更改

值得注意的是，方程(11)对应的配分函数 <math>Z(\delta)</math>是可以精确计算的，并可以证明其在任意计算层级取值相同<ref name="Garuccio_multiscale_2023">{{cite journal|author1=Garuccio, E.|author2=Lalli, M.|author3=Garlaschelli, D.|title=Multiscale network renormalization: scale-invariance without geometry|journal=Phys. Rev. Res.|year=2023|volume=5|url=}}</ref>，这证实了多尺度模型位于重整化流的不动点上（见附录）。与之前讨论的几何或拉普拉斯方法不同，多尺度模型既不需要带节点坐标的度量空间嵌入，也不需要扩散邻近性概念来进行重整化。实际上，该模型允许任意划分，其聚合不变性相对于邻近驱动的尺度不变性也更加显著。

值得注意的是，方程(11)对应的配分函数 <math>Z(\delta)</math>是可以精确计算的，并可以证明其在任意计算层级取值相同<ref name="Garuccio_multiscale_2023">{{cite journal|author1=Garuccio, E.|author2=Lalli, M.|author3=Garlaschelli, D.|title=Multiscale network renormalization: scale-invariance without geometry|journal=Phys. Rev. Res.|year=2023|volume=5|url=}}</ref>，这证实了多尺度模型位于重整化流的不动点上（见附录）。与之前讨论的几何或拉普拉斯方法不同，多尺度模型既不需要带节点坐标的度量空间嵌入，也不需要扩散邻近性概念来进行重整化。实际上，该模型允许任意划分，其聚合不变性相对于邻近驱动的尺度不变性也更加显著。

多尺度模型可以推广至加权<ref name="Verteletskyi_renormalization_2022">{{cite thesis|author1=Verteletskyi, V.|title=Renormalization of Networks with Weighted Links|publisher=MSc thesis, Leiden Univ.|year=2022|url=}}</ref>和定向<ref name="Lalli_geometry-free_2024">{{cite journal|author1=Lalli, M.|author2=Garlaschelli, D.|title=Geometry-free renormalization of directed networks: scale-invariance and reciprocity|journal=Preprint at arXiv|year=2024|url=}}</ref>网络。定向网络的推广是直接的：由于模型不依赖几何距离（但是必须要求对称，尽管定向网络具有通常不对称性），可通过为每个节点添加另一个适应度变量处理定向网络。为每个节点增加第三个适应度参数后，模型还能描述非平凡的互利度（reciprocity）<ref name="Domenico_complex_2016">{{cite journal|author1=Domenico, M.|author2=Biamonte, J.|title=Spectral entropies as information-theoretic tools for complex network comparison|journal=Phys. Rev. X|year=2016|volume=6|url=}}</ref>，从而能建模真实世界的有向网络[100-109]。更一般地，适应度变量 <math>x_{i,}</math>可以是任意维向量，上述表达式中的乘积 <math>x_{i,} x_{i,}</math>可解释为内积<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>。

多尺度模型可以推广至加权<ref name="Verteletskyi_renormalization_2022">{{cite thesis|author1=Verteletskyi, V.|title=Renormalization of Networks with Weighted Links|publisher=MSc thesis, Leiden Univ.|year=2022|url=}}</ref>和定向<ref name="Lalli_geometry-free_2024">{{cite journal|author1=Lalli, M.|author2=Garlaschelli, D.|title=Geometry-free renormalization of directed networks: scale-invariance and reciprocity|journal=Preprint at arXiv|year=2024|url=}}</ref>网络。定向网络的推广是直接的：由于模型不依赖几何距离（但是必须要求对称，尽管定向网络具有通常不对称性），可通过为每个节点添加另一个适应度变量处理定向网络。为每个节点增加第三个适应度参数后，模型还能描述非平凡的互利度（reciprocity）<ref name="Domenico_complex_2016">{{cite journal|author1=Domenico, M.|author2=Biamonte, J.|title=Spectral entropies as information-theoretic tools for complex network comparison|journal=Phys. Rev. X|year=2016|volume=6|url=}}</ref>，从而能建模真实世界的有向网络<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><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><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><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><ref name="Barrat_Architecture_weighted_networks">{{cite journal|author1=Barrat A|author2=Barthelemy M|author3=Pastor-Satorras R|author4=Vespignani A.|title=The architecture of complex weighted networks|journal=Proc. Natl Acad. Sci. USA|year=2004|volume=101|url=}}</ref><ref name="Serrano_Correlations_weighted_networks">{{cite journal|author1=Serrano M. Á.|author2=Boguñá M|author3=Pastor-Satorras R.|title=Correlations in weighted networks|journal=Phys. Rev. E|year=2006|volume=74|url=}}</ref><ref name="Nolan_Stable_Distributions">{{cite journal|author1=Nolan J. P.|title=Univariate Stable Distributions: Models for Heavy Tailed Data|journal=Springer Nature|year=2019|url=}}</ref><ref name="Caldarelli_heterogeneous_2024">{{cite journal|author1=Caldarelli, G.|author2=Gabrielli, A.|author3=Gili, T.|author4=Villegas, P.|title=Laplacian renormalization group: an introduction to heterogeneous coarse-graining|journal=J. Stat. Mech.|year=2024|url=}}</ref><ref name="Domenico_complex_2016">{{cite journal|author1=Domenico, M.|author2=Biamonte, J.|title=Spectral entropies as information-theoretic tools for complex network comparison|journal=Phys. Rev. X|year=2016|volume=6|url=}}</ref>。更一般地，适应度变量 <math>x_{i,}</math>可以是任意维向量，上述表达式中的乘积 <math>x_{i,} x_{i,}</math>可解释为内积<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>。

=== 真实世界网络重整化的淬火变体 ===

=== 真实世界网络重整化的淬火变体 ===