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<math>\mathbb{S}^D</math>模型具有纯几何形式—— <math>\mathbb{H}^{D+1}</math>模型<ref name="Krioukov_Hyperbolic_geometry">{{cite journal|author1=Krioukov D|author2=Papadopoulos F|author3=Kitsak M|author4=Vahdat A|author5=Boguñá M.|title=Hyperbolic geometry of complex networks|journal=Phys. Rev. E|year=2010|volume=82|url=}}</ref>，其中隐藏度 <math>\kappa</math>对应于径向坐标 <math>r</math>，从而将连接概率（方程(4)）转换为仅依赖于双曲空间中节点间距的费米-狄拉克函数。Mercator工具<ref name="García-Pérez_Mercator">{{cite journal|author1=García-Pérez G|author2=Allard A|author3=Serrano M. Á.|author4=Boguñá M.|title=Mercator: uncovering faithful hyperbolic embeddings of complex networks|journal=New J. Phys.|year=2019|volume=21|url=}}</ref><ref name="Jankowski_D_Mercator">{{cite journal|author1=Jankowski R|author2=Allard A|author3=Boguñá M|author4=Serrano M. Á.|title=The D-Mercator method for the multidimensional hyperbolic embedding of real networks|journal=Nat. Commun.|year=2023|volume=14|url=}}</ref>通过统计推断与机器学习技术，最大化观测网络结构与 <math>\mathbb{S}^D</math>（等价于 <math>\mathbb{H}^{D+1}</math>）模型<ref name="Boguñá_Sustaining_internet">{{cite journal|author1=Boguñá M|author2=Papadopoulos F|author3=Krioukov D.|title=Sustaining the internet with hyperbolic mapping|journal=Nat. Commun.|year=2010|volume=1|url=}}</ref>的契合度，将真实网络嵌入此空间。生成的空间映射可进一步用于可视化、分析、导航及重整化等下游任务。
<math>\mathbb{S}^D</math>模型具有纯几何形式—— <math>\mathbb{H}^{D+1}</math>模型<ref name="Krioukov_Hyperbolic_geometry">{{cite journal|author1=Krioukov D|author2=Papadopoulos F|author3=Kitsak M|author4=Vahdat A|author5=Boguñá M.|title=Hyperbolic geometry of complex networks|journal=Phys. Rev. E|year=2010|volume=82|url=}}</ref>，其中隐藏度 <math>\kappa</math>对应于径向坐标 <math>r</math>，从而将连接概率（方程(4)）转换为仅依赖于双曲空间中节点间距的费米-狄拉克函数。Mercator工具<ref name="García-Pérez_Mercator">{{cite journal|author1=García-Pérez G|author2=Allard A|author3=Serrano M. Á.|author4=Boguñá M.|title=Mercator: uncovering faithful hyperbolic embeddings of complex networks|journal=New J. Phys.|year=2019|volume=21|url=}}</ref><ref name="Jankowski_D_Mercator">{{cite journal|author1=Jankowski R|author2=Allard A|author3=Boguñá M|author4=Serrano M. Á.|title=The D-Mercator method for the multidimensional hyperbolic embedding of real networks|journal=Nat. Commun.|year=2023|volume=14|url=}}</ref>通过统计推断与机器学习技术，最大化观测网络结构与 <math>\mathbb{S}^D</math>（等价于 <math>\mathbb{H}^{D+1}</math>）模型<ref name="Boguñá_Sustaining_internet">{{cite journal|author1=Boguñá M|author2=Papadopoulos F|author3=Krioukov D.|title=Sustaining the internet with hyperbolic mapping|journal=Nat. Commun.|year=2010|volume=1|url=}}</ref>的契合度，将真实网络嵌入此空间。生成的空间映射可进一步用于可视化、分析、导航及重整化等下游任务。
