复杂网络重整化 - 版本历史

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相信未来：/* 实空间中的拉普拉斯重整化群 */

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实空间中的拉普拉斯重整化群

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 重整化群（renormalization group, RG）理论是研究[[多自由度系统]]的强有力框架，通过不同尺度下系统状态、模型参数和[[耦合常数]]的描述，实现分辨率层次转换，可识别[[相变临界点]]并分析临界行为。传统物理应用中，重整化群基于同质性、对称性、几何结构和局部性构建尺度变换方案。近年来，该理论向[[复杂网络]]拓展，但网络系统存在显式几何坐标缺失、节点与子图统计异质性及类格点对称性缺乏三大特征，导致统一重整化流程构建面临挑战。本文梳理网络重整化主流方法，评述理论突破，探讨关键科学问题。
-== 背景 ==
+== 历史 ==
 重整化和可重整化是现代物理学理解宇宙中物质现象的基础概念，该理论假定在足够大的时空尺度上，小尺度细节是可以被平均掉的，从而解释了为什么我们在日常生活中无需了解亚原子粒子水平的精确行为。重整化方法的形式化花费了物理学家数十年时间，并最终促成了对某些物理现象普遍性的惊人洞察。

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 重整化群理论通过在实空间可用的方法对微观细节进行形式化平均，从而定义了任意尺度下的有效自由度与相互作用，成为统计力学的重大突破。Kenneth Wilson后续将其发展为连续场论中的重整化方法<ref name="Wilson_renormalization_1974">{{cite journal|author1=Wilson, K. G.|author2=Kogut, J.|title=The renormalization group and the $\epsilon$ expansion|journal=Phys. Rep.|year=1974|volume=12|url=}}</ref><ref name="Castro_scaling">{{cite journal|author1=Castro C|author2=Jona-Lasinio G.|title=On the microscopic foundation of scaling laws|journal=Phys. Lett. A|year=1969|volume=29}}</ref>。Wilson的工作不仅为理解临界现象的标度行为奠定理论基础，更将场论中处理名义发散量的零散方法系统化。这些发散性源于数学上为解决任意小尺度问题的理想化操作，虽然研究者们对此曾提出减除方案<ref name="Gell-Mann_electrodynamics">{{cite journal|author1=Gell-Mann M|author2=Low F. E.|title=Quantum electrodynamics at small distances|journal=Phys. Rev.|year=1954|volume=95}}</ref>，但该方案在重整化群理论建立前缺乏概念基础。此后，重整化群方法催生了“理论之理论”，揭示微观细节迥异的物理系统在临界点附近如何出现普适行为<ref name="Zinn-Justin_field">{{cite journal|author1=Zinn-Justin J.|title=Quantum Field Theory and Critical Phenomena|journal=Oxford Univ. Press|year=2021}}</ref><ref name="Amit_field">{{cite journal|author1=Amit D. J|author2=Martin-Mayor V.|title=Field Theory, the Renormalization Group and Critical Phenomena|journal=World Scientific|year=2005}}</ref><ref name="Täuber_renormalization">{{cite journal|author1=Täuber U. C.|title=Renormalization group: applications in statistical physics|journal=Nucl. Phys. B|year=2012|volume=228}}</ref><ref name="Pelissetto_critical">{{cite journal|author1=Pelissetto A|author2=Vicari E.|title=Critical phenomena and renormalization-group theory|journal=Phys. Rep.|year=2002|volume=368}}</ref><ref name="Efrati_real-space">{{cite journal|author1=Efrati E|author2=Wang Z|author3=Kolan A|author4=Kadanoff L. P.|title=Real-space renormalization in statistical mechanics|journal=Rev. Mod. Phys.|year=2014|volume=86}}</ref>。
-过去五十年间，重整化群技术已扩展至从最大尺度的宇宙学到最小尺度的粒子物理等广泛领域。在统计力学范畴内，重整化群方法已推广至同质均匀的分形格点上的非平衡系统<ref name="Gefen_fractal">{{cite journal|author1=Gefen Y|author2=Mandelbrot B. B|author3=Aharony A.|title=Critical phenomena on fractal lattices|journal=Phys. Rev. Lett.|year=1980|volume=45}}</ref><ref name="Gefen_backbone">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B|author4=Kirkpatrick S.|title=Solvable fractal family, and its possible relation to the backbone at percolation|journal=Phys. Rev. Lett.|year=1981|volume=47}}</ref><ref name="Phani_percolation">{{cite journal|author1=Phani M. K|author2=Dhar D.|title=Real-space renormalisation group: application to directed percolation|journal=J. Phys. C|year=1982|volume=15}}</ref><ref name="Gefen_phase1">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. I. Quasi-linear lattices|journal=J. Phys. A|year=1983|volume=16}}</ref><ref name="Gefen_phase2">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Shapir Y|author4=Mandelbrot B. B.|title=Phase transitions on fractals. II. Sierpinski gaskets|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Gefen_phase3">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. III. Infinitely ramified lattices|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Das_critical">{{cite journal|author1=Das D|author2=Dey S|author3=Jacobsen J. L|author4=Dhar D.|title=Critical behavior of loops and biconnected clusters on fractals of dimension d \< 2|journal=J. Phys. A|year=2008|volume=41}}</ref>，其局部平移不变性能够简化计算。超越初始应用的重要范例包括：流行病动力学模型<ref name="Della_SIR">{{cite journal|author1=Della Morte M|author2=Sannino F.|title=Renormalization group approach to pandemics as a time-dependent SIR model|journal=Front. Phys.|year=2021}}</ref>、生物群体行为动力学<ref name="Cavagna_scaling">{{cite journal|author1=Cavagna A.|title=Dynamic scaling in natural swarms|journal=Nat. Phys.|year=2017|volume=13}}</ref><ref name="Cavagna_swarms">{{cite journal|author1=Cavagna A.|title=Dynamical renormalization group approach to the collective behavior of swarms|journal=Phys. Rev. Lett.|year=2019}}</ref><ref name="Cavagna_dimensions">{{cite journal|author1=Cavagna A.|title=Natural swarms in 3.99 dimensions|journal=Nat. Phys.|year=2023|volume=19}}</ref>、神经网络<ref name="Koch_information">{{cite journal|author1=Koch-Janusz M|author2=Ringel Z.|title=Mutual information, neural networks and the renormalization group|journal=Nat. Phys.|year=2018|volume=14}}</ref><ref name="Li_neural">{{cite journal|author1=Li S.-H|author2=Wang L.|title=Neural network renormalization group|journal=Phys. Rev. Lett.|year=2018}}</ref><ref name="Hu_machine">{{cite journal|author1=Hu H.-Y|author2=Li S.-H|author3=Wang L|author4=You Y.-Z.