1982|url=}}</ref><ref name="Samsel_fractal_communities">{{cite journal|author1=Samsel M|author2=Makulski K|author3=Łepek M|author4=Fronczak A|author5=Fronczak P.|title=Towards fractal origins of the community structure in complex networks: a model-based approach|journal=Preprint at arXiv|year=2023|url=}}</ref> 

实现迭代重整化群变换。该方法以最短路距离为度量，将指定距离内的节点聚合成粗粒化节点，生成重整化群序列<ref name="Song_Self_similarity">{{cite journal|author1=Song C|author2=Havlin S|author3=Makse H. A.|title=Self-similarity of complex networks|journal=Nature|year=2005|volume=433|url=}}</ref>，可揭示网络的自相似性与分形特征<ref name="Song_Self_similarity">{{cite journal|author1=Song C|author2=Havlin S|author3=Makse H. A.|title=Self-similarity of complex networks|journal=Nature|year=2005|volume=433|url=}}</ref><ref name="Goh_Skeleton_fractal">{{cite journal|author1=Goh K.-I|author2=Salvi G|author3=Kahng B|author4=Kim D.|title=Skeleton and fractal scaling in complex networks|journal=Phys. Rev. Lett.|year=2006|volume=96|url=}}</ref><ref name="Kim_Fractality_self_similarity">{{cite journal|author1=Kim J. S|author2=Goh K.-I|author3=Kahng B|author4=Kim D.|title=Fractality and self-similarity in scale-free networks|journal=New J. Phys.|year=2007|volume=9|url=}}</ref><ref name="Fronczak_Scaling_theory">{{cite journal|author1=Fronczak A.|title=Scaling theory of fractal complex networks|journal=Sci. Rep.|year=2024|url=}}</ref>

，并用于新生长机制[22]和网络普适类分类<ref name="Radicchi_Complex_networks_renormalization">{{cite journal|author1=Radicchi F|author2=Ramasco J|author3=Barrat A|author4=Fortunato S.|title=Complex networks renormalization: flows and fixed points|journal=Phys. Rev. Lett.|year=2008|volume=101|url=}}</ref><ref name="Rozenfeld_Small_world_fractal">{{cite journal|author1=Rozenfeld H. D.|author2=Song C|author3=Makse H. A.|title=Small-world to fractal transition in complex networks: a renormalization group approach|journal=Phys. Rev. Lett.|year=2010|volume=104|url=}}</ref>。但受小世界特性限制，该方法难以有效解耦和揭示标度行为，度分布自相似性不伴随关联函数标度律，尤其在网络集聚方面。

研究者还提出在规则格点网络添加长程连接实现实空间重整化群变换，用于研究Watts-Strogatz模型临界点行为[24]和格点无标度网络模型[25]，但需预知节点坐标或区分短程与长程连接，而真实网络中此类信息常未知。

研究者还提出在规则格点网络添加长程连接实现实空间重整化群变换，用于研究Watts-Strogatz模型临界点行为<ref name="Newman_Watts_RG_small_world">{{cite journal|author1=Newman M|author2=Watts D.|title=Renormalization group analysis of the small-world network model|journal=Phys. Lett. A|year=1999|volume=263|url=}}</ref>和格点无标度网络模型<ref name="Kim_Geographical_coarse_graining">{{cite journal|author1=Kim B. J.|title=Geographical coarse graining of complex networks|journal=Phys. Rev. Lett.|year=2004|volume=93|url=}}</ref>，但需预知节点坐标或区分短程与长程连接，而真实网络中此类信息常未知。

=== 谱方法 ===

=== 谱方法 ===