该模型族能有效解释真实网络连接性的关键特征，包括异质度分布<ref name="Serrano_Self_similarity_hidden_metric">{{cite journal|author1=Serrano M. Á.|author2=Krioukov D|author3=Boguñá M.|title=Self-similarity of complex networks and hidden metric spaces|journal=Phys. Rev. Lett.|year=2008|volume=100|url=}}</ref><ref name="Krioukov_Hyperbolic_geometry">{{cite journal|author1=Krioukov D|author2=Papadopoulos F|author3=Kitsak M|author4=Vahdat A|author5=Boguñá M.|title=Hyperbolic geometry of complex networks|journal=Phys. Rev. E|year=2010|volume=82|url=}}</ref><ref name="Gugelmann_Random_hyperbolic_graphs">{{cite journal|author1=Gugelmann L|author2=Panagiotou K|author3=Peter U.|title=Random hyperbolic graphs: degree sequence and clustering|journal=Proc. Automata, Languages and Programming|year=2012|pages=573–585|url=}}</ref>、显著集聚性<ref name="Serrano_Self_similarity_hidden_metric">{{cite journal|author1=Serrano M. Á.|author2=Krioukov D|author3=Boguñá M.|title=Self-similarity of complex networks and hidden metric spaces|journal=Phys. Rev. Lett.|year=2008|volume=100|url=}}</ref><ref name="Krioukov_Hyperbolic_geometry">{{cite journal|author1=Krioukov D|author2=Papadopoulos F|author3=Kitsak M|author4=Vahdat A|author5=Boguñá M.|title=Hyperbolic geometry of complex networks|journal=Phys. Rev. E|year=2010|volume=82|url=}}</ref><ref name="Gugelmann_Random_hyperbolic_graphs">{{cite journal|author1=Gugelmann L|author2=Panagiotou K|author3=Peter U.|title=Random hyperbolic graphs: degree sequence and clustering|journal=Proc. Automata, Languages and Programming|year=2012|pages=573–585|url=}}</ref><ref name="Candellero_Clustering_hyperbolic">{{cite journal|author1=Candellero E|author2=Fountoulakis N.|title=Clustering and the hyperbolic geometry of complex networks|journal=Internet Math.|year=2016|volume=12|url=}}</ref><ref name="Fountoulakis_Clustering_hyperbolic_model">{{cite journal|author1=Fountoulakis N|author2=Hoorn P|author3=Müller T|author4=Schepers M.|title=Clustering in a hyperbolic model of complex networks|journal=Electron. J. Probab.|year=2021|volume=26|url=}}</ref>、小世界现象<ref name="Serrano_Self_similarity_hidden_metric">{{cite journal|author1=Serrano M. Á.|author2=Krioukov D|author3=Boguñá M.|title=Self-similarity of complex networks and hidden metric spaces|journal=Phys. Rev. Lett.|year=2008|volume=100|url=}}</ref><ref name="Abdullah_Typical_distances">{{cite journal|author1=Abdullah M. A|author2=Fountoulakis N|author3=Bode M.|title=Typical distances in a geometric model for complex networks|journal=Internet Math.|year=2017|url=}}</ref><ref name="Friedrich_Diameter_hyperbolic">{{cite journal|author1=Friedrich T|author2=Krohmer A.|title=On the diameter of hyperbolic random graphs|journal=SIAM J. Discrete Math.|year=2018|volume=32|url=}}</ref><ref name="Müller_Diameter_KPKVB">{{cite journal|author1=Müller T|author2=Staps M.|title=The diameter of KPKVB random graphs|journal=Adv. Appl. Probab.|year=2019|volume=51|url=}}</ref>、渗流性质<ref name="Serrano_Percolation_self_similar">{{cite journal|author1=Serrano M. Á.|author2=Krioukov D|author3=Boguñá M.|title=Percolation in self-similar networks|journal=Phys. Rev. Lett.|year=2011|volume=106|url=}}</ref><ref name="Fountoulakis_Law_large_numbers">{{cite journal|author1=Fountoulakis N|author2=Müller T.|title=Law of large numbers for the largest component in a hyperbolic model of complex networks|journal=Ann. Appl. Probab.|year=2018|volume=28|url=}}</ref><ref name="Bianconi_Topological_percolation">{{cite journal|author1=Bianconi G|author2=Ziff R.