|title=Machine learning holographic mapping by neural network renormalization group|journal=Phys. Rev. Res.|year=2020|volume=2}}</ref><ref name="Caso_graph">{{cite journal|author1=Caso F|author2=Trappolini G|author3=Bacciu A|author4=Liò P|author5=Silvestri F.|title=Renormalized graph neural networks|journal=Preprint at arXiv|year=2023}}</ref>、动态与有向逾渗<ref name="Young_percolation">{{cite journal|author1=Young A. P|author2=Stinchcombe R. B.|title=A renormalization group theory for percolation problems|journal=J. Phys. C|year=1975|volume=8}}</ref><ref name="Reynolds_percolation">{{cite journal|author1=Reynolds P. J|author2=Stanley H. E|author3=Klein W.|title=A real-space renormalization group for site and bond percolation|journal=J. Phys. C|year=1977|volume=10}}</ref>、观点动力学的选民模型<ref name="Galam_voting">{{cite journal|author1=Galam S.|title=Real space renormalization group and totalitarian paradox of majority rule voting|journal=Physica A|year=2000|volume=285}}</ref>，以及一维链<ref name="Kogan_synchronization">{{cite journal|author1=Kogan O|author2=Rogers J. L|author3=Cross M. C|author4=Refael G.|title=Renormalization group approach to oscillator synchronization|journal=Phys. Rev. E|year=2009|volume=80}}</ref>、格点<ref name="Östborn_disorder">{{cite journal|author1=Östborn P.|title=Renormalization of oscillator lattices with disorder|journal=Phys. Rev. E|year=2009|volume=79}}</ref>和层级树<ref name="Garlaschelli_synchronization">{{cite journal|author1=Garlaschelli D|author2=Hollander F|author3=Meylahn J|author4=Zeegers B.|title=Synchronization of phase oscillators on the hierarchical lattice|journal=J. Stat. Phys.|year=2019|volume=174}}</ref>上非线性振子模型（如Kuramoto模型<ref name="Hinrichsen_non-equilibrium_2000">{{cite journal|author1=Hinrichsen, H.|title=Non-equilibrium critical phenomena and phase transitions into absorbing states|journal=Adv. Phys.|year=2000|volume=49|url=}}</ref><ref name="Dornic_Langevin_2005">{{cite journal|author1=Dornic, I.|author2=Chaté, H.|author3=Muñoz, M. A.|title=Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions|journal=Phys. Rev. Lett.|year=2005|volume=94|url=}}</ref>）的同步模型。此外，重整化群方法在张量网络<ref name="Li_neural">{{cite journal|author1=Li S.-H|author2=Wang L.|title=Neural network renormalization group|journal=Phys. Rev. Lett.|year=2018}}</ref><ref name="Levin_tensor">{{cite journal|author1=Levin M|author2=Nave C. P.|title=Tensor renormalization group approach to two-dimensional classical lattice models|journal=Phys. Rev. Lett.|year=2007}}</ref><ref name="Evenbly_tensor">{{cite journal|author1=Evenbly G|author2=Vidal G.|title=Tensor network renormalization|journal=Phys. Rev. Lett.|year=2015}}</ref><ref name="Bal_flows">{{cite journal|author1=Bal M|author2=Mariën M|author3=Haegeman J|author4=Verstraete F.|title=Renormalization group flows of Hamiltonians using tensor networks|journal=Phys. Rev. Lett.|year=2017}}</ref>及信息论框架<ref name="Lenggenhager_optimal">{{cite journal|author1=Lenggenhager P. M|author2=Gökmen D. E|author3=Ringel Z|author4=Huber S. D|author5=Koch-Janusz M.|title=Optimal renormalization group transformation from information theory|journal=Phys. Rev. X|year=2020|volume=10}}</ref><ref name="Gökmen_statistical">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Statistical physics through the lens of real-space mutual information|journal=Phys. Rev. Lett.|year=2021|volume=127}}</ref><ref name="Gordon_relevance">{{cite journal|author1=Gordon A|author2=Banerjee A|author3=Koch-Janusz M|author4=Ringel Z.|title=Relevance in the renormalization group and in information theory|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Sarra_mutual">{{cite journal|author1=Sarra L|author2=Aiello A|author3=Marquardt F.|title=Renormalized mutual information for artificial scientific discovery|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Gökmen_symmetries">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Symmetries and phase diagrams with real-space mutual information neural estimation|journal=Phys. Rev. E|year=2021|volume=104}}</ref>中亦获成功应用。
+过去五十年间，重整化群技术已扩展至从最大尺度的宇宙学到最小尺度的粒子物理等广泛领域。在统计力学范畴内，重整化群方法已推广至同质均匀的分形格点上的非平衡系统<ref name="Gefen_fractal">{{cite journal|author1=Gefen Y|author2=Mandelbrot B. B|author3=Aharony A.|title=Critical phenomena on fractal lattices|journal=Phys. Rev. Lett.|year=1980|volume=45}}</ref><ref name="Gefen_backbone">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B|author4=Kirkpatrick S.|title=Solvable fractal family, and its possible relation to the backbone at percolation|journal=Phys. Rev. Lett.|year=1981|volume=47}}</ref><ref name="Phani_percolation">{{cite journal|author1=Phani M. K|author2=Dhar D.|title=Real-space renormalisation group: application to directed percolation|journal=J. Phys. C|year=1982|volume=15}}</ref><ref name="Gefen_phase1">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. I. Quasi-linear lattices|journal=J. Phys. A|year=1983|volume=16}}</ref><ref name="Gefen_phase2">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Shapir Y|author4=Mandelbrot B. B.|title=Phase transitions on fractals. II. Sierpinski gaskets|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Gefen_phase3">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. III. Infinitely ramified lattices|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Das_critical">{{cite journal|author1=Das D|author2=Dey S|author3=Jacobsen J. L|author4=Dhar D.