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复杂网络重整化 (查看源代码)

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本文系统综述重整化方法在复杂网络中的应用，分析真实网络缺乏显式几何结构、多尺度异质性和跨尺度交互连接的特征，揭示传统重整化方法的改造需求，回顾粗粒化方案，探讨将重整化群核心原理推广至异质性网络的路径，厘清开放问题。

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|知识框1|统计物理和场论中的重整化 物理系统中相互作用的局部性与因果性限制小尺度涨落影响，使宏观尺度仅需少量参数刻画。该思想规范化始于20世纪60年代，Leo Kadanoff将粗粒化概念应用于统计力学系统，理解二阶相变临界点邻域行为~~[118]~~。 Ising模型是经典范例，晶格自旋经历无序态与有序态相变，通过引入“块自旋”实现粗粒化，重整化群理论形式化块自旋相互作用的标度行为~~[111]~~。早期基于晶格的数值计算后发展为变分方法~~[169–172]~~。 重整化群理论通过实空间平均微观细节，定义任意尺度的有效自由度与相互作用，成为统计力学突破。Kenneth Wilson将其发展为连续场论方法~~[113~~,~~173]~~，不仅为临界现象标度行为奠定基础，还系统化处理场论中的发散量。此后，重整化群方法揭示微观细节迥异的系统在临界点附近的普适行为~~[175–179]~~。 过去五十年，重整化群技术扩展至从宇宙学到粒子物理的广泛领域。统计力学中，推广至同质分形格点的非平衡系统~~[180–186]~~，其局部平移不变性简化计算。重要范例包括流行病动力学~~[187]~~、生物群体行为~~[188–190]~~、神经网络~~[191–194]~~、动态与有向逾渗~~[195,196]~~、观点动力学的选民模型~~[197]~~，以及一维链~~[198]~~、格点~~[199]~~和层级树~~[200]~~上的非线性振子同步模型。此外，在张量网络~~[192,201–203]~~及信息论框架~~[204–208]~~中亦获应用。 非平衡系统中决定普适类的要素往往未知，高阶重整化群计算通常不可行。非微扰重整化群作为替代方案~~[209–212]，已应用于格点流行病扩散动力学[213]。~~|

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|知识框1|统计物理和场论中的重整化 物理系统中相互作用的局部性与因果性限制小尺度涨落影响，使宏观尺度仅需少量参数刻画。该思想规范化始于20世纪60年代，Leo Kadanoff将粗粒化概念应用于统计力学系统，理解二阶相变临界点邻域行为<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，晶格自旋经历无序态与有序态相变，通过引入“块自旋”实现粗粒化，重整化群理论形式化块自旋相互作用的标度行为<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>。早期基于晶格的数值计算后发展为变分方法<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>。 重整化群理论通过实空间平均微观细节，定义任意尺度的有效自由度与相互作用，成为统计力学突破。Kenneth Wilson将其发展为连续场论方法<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>，不仅为临界现象标度行为奠定基础，还系统化处理场论中的发散量。此后，重整化群方法揭示微观细节迥异的系统在临界点附近的普适行为<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，重整化群技术扩展至从宇宙学到粒子物理的广泛领域。统计力学中，推广至同质分形格点的非平衡系统<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>，其局部平移不变性简化计算。重要范例包括流行病动力学<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<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>上的非线性振子同步模型。此外，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>中亦获应用。 非平衡系统中决定普适类的要素往往未知，高阶重整化群计算通常不可行。非微扰重整化群作为替代方案<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>，已应用于格点流行病扩散动力学<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>。

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== 复杂网络重整化的挑战 ==

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