|title=Topological percolation on hyperbolic simplicial complexes|journal=Phys. Rev. E|year=2018|volume=98|url=}}</ref>、谱特征<ref name="Kiwi_Spectral_gap">{{cite journal|author1=Kiwi M|author2=Mitsche D.|title=Spectral gap of random hyperbolic graphs and related parameters|journal=Ann. Appl. Probab.|year=2018|volume=28|url=}}</ref>及自相似性<ref name="Serrano_Self_similarity_hidden_metric">{{cite journal|author1=Serrano M. Á.|author2=Krioukov D|author3=Boguñá M.|title=Self-similarity of complex networks and hidden metric spaces|journal=Phys. Rev. Lett.|year=2008|volume=100|url=}}</ref>。该框架还可以进一步扩展至生长网络的优先连接机制<ref name="Papadopoulos_Popularity_similarity">{{cite journal|author1=Papadopoulos F|author2=Kitsak M|author3=Serrano M. Á.|author4=Boguñá M|author5=Krioukov D.|title=Popularity versus similarity in growing networks|journal=Nature|year=2012|volume=489|url=}}</ref>、加权网络<ref name="Allard_Geometric_nature_weights">{{cite journal|author1=Allard A|author2=Serrano M. Á.|author3=García-Pérez G|author4=Boguñá M.|title=The geometric nature of weights in real complex networks|journal=Nat. Commun.|year=2017|volume=8|url=}}</ref>、二部网络<ref name="Serrano_Metabolic_networks">{{cite journal|author1=Serrano M. Á.|author2=Boguñá M|author3=Sagués F.|title=Uncovering the hidden geometry behind metabolic networks|journal=Mol. Biosyst.|year=2012|volume=8|url=}}</ref><ref name="Kitsak_Latent_geometry_bipartite">{{cite journal|author1=Kitsak M|author2=Papadopoulos F|author3=Krioukov D.|title=Latent geometry of bipartite networks|journal=Phys. Rev. E|year=2017|volume=95|url=}}</ref>、互补性驱动网络<ref name="Budel_Complementarity_networks">{{cite journal|author1=Budel G|author2=Kitsak M.|title=Complementarity in complex networks|journal=Preprint at arXiv|year=2023|url=}}</ref>、多层网络<ref name="Kleineberg_Hidden_geometric_correlations">{{cite journal|author1=Kleineberg K.-K|author2=Boguñá M|author3=Serrano M. Á.|author4=Papadopoulos F.|title=Hidden geometric correlations in real multiplex networks|journal=Nat. Phys.|year=2016|volume=12|url=}}</ref><ref name="Kleineberg_Geometric_correlations_multiplex">{{cite journal|author1=Kleineberg K.-K|author2=Buzna L|author3=Papadopoulos F|author4=Boguñá M|author5=Serrano M. Á.|title=Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks|journal=Phys. Rev. Lett.|year=2017|volume=118|url=}}</ref>、具有社区结构的网络<ref name="Zuev_Soft_communities">{{cite journal|author1=Zuev K|author2=Boguñá M|author3=Bianconi G|author4=Krioukov D.|title=Emergence of soft communities from geometric preferential attachment|journal=Sci. Rep.|year=2015|volume=5|url=}}</ref><ref name="García-Pérez_Soft_communities_similarity">{{cite journal|author1=García-Pérez G|author2=Serrano M. Á.|author3=Boguñá M.|title=Soft communities in similarity space|journal=J. Stat. Phys.|year=2018|volume=173|url=}}</ref><ref name="Muscoloni_nPSO_model">{{cite journal|author1=Muscoloni A|author2=Cannistraci C. V.|title=A nonuniform popularity-similarity optimization (nPSO) model to efficiently generate realistic complex networks with communities|journal=New J. Phys.