|title=Critical behavior of loops and biconnected clusters on fractals of dimension d \< 2|journal=J. Phys. A|year=2008|volume=41}}</ref>，其局部平移不变性能够简化计算。超越初始应用的重要范例包括：流行病动力学模型<ref name="Della_SIR">{{cite journal|author1=Della Morte M|author2=Sannino F.|title=Renormalization group approach to pandemics as a time-dependent SIR model|journal=Front. Phys.|year=2021}}</ref>、生物群体行为动力学<ref name="Cavagna_scaling">{{cite journal|author1=Cavagna A.|title=Dynamic scaling in natural swarms|journal=Nat. Phys.|year=2017|volume=13}}</ref><ref name="Cavagna_swarms">{{cite journal|author1=Cavagna A.|title=Dynamical renormalization group approach to the collective behavior of swarms|journal=Phys. Rev. Lett.|year=2019}}</ref><ref name="Cavagna_dimensions">{{cite journal|author1=Cavagna A.|title=Natural swarms in 3.99 dimensions|journal=Nat. Phys.|year=2023|volume=19}}</ref>、神经网络<ref name="Koch_information">{{cite journal|author1=Koch-Janusz M|author2=Ringel Z.|title=Mutual information, neural networks and the renormalization group|journal=Nat. Phys.|year=2018|volume=14}}</ref><ref name="Li_neural">{{cite journal|author1=Li S.-H|author2=Wang L.|title=Neural network renormalization group|journal=Phys. Rev. Lett.|year=2018}}</ref><ref name="Hu_machine">{{cite journal|author1=Hu H.-Y|author2=Li S.-H|author3=Wang L|author4=You Y.-Z.|title=Machine learning holographic mapping by neural network renormalization group|journal=Phys. Rev. Res.|year=2020|volume=2}}</ref><ref name="Caso_graph">{{cite journal|author1=Caso F|author2=Trappolini G|author3=Bacciu A|author4=Liò P|author5=Silvestri F.|title=Renormalized graph neural networks|journal=Preprint at arXiv|year=2023}}</ref>、动态与有向逾渗<ref name="Young_percolation">{{cite journal|author1=Young A. P|author2=Stinchcombe R. B.|title=A renormalization group theory for percolation problems|journal=J. Phys. C|year=1975|volume=8}}</ref><ref name="Reynolds_percolation">{{cite journal|author1=Reynolds P. J|author2=Stanley H. E|author3=Klein W.|title=A real-space renormalization group for site and bond percolation|journal=J. Phys. C|year=1977|volume=10}}</ref>、观点动力学的选民模型<ref name="Galam_voting">{{cite journal|author1=Galam S.|title=Real space renormalization group and totalitarian paradox of majority rule voting|journal=Physica A|year=2000|volume=285}}</ref>，以及一维链<ref name="Kogan_synchronization">{{cite journal|author1=Kogan O|author2=Rogers J. L|author3=Cross M. C|author4=Refael G.|title=Renormalization group approach to oscillator synchronization|journal=Phys. Rev. E|year=2009|volume=80}}</ref>、格点<ref name="Östborn_disorder">{{cite journal|author1=Östborn P.|title=Renormalization of oscillator lattices with disorder|journal=Phys. Rev. E|year=2009|volume=79}}</ref>和层级树<ref name="Garlaschelli_synchronization">{{cite journal|author1=Garlaschelli D|author2=Hollander F|author3=Meylahn J|author4=Zeegers B.|title=Synchronization of phase oscillators on the hierarchical lattice|journal=J. Stat. Phys.|year=2019|volume=174}}</ref>上[[非线性振子模型]]（如Kuramoto模型<ref name="Hinrichsen_non-equilibrium_2000">{{cite journal|author1=Hinrichsen, H.|title=Non-equilibrium critical phenomena and phase transitions into absorbing states|journal=Adv. Phys.|year=2000|volume=49|url=}}</ref><ref name="Dornic_Langevin_2005">{{cite journal|author1=Dornic, I.|author2=Chaté, H.|author3=Muñoz, M. A.|title=Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions|journal=Phys. Rev. Lett.|year=2005|volume=94|url=}}</ref>）的同步模型。此外，重整化群方法在[[张量网络]]<ref name="Li_neural">{{cite journal|author1=Li S.-H|author2=Wang L.|title=Neural network renormalization group|journal=Phys. Rev. Lett.|year=2018}}</ref><ref name="Levin_tensor">{{cite journal|author1=Levin M|author2=Nave C. P.|title=Tensor renormalization group approach to two-dimensional classical lattice models|journal=Phys. Rev. Lett.|year=2007}}</ref><ref name="Evenbly_tensor">{{cite journal|author1=Evenbly G|author2=Vidal G.|title=Tensor network renormalization|journal=Phys. Rev. Lett.|year=2015}}</ref><ref name="Bal_flows">{{cite journal|author1=Bal M|author2=Mariën M|author3=Haegeman J|author4=Verstraete F.|title=Renormalization group flows of Hamiltonians using tensor networks|journal=Phys. Rev. Lett.|year=2017}}</ref>及[[信息论]]框架<ref name="Lenggenhager_optimal">{{cite journal|author1=Lenggenhager P. M|author2=Gökmen D. E|author3=Ringel Z|author4=Huber S. D|author5=Koch-Janusz M.|title=Optimal renormalization group transformation from information theory|journal=Phys. Rev. X|year=2020|volume=10}}</ref><ref name="Gökmen_statistical">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Statistical physics through the lens of real-space mutual information|journal=Phys. Rev. Lett.|year=2021|volume=127}}</ref><ref name="Gordon_relevance">{{cite journal|author1=Gordon A|author2=Banerjee A|author3=Koch-Janusz M|author4=Ringel Z.|title=Relevance in the renormalization group and in information theory|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Sarra_mutual">{{cite journal|author1=Sarra L|author2=Aiello A|author3=Marquardt F.|title=Renormalized mutual information for artificial scientific discovery|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Gökmen_symmetries">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Symmetries and phase diagrams with real-space mutual information neural estimation|journal=Phys. Rev. E|year=2021|volume=104}}</ref>中亦获成功应用。
