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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>。
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|1,280 × 693px|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]]
<br>[[文件:D26cc9a6-ef2d-4022-8c1b-8080bbc9e67b.png|1,280 × 693px|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]]<br>