|year=2018|volume=20|url=}}</ref>、有向网络<ref name="Allard_Geometric_description_directed">{{cite journal|author1=Allard A|author2=Serrano M. Á.|author3=Boguñá M.|title=Geometric description of clustering in directed networks|journal=Nat. Phys.|year=2024|volume=20|url=}}</ref>以及节点含关联特征的网络<ref name="Aliakbarisani_Feature_enriched_hyperbolic">{{cite journal|author1=Aliakbarisani R|author2=Ángeles Serrano M|author3=Boguñá M.|title=Feature-enriched hyperbolic network geometry|journal=Preprint at arXiv|year=2023|url=}}</ref>。[[文件:D26cc9a6-ef2d-4022-8c1b-8080bbc9e67b.png|320×173像素|有框|左|图1 真实网络上几何重整化的直接与逆向方法及其结果]]
该模型族能有效解释真实网络连接性的关键特征，包括异质度分布<ref name="Serrano_Self_similarity_hidden_metric">{{cite journal|author1=Serrano M. Á.|author2=Krioukov D|author3=Boguñá M.|title=Self-similarity of complex networks and hidden metric spaces|journal=Phys. Rev. Lett.|year=2008|volume=100|url=}}</ref><ref name="Krioukov_Hyperbolic_geometry">{{cite journal|author1=Krioukov D|author2=Papadopoulos F|author3=Kitsak M|author4=Vahdat A|author5=Boguñá M.|title=Hyperbolic geometry of complex networks|journal=Phys. Rev. E|year=2010|volume=82|url=}}</ref><ref name="Gugelmann_Random_hyperbolic_graphs">{{cite journal|author1=Gugelmann L|author2=Panagiotou K|author3=Peter U.|title=Random hyperbolic graphs: degree sequence and clustering|journal=Proc. Automata, Languages and Programming|year=2012|pages=573–585|url=}}</ref>、显著集聚性<ref name="Serrano_Self_similarity_hidden_metric">{{cite journal|author1=Serrano M. Á.|author2=Krioukov D|author3=Boguñá M.|title=Self-similarity of complex networks and hidden metric spaces|journal=Phys. Rev. Lett.|year=2008|volume=100|url=}}</ref><ref name="Krioukov_Hyperbolic_geometry">{{cite journal|author1=Krioukov D|author2=Papadopoulos F|author3=Kitsak M|author4=Vahdat A|author5=Boguñá M.|title=Hyperbolic geometry of complex networks|journal=Phys. Rev. E|year=2010|volume=82|url=}}</ref><ref name="Gugelmann_Random_hyperbolic_graphs">{{cite journal|author1=Gugelmann L|author2=Panagiotou K|author3=Peter U.|title=Random hyperbolic graphs: degree sequence and clustering|journal=Proc. Automata, Languages and Programming|year=2012|pages=573–585|url=}}</ref><ref name="Candellero_Clustering_hyperbolic">{{cite journal|author1=Candellero E|author2=Fountoulakis N.|title=Clustering and the hyperbolic geometry of complex networks|journal=Internet Math.|year=2016|volume=12|url=}}</ref><ref name="Fountoulakis_Clustering_hyperbolic_model">{{cite journal|author1=Fountoulakis N|author2=Hoorn P|author3=Müller T|author4=Schepers M.|title=Clustering in a hyperbolic model of complex networks|journal=Electron. J. Probab.|year=2021|volume=26|url=}}</ref>、小世界现象<ref name="Serrano_Self_similarity_hidden_metric">{{cite journal|author1=Serrano M. Á.|author2=Krioukov D|author3=Boguñá M.|title=Self-similarity of complex networks and hidden metric spaces|journal=Phys. Rev. Lett.|year=2008|volume=100|url=}}</ref><ref name="Abdullah_Typical_distances">{{cite journal|author1=Abdullah M. A|author2=Fountoulakis N|author3=Bode M.