 与平衡态不同，非平衡系统中决定普适类的要素往往未知，且微扰处理中高阶重整化群计算通常不可行。非微扰重整化群作为有潜力的替代方案<ref name="Canet_reaction">{{cite journal|author1=Canet L|author2=Delamotte B|author3=Deloubrière O|author4=Wschebor N.|title=Nonperturbative renormalization-group study of reaction--diffusion processes|journal=Phys. Rev. Lett.|year=2004|volume=92}}</ref><ref name="Canet_branching">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B.|title=Quantitative phase diagrams of branching and annihilating random walks|journal=Phys. Rev. Lett.|year=2004|volume=92}}</ref><ref name="Canet_fixed">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B|author4=Dornic I|author5=Muñoz M. A.|title=Nonperturbative fixed point in a nonequilibrium phase transition|journal=Phys. Rev. Lett.|year=2005|volume=95}}</ref><ref name="Canet_framework">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B.|title=General framework of the non-perturbative renormalization group for non-equilibrium steady states|journal=J. Phys. A|year=2011|volume=44}}</ref>，已应用于格点上的流行病扩散动力学模型等研究<ref name="Tarpin_epidemic">{{cite journal|author1=Tarpin M|author2=Benitez F|author3=Canet L|author4=Wschebor N.|title=Nonperturbative renormalization group for the diffusive epidemic process|journal=Phys. Rev. E|year=2017|volume=96}}</ref>。统计物理和场论中的重整化总结了传统应用于物理系统的重整化和重整化群变换的基本概念。

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 重整化群理论通过在实空间可用的方法对微观细节进行形式化平均，从而定义了任意尺度下的有效自由度与相互作用，成为统计力学的重大突破。Kenneth Wilson后续将其发展为连续场论中的重整化方法<ref name="Wilson_renormalization_1974">{{cite journal|author1=Wilson, K. G.|author2=Kogut, J.|title=The renormalization group and the $\epsilon$ expansion|journal=Phys. Rep.|year=1974|volume=12|url=}}</ref><ref name="Castro_scaling">{{cite journal|author1=Castro C|author2=Jona-Lasinio G.|title=On the microscopic foundation of scaling laws|journal=Phys. Lett. A|year=1969|volume=29}}</ref>。Wilson的工作不仅为理解临界现象的标度行为奠定理论基础，更将场论中处理名义发散量的零散方法系统化。这些发散性源于数学上为解决任意小尺度问题的理想化操作，虽然研究者们对此曾提出减除方案<ref name="Gell-Mann_electrodynamics">{{cite journal|author1=Gell-Mann M|author2=Low F. E.|title=Quantum electrodynamics at small distances|journal=Phys. Rev.|year=1954|volume=95}}</ref>，但该方案在重整化群理论建立前缺乏概念基础。此后，重整化群方法催生了“理论之理论”，揭示微观细节迥异的物理系统在临界点附近如何出现普适行为<ref name="Zinn-Justin_field">{{cite journal|author1=Zinn-Justin J.|title=Quantum Field Theory and Critical Phenomena|journal=Oxford Univ. Press|year=2021}}</ref><ref name="Amit_field">{{cite journal|author1=Amit D. J|author2=Martin-Mayor V.|title=Field Theory, the Renormalization Group and Critical Phenomena|journal=World Scientific|year=2005}}</ref><ref name="Täuber_renormalization">{{cite journal|author1=Täuber U. C.|title=Renormalization group: applications in statistical physics|journal=Nucl. Phys. B|year=2012|volume=228}}</ref><ref name="Pelissetto_critical">{{cite journal|author1=Pelissetto A|author2=Vicari E.|title=Critical phenomena and renormalization-group theory|journal=Phys. Rep.|year=2002|volume=368}}</ref><ref name="Efrati_real-space">{{cite journal|author1=Efrati E|author2=Wang Z|author3=Kolan A|author4=Kadanoff L. P.|title=Real-space renormalization in statistical mechanics|journal=Rev. Mod. Phys.|year=2014|volume=86}}</ref>。
-过去五十年间，重整化群技术已扩展至从最大尺度的宇宙学到最小尺度的粒子物理等广泛领域。在统计力学范畴内，重整化群方法已推广至同质均匀的分形格点上的非平衡系统<ref name="Gefen_fractal">{{cite journal|author1=Gefen Y|author2=Mandelbrot B. B|author3=Aharony A.|title=Critical phenomena on fractal lattices|journal=Phys. Rev. Lett.|year=1980|volume=45}}</ref><ref name="Gefen_backbone">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B|author4=Kirkpatrick S.|title=Solvable fractal family, and its possible relation to the backbone at percolation|journal=Phys. Rev. Lett.|year=1981|volume=47}}</ref><ref name="Phani_percolation">{{cite journal|author1=Phani M. K|author2=Dhar D.|title=Real-space renormalisation group: application to directed percolation|journal=J. Phys. C|year=1982|volume=15}}</ref><ref name="Gefen_phase1">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. I. Quasi-linear lattices|journal=J. Phys. A|year=1983|volume=16}}</ref><ref name="Gefen_phase2">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Shapir Y|author4=Mandelbrot B. B.|title=Phase transitions on fractals. II. Sierpinski gaskets|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Gefen_phase3">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. III. Infinitely ramified lattices|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Das_critical">{{cite journal|author1=Das D|author2=Dey S|author3=Jacobsen J. L|author4=Dhar D.|title=Critical behavior of loops and biconnected clusters on fractals of dimension d \< 2|journal=J. Phys. A|year=2008|volume=41}}</ref>，其局部平移不变性能够简化计算。超越初始应用的重要范例包括：流行病动力学模型<ref name="Della_SIR">{{cite journal|author1=Della Morte M|author2=Sannino F.|title=Renormalization group approach to pandemics as a time-dependent SIR model|journal=Front. Phys.|year=2021}}</ref>、生物群体行为动力学<ref name="Cavagna_scaling">{{cite journal|author1=Cavagna A.|title=Dynamic scaling in natural swarms|journal=Nat. Phys.|year=2017|volume=13}}</ref><ref name="Cavagna_swarms">{{cite journal|author1=Cavagna A.|title=Dynamical renormalization group approach to the collective behavior of swarms|journal=Phys. Rev. Lett.|year=2019}}</ref><ref name="Cavagna_dimensions">{{cite journal|author1=Cavagna A.|title=Natural swarms in 3.99 dimensions|journal=Nat. Phys.|year=2023|volume=19}}</ref>、神经网络<ref name="Koch_information">{{cite journal|author1=Koch-Janusz M|author2=Ringel Z.|title=Mutual information, neural networks and the renormalization group|journal=Nat. Phys.|year=2018|volume=14}}</ref><ref name="Li_neural">{{cite journal|author1=Li S.-H|author2=Wang L.|title=Neural network renormalization group|journal=Phys. Rev. Lett.|year=2018}}</ref><ref name="Hu_machine">{{cite journal|author1=Hu H.-Y|author2=Li S.-H|author3=Wang L|author4=You Y.-Z.|title=Machine learning holographic mapping by neural network renormalization group|journal=Phys. Rev. Res.|year=2020|volume=2}}</ref><ref name="Caso_graph">{{cite journal|author1=Caso F|author2=Trappolini G|author3=Bacciu A|author4=Liò P|author5=Silvestri F.|title=Renormalized graph neural networks|journal=Preprint at arXiv|year=2023}}</ref>、动态与有向逾渗<ref name="Young_percolation">{{cite journal|author1=Young A. P|author2=Stinchcombe R. B.|title=A renormalization group theory for percolation problems|journal=J. Phys. C|year=1975|volume=8}}</ref><ref name="Reynolds_percolation">{{cite journal|author1=Reynolds P. J|author2=Stanley H. E|author3=Klein W.|title=A real-space renormalization group for site and bond percolation|journal=J. Phys. C|year=1977|volume=10}}</ref>、观点动力学的选民模型<ref name="Galam_voting">{{cite journal|author1=Galam S.|title=Real space renormalization group and totalitarian paradox of majority rule voting|journal=Physica A|year=2000|volume=285}}</ref>，以及一维链<ref name="Kogan_synchronization">{{cite journal|author1=Kogan O|author2=Rogers J. L|author3=Cross M. C|author4=Refael G.|title=Renormalization group approach to oscillator synchronization|journal=Phys. Rev. E|year=2009|volume=80}}</ref>、格点<ref name="Östborn_disorder">{{cite journal|author1=Östborn P.|title=Renormalization of oscillator lattices with disorder|journal=Phys. Rev. E|year=2009|volume=79}}</ref>和层级树<ref name="Garlaschelli_synchronization">{{cite journal|author1=Garlaschelli D|author2=Hollander F|author3=Meylahn J|author4=Zeegers B.|title=Synchronization of phase oscillators on the hierarchical lattice|journal=J. Stat. Phys.|year=2019|volume=174}}</ref>上非线性振子模型（如Kuramoto模型<ref name="Hinrichsen_non-equilibrium_2000">{{cite journal|author1=Hinrichsen, H.|title=Non-equilibrium critical phenomena and phase transitions into absorbing states|journal=Adv. Phys.|year=2000|volume=49|url=}}</ref><ref name="Dornic_Langevin_2005">{{cite journal|author1=Dornic, I.|author2=Chaté, H.|author3=Muñoz, M. A.|title=Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions|journal=Phys. Rev. Lett.|year=2005|volume=94|url=}}</ref>）的同步模型。此外，重整化群方法在张量网络[192,201–203]及信息论框架<ref name="Lenggenhager_optimal">{{cite journal|author1=Lenggenhager P. M|author2=Gökmen D. E|author3=Ringel Z|author4=Huber S. D|author5=Koch-Janusz M.|title=Optimal renormalization group transformation from information theory|journal=Phys. Rev. X|year=2020|volume=10}}</ref><ref name="Gökmen_statistical">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Statistical physics through the lens of real-space mutual information|journal=Phys. Rev. Lett.|year=2021|volume=127}}</ref><ref name="Gordon_relevance">{{cite journal|author1=Gordon A|author2=Banerjee A|author3=Koch-Janusz M|author4=Ringel Z.|title=Relevance in the renormalization group and in information theory|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Sarra_mutual">{{cite journal|author1=Sarra L|author2=Aiello A|author3=Marquardt F.|title=Renormalized mutual information for artificial scientific discovery|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Gökmen_symmetries">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Symmetries and phase diagrams with real-space mutual information neural estimation|journal=Phys. Rev. E|year=2021|volume=104}}</ref>中亦获成功应用。
+过去五十年间，重整化群技术已扩展至从最大尺度的宇宙学到最小尺度的粒子物理等广泛领域。在统计力学范畴内，重整化群方法已推广至同质均匀的分形格点上的非平衡系统<ref name="Gefen_fractal">{{cite journal|author1=Gefen Y|author2=Mandelbrot B. B|author3=Aharony A.|title=Critical phenomena on fractal lattices|journal=Phys. Rev. Lett.|year=1980|volume=45}}</ref><ref name="Gefen_backbone">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B|author4=Kirkpatrick S.|title=Solvable fractal family, and its possible relation to the backbone at percolation|journal=Phys. Rev. Lett.|year=1981|volume=47}}</ref><ref name="Phani_percolation">{{cite journal|author1=Phani M. K|author2=Dhar D.|title=Real-space renormalisation group: application to directed percolation|journal=J. Phys. C|year=1982|volume=15}}</ref><ref name="Gefen_phase1">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. I. Quasi-linear lattices|journal=J. Phys. A|year=1983|volume=16}}</ref><ref name="Gefen_phase2">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Shapir Y|author4=Mandelbrot B. B.