|title=Typical distances in a geometric model for complex networks|journal=Internet Math.|year=2017|url=}}</ref><ref name="Friedrich_Diameter_hyperbolic">{{cite journal|author1=Friedrich T|author2=Krohmer A.|title=On the diameter of hyperbolic random graphs|journal=SIAM J. Discrete Math.|year=2018|volume=32|url=}}</ref><ref name="Müller_Diameter_KPKVB">{{cite journal|author1=Müller T|author2=Staps M.|title=The diameter of KPKVB random graphs|journal=Adv. Appl. Probab.|year=2019|volume=51|url=}}</ref>、渗流性质<ref name="Serrano_Percolation_self_similar">{{cite journal|author1=Serrano M. Á.|author2=Krioukov D|author3=Boguñá M.|title=Percolation in self-similar networks|journal=Phys. Rev. Lett.|year=2011|volume=106|url=}}</ref><ref name="Fountoulakis_Law_large_numbers">{{cite journal|author1=Fountoulakis N|author2=Müller T.|title=Law of large numbers for the largest component in a hyperbolic model of complex networks|journal=Ann. Appl. Probab.|year=2018|volume=28|url=}}</ref><ref name="Bianconi_Topological_percolation">{{cite journal|author1=Bianconi G|author2=Ziff R.|title=Topological percolation on hyperbolic simplicial complexes|journal=Phys. Rev. E|year=2018|volume=98|url=}}</ref>、谱特征<ref name="Kiwi_Spectral_gap">{{cite journal|author1=Kiwi M|author2=Mitsche D.|title=Spectral gap of random hyperbolic graphs and related parameters|journal=Ann. Appl. Probab.|year=2018|volume=28|url=}}</ref>及自相似性<ref name="Serrano_Self_similarity_hidden_metric">{{cite journal|author1=Serrano M. Á.|author2=Krioukov D|author3=Boguñá M.|title=Self-similarity of complex networks and hidden metric spaces|journal=Phys. Rev. Lett.|year=2008|volume=100|url=}}</ref>。该框架还可以进一步扩展至生长网络的优先连接机制<ref name="Papadopoulos_Popularity_similarity">{{cite journal|author1=Papadopoulos F|author2=Kitsak M|author3=Serrano M. Á.|author4=Boguñá M|author5=Krioukov D.|title=Popularity versus similarity in growing networks|journal=Nature|year=2012|volume=489|url=}}</ref>、加权网络<ref name="Allard_Geometric_nature_weights">{{cite journal|author1=Allard A|author2=Serrano M. Á.|author3=García-Pérez G|author4=Boguñá M.|title=The geometric nature of weights in real complex networks|journal=Nat. Commun.|year=2017|volume=8|url=}}</ref>、二部网络<ref name="Serrano_Metabolic_networks">{{cite journal|author1=Serrano M. Á.|author2=Boguñá M|author3=Sagués F.|title=Uncovering the hidden geometry behind metabolic networks|journal=Mol. Biosyst.|year=2012|volume=8|url=}}</ref><ref name="Kitsak_Latent_geometry_bipartite">{{cite journal|author1=Kitsak M|author2=Papadopoulos F|author3=Krioukov D.|title=Latent geometry of bipartite networks|journal=Phys. Rev. E|year=2017|volume=95|url=}}</ref>、互补性驱动网络<ref name="Budel_Complementarity_networks">{{cite journal|author1=Budel G|author2=Kitsak M.|title=Complementarity in complex networks|journal=Preprint at arXiv|year=2023|url=}}</ref>、多层网络<ref name="Kleineberg_Hidden_geometric_correlations">{{cite journal|author1=Kleineberg K.-K|author2=Boguñá M|author3=Serrano M. Á.