|title=Phase transitions on fractals. II. Sierpinski gaskets|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Gefen_phase3">{{cite journal|author1=Gefen Y|author2=Aharony A|author3=Mandelbrot B. B.|title=Phase transitions on fractals. III. Infinitely ramified lattices|journal=J. Phys. A|year=1984|volume=17}}</ref><ref name="Das_critical">{{cite journal|author1=Das D|author2=Dey S|author3=Jacobsen J. L|author4=Dhar D.|title=Critical behavior of loops and biconnected clusters on fractals of dimension d \< 2|journal=J. Phys. A|year=2008|volume=41}}</ref>，其局部平移不变性能够简化计算。超越初始应用的重要范例包括：流行病动力学模型<ref name="Della_SIR">{{cite journal|author1=Della Morte M|author2=Sannino F.|title=Renormalization group approach to pandemics as a time-dependent SIR model|journal=Front. Phys.|year=2021}}</ref>、生物群体行为动力学<ref name="Cavagna_scaling">{{cite journal|author1=Cavagna A.|title=Dynamic scaling in natural swarms|journal=Nat. Phys.|year=2017|volume=13}}</ref><ref name="Cavagna_swarms">{{cite journal|author1=Cavagna A.|title=Dynamical renormalization group approach to the collective behavior of swarms|journal=Phys. Rev. Lett.|year=2019}}</ref><ref name="Cavagna_dimensions">{{cite journal|author1=Cavagna A.|title=Natural swarms in 3.99 dimensions|journal=Nat. Phys.|year=2023|volume=19}}</ref>、神经网络<ref name="Koch_information">{{cite journal|author1=Koch-Janusz M|author2=Ringel Z.|title=Mutual information, neural networks and the renormalization group|journal=Nat. Phys.|year=2018|volume=14}}</ref><ref name="Li_neural">{{cite journal|author1=Li S.-H|author2=Wang L.|title=Neural network renormalization group|journal=Phys. Rev. Lett.|year=2018}}</ref><ref name="Hu_machine">{{cite journal|author1=Hu H.-Y|author2=Li S.-H|author3=Wang L|author4=You Y.-Z.|title=Machine learning holographic mapping by neural network renormalization group|journal=Phys. Rev. Res.|year=2020|volume=2}}</ref><ref name="Caso_graph">{{cite journal|author1=Caso F|author2=Trappolini G|author3=Bacciu A|author4=Liò P|author5=Silvestri F.|title=Renormalized graph neural networks|journal=Preprint at arXiv|year=2023}}</ref>、动态与有向逾渗<ref name="Young_percolation">{{cite journal|author1=Young A. P|author2=Stinchcombe R. B.|title=A renormalization group theory for percolation problems|journal=J. Phys. C|year=1975|volume=8}}</ref><ref name="Reynolds_percolation">{{cite journal|author1=Reynolds P. J|author2=Stanley H. E|author3=Klein W.|title=A real-space renormalization group for site and bond percolation|journal=J. Phys. C|year=1977|volume=10}}</ref>、观点动力学的选民模型<ref name="Galam_voting">{{cite journal|author1=Galam S.|title=Real space renormalization group and totalitarian paradox of majority rule voting|journal=Physica A|year=2000|volume=285}}</ref>，以及一维链<ref name="Kogan_synchronization">{{cite journal|author1=Kogan O|author2=Rogers J. L|author3=Cross M. C|author4=Refael G.|title=Renormalization group approach to oscillator synchronization|journal=Phys. Rev. E|year=2009|volume=80}}</ref>、格点<ref name="Östborn_disorder">{{cite journal|author1=Östborn P.|title=Renormalization of oscillator lattices with disorder|journal=Phys. Rev. E|year=2009|volume=79}}</ref>和层级树<ref name="Garlaschelli_synchronization">{{cite journal|author1=Garlaschelli D|author2=Hollander F|author3=Meylahn J|author4=Zeegers B.|title=Synchronization of phase oscillators on the hierarchical lattice|journal=J. Stat. Phys.|year=2019|volume=174}}</ref>上非线性振子模型（如Kuramoto模型<ref name="Hinrichsen_non-equilibrium_2000">{{cite journal|author1=Hinrichsen, H.|title=Non-equilibrium critical phenomena and phase transitions into absorbing states|journal=Adv. Phys.|year=2000|volume=49|url=}}</ref><ref name="Dornic_Langevin_2005">{{cite journal|author1=Dornic, I.|author2=Chaté, H.|author3=Muñoz, M. A.|title=Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions|journal=Phys. Rev. Lett.|year=2005|volume=94|url=}}</ref>）的同步模型。此外，重整化群方法在张量网络<ref name="Li_neural">{{cite journal|author1=Li S.-H|author2=Wang L.|title=Neural network renormalization group|journal=Phys. Rev. Lett.|year=2018}}</ref><ref name="Levin_tensor">{{cite journal|author1=Levin M|author2=Nave C. P.|title=Tensor renormalization group approach to two-dimensional classical lattice models|journal=Phys. Rev. Lett.|year=2007}}</ref><ref name="Evenbly_tensor">{{cite journal|author1=Evenbly G|author2=Vidal G.|title=Tensor network renormalization|journal=Phys. Rev. Lett.|year=2015}}</ref><ref name="Bal_flows">{{cite journal|author1=Bal M|author2=Mariën M|author3=Haegeman J|author4=Verstraete F.|title=Renormalization group flows of Hamiltonians using tensor networks|journal=Phys. Rev. Lett.|year=2017}}</ref>及信息论框架<ref name="Lenggenhager_optimal">{{cite journal|author1=Lenggenhager P. M|author2=Gökmen D. E|author3=Ringel Z|author4=Huber S. D|author5=Koch-Janusz M.|title=Optimal renormalization group transformation from information theory|journal=Phys. Rev. X|year=2020|volume=10}}</ref><ref name="Gökmen_statistical">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Statistical physics through the lens of real-space mutual information|journal=Phys. Rev. Lett.|year=2021|volume=127}}</ref><ref name="Gordon_relevance">{{cite journal|author1=Gordon A|author2=Banerjee A|author3=Koch-Janusz M|author4=Ringel Z.|title=Relevance in the renormalization group and in information theory|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Sarra_mutual">{{cite journal|author1=Sarra L|author2=Aiello A|author3=Marquardt F.|title=Renormalized mutual information for artificial scientific discovery|journal=Phys. Rev. Lett.|year=2021|volume=126}}</ref><ref name="Gökmen_symmetries">{{cite journal|author1=Gökmen D. E|author2=Ringel Z|author3=Huber S. D|author4=Koch-Janusz M.|title=Symmetries and phase diagrams with real-space mutual information neural estimation|journal=Phys. Rev. E|year=2021|volume=104}}</ref>中亦获成功应用。
 与平衡态不同，非平衡系统中决定普适类的要素往往未知，且微扰处理中高阶重整化群计算通常不可行。非微扰重整化群作为有潜力的替代方案<ref name="Canet_reaction">{{cite journal|author1=Canet L|author2=Delamotte B|author3=Deloubrière O|author4=Wschebor N.|title=Nonperturbative renormalization-group study of reaction--diffusion processes|journal=Phys. Rev. Lett.|year=2004|volume=92}}</ref><ref name="Canet_branching">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B.|title=Quantitative phase diagrams of branching and annihilating random walks|journal=Phys. Rev. Lett.|year=2004|volume=92}}</ref><ref name="Canet_fixed">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B|author4=Dornic I|author5=Muñoz M. A.|title=Nonperturbative fixed point in a nonequilibrium phase transition|journal=Phys. Rev. Lett.|year=2005|volume=95}}</ref><ref name="Canet_framework">{{cite journal|author1=Canet L|author2=Chaté H|author3=Delamotte B.|title=General framework of the non-perturbative renormalization group for non-equilibrium steady states|journal=J. Phys. A|year=2011|volume=44}}</ref>，已应用于格点上的流行病扩散动力学模型等研究<ref name="Tarpin_epidemic">{{cite journal|author1=Tarpin M|author2=Benitez F|author3=Canet L|author4=Wschebor N.|title=Nonperturbative renormalization group for the diffusive epidemic process|journal=Phys. Rev. E|year=2017|volume=96}}</ref>。统计物理和场论中的重整化总结了传统应用于物理系统的重整化和重整化群变换的基本概念。

@@ 第2行： / 第2行： @@
 |keywords=网络重整化,复杂网络,重整化群,粗粒化方法,几何网络,拉普拉斯重整化,多尺度网络
 |description=本文系统综述了网络重整化领域的主流方法，探讨复杂网络缺乏几何坐标、异质性及对称性缺失带来的挑战，涵盖几何重整化、拉普拉斯重整化及多尺度重整化等方法，分析其应用与开放问题。}}
-重整化群（renormalization group, RG）理论是研究多自由度系统的强有力框架，通过不同尺度下系统状态、模型参数和耦合常数的描述，实现分辨率层次转换，可识别相变临界点并分析临界行为。传统物理应用中，重整化群基于同质性、对称性、几何结构和局部性构建尺度变换方案。近年来，该理论向复杂网络拓展，但网络系统存在显式几何坐标缺失、节点与子图统计异质性及类格点对称性缺乏三大特征，导致统一重整化流程构建面临挑战。本文梳理网络重整化主流方法，评述理论突破，探讨关键科学问题。
+重整化群（renormalization group, RG）理论是研究[[多自由度系统]]的强有力框架，通过不同尺度下系统状态、模型参数和[[耦合常数]]的描述，实现分辨率层次转换，可识别[[相变临界点]]并分析临界行为。传统物理应用中，重整化群基于同质性、对称性、几何结构和局部性构建尺度变换方案。近年来，该理论向[[复杂网络]]拓展，但网络系统存在显式几何坐标缺失、节点与子图统计异质性及类格点对称性缺乏三大特征，导致统一重整化流程构建面临挑战。本文梳理网络重整化主流方法，评述理论突破，探讨关键科学问题。
 == 背景 ==
@@ 第8行： / 第8行： @@
 重整化和可重整化是现代物理学理解宇宙中物质现象的基础概念，该理论假定在足够大的时空尺度上，小尺度细节是可以被平均掉的，从而解释了为什么我们在日常生活中无需了解亚原子粒子水平的精确行为。重整化方法的形式化花费了物理学家数十年时间，并最终促成了对某些物理现象普遍性的惊人洞察。
-物理系统中相互作用的局部性与因果性协同作用会限制小尺度涨落的影响，使得宏观尺度下仅需少量参数即可刻画该系统。该思想的规范化始于20世纪60年代，Leo Kadanoff将粗粒化概念（早期由Paul与Tatiana Ehrenfest<ref name="Ehrenfest_statistical">{{cite journal|author1=Ehrenfest P|author2=Ehrenfest T.|title=Begriffliche Grundlagen der statistischen Auffassung in der Mechanik|journal=Springer|year=1907}}</ref>研究）应用于统计力学系统，以理解其二阶相变临界点邻域的行为<ref name="Kadanoff_scaling_1966">{{cite journal|author1=Kadanoff, L. P.|title=Scaling laws for Ising models near Tc|journal=Phys. Phys. Fiz.|year=1966|volume=2|url=}}</ref>。
+物理系统中相互作用的局部性与因果性协同作用会限制小尺度涨落的影响，使得宏观尺度下仅需少量参数即可刻画该系统。该思想的规范化始于20世纪60年代，Leo Kadanoff将粗粒化概念（早期由Paul与Tatiana Ehrenfest<ref name="Ehrenfest_statistical">{{cite journal|author1=Ehrenfest P|author2=Ehrenfest T.|title=Begriffliche Grundlagen der statistischen Auffassung in der Mechanik|journal=Springer|year=1907}}</ref>研究）应用于[[统计力学]]系统，以理解其二阶[[相变]]临界点邻域的行为<ref name="Kadanoff_scaling_1966">{{cite journal|author1=Kadanoff, L. P.|title=Scaling laws for Ising models near Tc|journal=Phys. Phys. Fiz.|year=1966|volume=2|url=}}</ref>。
-Ising模型是经典范例，其晶格自旋经历无序态与有序态（磁化态）之间的相变。通过引入“块自旋”自由度可实现模型的粗粒化。块自旋变量间相互作用的标度行为通过重整化群理论<ref name="Kadanoff_recursion_1976">{{cite journal|author1=Kadanoff, L. P.|title=Notes on Migdal's recursion formulas|journal=Ann. Phys.|year=1976|volume=100|url=}}</ref>得以形式化。早期基于晶格的数值计算<ref name="Niemeijer_Wilson">{{cite journal|author1=Niemeijer T|author2=Leeuwen J. M. J.|title=Wilson theory for spin systems on a triangular lattice|journal=Phys. Rev. Lett.|year=1973|volume=31}}</ref>之后发展为更普适的变分方法<ref name="Kadanoff_variational">{{cite journal|author1=Kadanoff L. P.|title=Variational principles and approximate renormalization group calculations|journal=Phys. Rev. Lett.|year=1975|volume=34}}</ref><ref name="Southern_Ising">{{cite journal|author1=Southern B. W.|title=Kadanoff's variational renormalisation group method: the Ising model on the square and triangular lattices|journal=J. Phys. A|year=1978|volume=11}}</ref><ref name="Jan_Kadanoff">{{cite journal|author1=Jan N|author2=Glazer A.|title=Kadanoff's approximate renormalization group transformation applied to the triangular Ising lattice|journal=Physica A|year=1978|volume=91}}</ref><ref name="Nijs_Potts">{{cite journal|author1=Nijs M|author2=Knops H.|title=Variational renormalization method and the Potts model|journal=Physica A|year=1978|volume=93}}</ref>。