|author4=Papadopoulos F.|title=Hidden geometric correlations in real multiplex networks|journal=Nat. Phys.|year=2016|volume=12|url=}}</ref><ref name="Kleineberg_Geometric_correlations_multiplex">{{cite journal|author1=Kleineberg K.-K|author2=Buzna L|author3=Papadopoulos F|author4=Boguñá M|author5=Serrano M. Á.|title=Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks|journal=Phys. Rev. Lett.|year=2017|volume=118|url=}}</ref>、具有社区结构的网络<ref name="Zuev_Soft_communities">{{cite journal|author1=Zuev K|author2=Boguñá M|author3=Bianconi G|author4=Krioukov D.|title=Emergence of soft communities from geometric preferential attachment|journal=Sci. Rep.|year=2015|volume=5|url=}}</ref><ref name="García-Pérez_Soft_communities_similarity">{{cite journal|author1=García-Pérez G|author2=Serrano M. Á.|author3=Boguñá M.|title=Soft communities in similarity space|journal=J. Stat. Phys.|year=2018|volume=173|url=}}</ref><ref name="Muscoloni_nPSO_model">{{cite journal|author1=Muscoloni A|author2=Cannistraci C. V.|title=A nonuniform popularity-similarity optimization (nPSO) model to efficiently generate realistic complex networks with communities|journal=New J. Phys.|year=2018|volume=20|url=}}</ref>、有向网络<ref name="Allard_Geometric_description_directed">{{cite journal|author1=Allard A|author2=Serrano M. Á.|author3=Boguñá M.|title=Geometric description of clustering in directed networks|journal=Nat. Phys.|year=2024|volume=20|url=}}</ref>以及节点含关联特征的网络<ref name="Aliakbarisani_Feature_enriched_hyperbolic">{{cite journal|author1=Aliakbarisani R|author2=Ángeles Serrano M|author3=Boguñá M.|title=Feature-enriched hyperbolic network geometry|journal=Preprint at arXiv|year=2023|url=}}</ref>。
[[文件:D26cc9a6-ef2d-4022-8c1b-8080bbc9e67b.png|320×173像素|无框|左|图1 真实网络上几何重整化的直接与逆向方法及其结果]]
图1 真实网络上几何重整化的直接与逆向方法及其结果。 a. 几何重整化（GR）变换示意图。在层级 <math>\ell = 0</math> 中，沿相似性圆定义 <math>r = 2</math> 个连续节点组成的非重叠蓝色块，从而将原 <math>N_0</math> 个节点划分为 <math>N_1 = N_0/r</math> 个块，形成分区 <math>\Omega_0</math>。这些块被粗粒化为层级 <math>\ell = 1</math> 中的超节点，每个超节点的角坐标由其 <math>\ell = 0</math> 层构成节点所在区域确定。如方程(2)所述连接 <math>\ell = 1</math> 层的超节点，该过程可迭代生成 <math>\ell = 2</math> 层。由于半群性质， <math>\ell = 2</math> 层亦可直接通过 <math>r = 4</math> 的块从 <math>\ell = 0</math> 层生成。b. GR 复现人脑连接组的多尺度连通结构。基于解剖数据重建五级长度尺度的多尺度层级表示（上图）。各尺度 <math>\ell</math> 下连接组的拓扑特性（符号）与基于最高分辨率几何映射的 GR 预测（曲线）高度吻合（下图）。从左至右依次为：互补累积度分布 <math>P_c(k_{\text{res}})</math>、聚类系数 <math>\bar{c}(k_{\text{res}})</math>、归一化最近邻平均度 <math>\bar k_{\text{nn}} (k_{\text{res}}) = \bar k_{\text{nn}}(k_\text{res}) \langle k \rangle/\langle k \rangle^2</math> 及富俱乐部系数 <math>r(k_{\text{res}})</math>。计算中采用重整化度 <math>k_{\text{res}} = k/\langle k \rangle</math> 以消除各层级平均度差异的影响。 c. 几何分支生长过程（GBG）以自相似方式演化网络。d. GBG 是 GR 的统计逆过程：对互联网重整化层级应用 GBG 可恢复原始网络的统计等效结构；反之，对 GBG 生成的人体代谢网络替代体应用 GR 可复原原始架构。e. GR 与 GBG 的结合为单实例真实网络的有限尺寸标度研究提供方法，可数值估算临界指数。以互联网链路随机失效的键逾渗模拟为例，网络尺寸 <math>N</math>、临界键占据概率 <math>p_c</math> 及磁化率最大值 <math>\xi_{\text{max}}</math> 均符合幂律关系（黑色符号为原始网络数据）。GBG 与 GR 壳层参数为分支率 <math>b = 2</math>、块尺寸 <math>r = 2</math>，结果表明互联网图存在逾渗阈值消失现象，符合无标度网络预期。