+[[Ising模型]]是经典范例，其晶格自旋经历无序态与有序态（磁化态）之间的相变。通过引入“块自旋”自由度可实现模型的[[粗粒化]]。块自旋变量间相互作用的标度行为通过重整化群理论<ref name="Kadanoff_recursion_1976">{{cite journal|author1=Kadanoff, L. P.|title=Notes on Migdal's recursion formulas|journal=Ann. Phys.|year=1976|volume=100|url=}}</ref>得以形式化。早期基于晶格的数值计算<ref name="Niemeijer_Wilson">{{cite journal|author1=Niemeijer T|author2=Leeuwen J. M. J.|title=Wilson theory for spin systems on a triangular lattice|journal=Phys. Rev. Lett.|year=1973|volume=31}}</ref>之后发展为更普适的变分方法<ref name="Kadanoff_variational">{{cite journal|author1=Kadanoff L. P.|title=Variational principles and approximate renormalization group calculations|journal=Phys. Rev. Lett.|year=1975|volume=34}}</ref><ref name="Southern_Ising">{{cite journal|author1=Southern B. W.|title=Kadanoff's variational renormalisation group method: the Ising model on the square and triangular lattices|journal=J. Phys. A|year=1978|volume=11}}</ref><ref name="Jan_Kadanoff">{{cite journal|author1=Jan N|author2=Glazer A.|title=Kadanoff's approximate renormalization group transformation applied to the triangular Ising lattice|journal=Physica A|year=1978|volume=91}}</ref><ref name="Nijs_Potts">{{cite journal|author1=Nijs M|author2=Knops H.|title=Variational renormalization method and the Potts model|journal=Physica A|year=1978|volume=93}}</ref>。
 重整化群理论通过在实空间可用的方法对微观细节进行形式化平均，从而定义了任意尺度下的有效自由度与相互作用，成为统计力学的重大突破。Kenneth Wilson后续将其发展为连续场论中的重整化方法<ref name="Wilson_renormalization_1974">{{cite journal|author1=Wilson, K. G.|author2=Kogut, J.|title=The renormalization group and the $\epsilon$ expansion|journal=Phys. Rep.|year=1974|volume=12|url=}}</ref><ref name="Castro_scaling">{{cite journal|author1=Castro C|author2=Jona-Lasinio G.|title=On the microscopic foundation of scaling laws|journal=Phys. Lett. A|year=1969|volume=29}}</ref>。Wilson的工作不仅为理解临界现象的标度行为奠定理论基础，更将场论中处理名义发散量的零散方法系统化。这些发散性源于数学上为解决任意小尺度问题的理想化操作，虽然研究者们对此曾提出减除方案<ref name="Gell-Mann_electrodynamics">{{cite journal|author1=Gell-Mann M|author2=Low F. E.|title=Quantum electrodynamics at small distances|journal=Phys. Rev.|year=1954|volume=95}}</ref>，但该方案在重整化群理论建立前缺乏概念基础。此后，重整化群方法催生了“理论之理论”，揭示微观细节迥异的物理系统在临界点附近如何出现普适行为<ref name="Zinn-Justin_field">{{cite journal|author1=Zinn-Justin J.|title=Quantum Field Theory and Critical Phenomena|journal=Oxford Univ. Press|year=2021}}</ref><ref name="Amit_field">{{cite journal|author1=Amit D. J|author2=Martin-Mayor V.|title=Field Theory, the Renormalization Group and Critical Phenomena|journal=World Scientific|year=2005}}</ref><ref name="Täuber_renormalization">{{cite journal|author1=Täuber U. C.|title=Renormalization group: applications in statistical physics|journal=Nucl. Phys. B|year=2012|volume=228}}</ref><ref name="Pelissetto_critical">{{cite journal|author1=Pelissetto A|author2=Vicari E.|title=Critical phenomena and renormalization-group theory|journal=Phys. Rep.|year=2002|volume=368}}</ref><ref name="Efrati_real-space">{{cite journal|author1=Efrati E|author2=Wang Z|author3=Kolan A|author4=Kadanoff L. P.|title=Real-space renormalization in statistical mechanics|journal=Rev. Mod. Phys.|year=2014|volume=86}}</ref>。
@@ 第20行： / 第20行： @@
 除了“传统”物理系统，自然界和社会中的大型复杂系统也包含多个相互作用的尺度，且没有先验的最佳分辨率尺度。因而自然会问，在这些情况下，重整化是否同样适用？例如，在建模企业间商品和服务流动时，当将企业聚集成更广泛的类别（如经济部门）时，其关系是否能被一致地表示？同样，在考虑网络上的流行病传播时，人们希望探索当模型和参数具有不确定性时，基于聚合的分析结果在长期行为上是否与仓室模型的结果有所不同。
-尽管将物理系统的研究经验直接移植到复杂网络上可能很诱人，但必须批判性地重新审视其中的关键假设，包括在物质通常具备的相互作用的局部性和空间规则性——其中无序性通常可视为小扰动。此外，在考虑网络上的动态过程时，必须确保无论是对于特定个体还是对统计系综，任何粗粒化方法能够都能搞保持原先的动力学性质。因此，有必要重新审视重整化的基本前提，即通过对小尺度的平均来理解整体的行为，并且在复杂系统的意义下重新表述重整化方法也十分重要。
+尽管将物理系统的研究经验直接移植到复杂网络上可能很诱人，但必须批判性地重新审视其中的关键假设，包括在物质通常具备的相互作用的局部性和空间规则性——其中无序性通常可视为小扰动。此外，在考虑网络上的动态过程时，必须确保无论是对于特定个体还是对统计系统，任何粗粒化方法能够都能够保持原先的动力学性质。因此，有必要重新审视重整化的基本前提，即通过对小尺度的平均来理解整体的行为，并且在复杂系统的意义下重新表述重整化方法也十分重要。
 == 简介 ==

@@ 第2行： / 第2行： @@
 |keywords=网络重整化,复杂网络,重整化群,粗粒化方法,几何网络,拉普拉斯重整化,多尺度网络
 |description=本文系统综述了网络重整化领域的主流方法，探讨复杂网络缺乏几何坐标、异质性及对称性缺失带来的挑战，涵盖几何重整化、拉普拉斯重整化及多尺度重整化等方法，分析其应用与开放问题。}}
-重整化群（renormalization group, RG）理论是研究<nowiki>[[多自由度系统]]</nowiki>的强有力框架，通过不同尺度下系统状态、模型参数和耦合常数的描述，实现分辨率层次转换，可识别相变临界点并分析临界行为。传统物理应用中，重整化群基于同质性、对称性、几何结构和局部性构建尺度变换方案。近年来，该理论向复杂网络拓展，但网络系统存在显式几何坐标缺失、节点与子图统计异质性及类格点对称性缺乏三大特征，导致统一重整化流程构建面临挑战。本文梳理网络重整化主流方法，评述理论突破，探讨关键科学问题。
+重整化群（renormalization group, RG）理论是研究多自由度系统的强有力框架，通过不同尺度下系统状态、模型参数和耦合常数的描述，实现分辨率层次转换，可识别相变临界点并分析临界行为。传统物理应用中，重整化群基于同质性、对称性、几何结构和局部性构建尺度变换方案。近年来，该理论向复杂网络拓展，但网络系统存在显式几何坐标缺失、节点与子图统计异质性及类格点对称性缺乏三大特征，导致统一重整化流程构建面临挑战。本文梳理网络重整化主流方法，评述理论突破，探讨关键科学问题。
 == 背景 ==

@@ 第130行： / 第130行： @@
 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>。
 <br>
-[[文件:D26cc9a6-ef2d-4022-8c1b-8080bbc9e67b.png|640px|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]]
+[[文件:D26cc9a6-ef2d-4022-8c1b-8080bbc9e67b.png|替代=|无框|640x640像素]]
 </br>
@@ 第188行： / 第189行： @@
-[[文件:图2 实空间中的拉普拉斯网络重整化方法的构造.png|400*150px|有框|左|图2 实空间中的拉普拉斯网络重整化方法的构造]]
+[[文件:图2 实空间中的拉普拉斯网络重整化方法的构造.png|有框|图2 实空间中的拉普拉斯网络重整化方法的构造|替代=|无]]

@@ 第129行： / 第129行： @@
 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. 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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. 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 [[文件:D26cc9a6-ef2d-4022-8c1b-8080bbc9e67b.png|640px|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）。
 === 几何重整化技术 ===