图a改编自文献<ref name="Zheng_Geometric_renormalization_weighted">{{cite journal|author1=Zheng M|author2=García-Pérez G|author3=Boguñá M|author4=Serrano M. Á.|title=Geometric renormalization of weighted networks|journal=Commun. Phys.|year=2024|volume=7|url=}}</ref>（Springer Nature Limited）；图b转载自文献<ref name="Zheng_Geometric_renormalization_connectome">{{cite journal|author1=Zheng M|author2=Allard A|author3=Hagmann P|author4=Alemán-Gómez Y|author5=Serrano M. Á.|title=Geometric renormalization unravels self-similarity of the multiscale human connectome|journal=Proc. Natl Acad. Sci. USA|year=2020|volume=117|url=}}</ref>（PNAS）；图c、d、e改编自文献<ref name="Zheng_Geometric_branching_growth">{{cite journal|author1=Zheng M|author2=García-Pérez G|author3=Boguñá M|author4=Serrano M. Á.|title=Scaling up real networks by geometric branching growth|journal=Proc. Natl Acad. Sci. USA|year=2021|url=}}</ref>（PNAS）。
图1 真实网络上几何重整化的直接与逆向方法及其结果。 a. 几何重整化（GR）变换示意图。在层级 <math>\ell = 0</math> 中，沿相似性圆定义 <math>r = 2</math> 个连续节点组成的非重叠蓝色块，从而将原 <math>N_0</math> 个节点划分为 <math>N_1 = N_0/r</math> 个块，形成分区 <math>\Omega_0</math>。这些块被粗粒化为层级 <math>\ell = 1</math> 中的超节点，每个超节点的角坐标由其 <math>\ell = 0</math> 层构成节点所在区域确定。如方程(2)所述连接 <math>\ell = 1</math> 层的超节点，该过程可迭代生成 <math>\ell = 2</math> 层。由于半群性质， <math>\ell = 2</math> 层亦可直接通过 <math>r = 4</math> 的块从 <math>\ell = 0</math> 层生成。b. GR 复现人脑连接组的多尺度连通结构。基于解剖数据重建五级长度尺度的多尺度层级表示（上图）。各尺度 <math>\ell</math> 下连接组的拓扑特性（符号）与基于最高分辨率几何映射的 GR 预测（曲线）高度吻合（下图）。从左至右依次为：互补累积度分布 <math>P_c(k_{\text{res}})</math>、聚类系数 <math>\bar{c}(k_{\text{res}})</math>、归一化最近邻平均度 <math>\bar k_{\text{nn}} (k_{\text{res}}) = \bar k_{\text{nn}}(k_\text{res}) \langle k \rangle/\langle k \rangle^2</math> 及富俱乐部系数 <math>r(k_{\text{res}})</math>。计算中采用重整化度 <math>k_{\text{res}} = k/\langle k \rangle</math> 以消除各层级平均度差异的影响。 c. 几何分支生长过程（GBG）以自相似方式演化网络。d. GBG 是 GR 的统计逆过程：对互联网重整化层级应用 GBG 可恢复原始网络的统计等效结构；反之，对 GBG 生成的人体代谢网络替代体应用 GR 可复原原始架构。e. GR 与 GBG 的结合为单实例真实网络的有限尺寸标度研究提供方法，可数值估算临界指数。以互联网链路随机失效的键逾渗模拟为例，网络尺寸 <math>N</math>、临界键占据概率 <math>p_c</math> 及磁化率最大值 <math>\xi_{\text{max}}</math> 均符合幂律关系（黑色符号为原始网络数据）。GBG 与 GR 壳层参数为分支率 <math>b = 2</math>、块尺寸 <math>r = 2</math>，结果表明互联网图存在逾渗阈值消失现象，符合无标度网络预期。图a改编自文献<ref name="Zheng_Geometric_renormalization_weighted">{{cite journal|author1=Zheng M|author2=García-Pérez G|author3=Boguñá M|author4=Serrano M. Á.|title=Geometric renormalization of weighted networks|journal=Commun. Phys.|year=2024|volume=7|url=}}</ref>（Springer Nature Limited）；图b转载自文献<ref name="Zheng_Geometric_renormalization_connectome">{{cite journal|author1=Zheng M|author2=Allard A|author3=Hagmann P|author4=Alemán-Gómez Y|author5=Serrano M. Á.|title=Geometric renormalization unravels self-similarity of the multiscale human connectome|journal=Proc. Natl Acad. Sci. USA|year=2020|volume=117|url=}}</ref>（PNAS）；图c、d、e改编自文献<ref name="Zheng_Geometric_branching_growth">{{cite journal|author1=Zheng M|author2=García-Pérez G|author3=Boguñá M|author4=Serrano M. Á.|title=Scaling up real networks by geometric branching growth|journal=Proc. Natl Acad. Sci. USA|year=2021|url=}}</ref>（PNAS）。