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== 参考文献 ==
== 参考文献 ==
1. Pósfai, M. & Barabási, A.-L. Network Science (Cambridge Univ. Press, 2016).
2. Squartini, T. & Garlaschelli, D. Maximum-Entropy Networks: Pattern Detection, Network Reconstruction and Graph Combinatorics (Springer, 2017).
3. Newman, M. Networks (Oxford Univ. Press, 2018).
4. Cimini, G. The statistical physics of real-world networks. Nat. Rev. Phys. 1 (2019).
5. Radicchi, F., Ramasco, J., Barrat, A. & Fortunato, S. Complex networks renormalization: flows and fixed points. Phys. Rev. Lett. 101 (2008).
6. Radicchi, F., Barrat, A., Fortunato, S. & Ramasco, J. J. Renormalization flows in complex networks. Phys. Rev. E 79 (2009).
7. Chen, D., Su, H., Wang, X., Pan, G.-J. & Chen, G. Finite-size scaling of geometric renormalization flows in complex networks. Phys. Rev. E 104 (2021).
8. Chen, D., Cai, D. & Su, H. Scaling properties of scale-free networks in degree-thresholding renormalization flows. IEEE Trans. Netw. Sci. Eng. 10 (2023).
9. Serafino, M. True scale-free networks hidden by finite size effects. Proc. Natl Acad. Sci. USA 118 (2021).
10. Itzykson, C. & Drouffe, J. M. Statistical Field Theory: Volume 1, From Brownian Motion to Renormalization and Lattice Gauge Theory (Cambridge Univ. Press, 1989).
...（此处省略部分参考文献，完整列表见原文）...
11. Burgess, C. P. Introduction to Effective Field Theory (Cambridge Univ. Press, 2020).
12. Bardoscia, M. The physics of financial networks. Nat. Rev. Phys. 3 (2021).
13. Soriano-Paños, D. Modeling communicable diseases, human mobility, and epidemics: a review. Ann. Phys. 534 (2022).
14. Song, C., Havlin, S. & Makse, H. A. Self-similarity of complex networks. Nature 433 (2005).
15. Binder, K. & Young, A. P. Spin glasses: experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58 (1986).
16. Bar-Yam, Y. & Patil, S. P. Renormalization of sparse disorder in the Ising model. Preprint at arXiv (2018).
17. Mandelbrot, B. B. The Fractal Geometry of Nature Vol. 2 (Freeman, 1982).
18. Samsel, M., Makulski, K., Łepek, M., Fronczak, A. & Fronczak, P. Towards fractal origins of the community structure in complex networks: a model-based approach. Preprint at arXiv (2023).
19. Goh, K.-I., Salvi, G., Kahng, B. & Kim, D. Skeleton and fractal scaling in complex networks. Phys. Rev. Lett. 96 (2006).
20. Kim, J. S., Goh, K.-I., Kahng, B. & Kim, D. Fractality and self-similarity in scale-free networks. New J. Phys. 9 (2007).
21. Fronczak, A. Scaling theory of fractal complex networks. Sci. Rep. (2024).
22. Song, C., Havlin, S. & Makse, H. A. Origins of fractality in the growth of complex networks. Nat. Phys. 2 (2006).
23. Rozenfeld, H. D., Song, C. & Makse, H. A. Small-world to fractal transition in complex networks: a renormalization group approach. Phys. Rev. Lett. 104 (2010).
24. Newman, M. & Watts, D. Renormalization group analysis of the small-world network model. Phys. Lett. A 263 (1999).
25. Kim, B. J. Geographical coarse graining of complex networks. Phys. Rev. Lett. 93 (2004).
26. Gfeller, D. & Los Rios, P. Spectral coarse graining of complex networks. Phys. Rev. Lett. 99 (2007).
27. Wang, Y., Zeng, A., Di, Z. & Fan, Y. Spectral coarse graining for random walks in bipartite networks. Chaos 23 (2013).
28. Gfeller, D. & Los Rios, P. Spectral coarse graining and synchronization in oscillator networks. Phys. Rev. Lett. 100 (2008).
29. Chen, J., an Lu, J., Lu, X., Wu, X. & Chen, G. Spectral coarse graining of complex clustered networks. Commun. Nonlinear Sci. Numer. Simul. 18 (2013).
30. Jia, Z., Zeng, L., Wang, Y.-Y. & Wang, P. Optimization algorithms for spectral coarse-graining of complex networks. Physica A 514 (2019).
31. Zeng, A. & Lü, L. Coarse graining for synchronization in directed networks. Phys. Rev. E 83 (2011).
32. Wang, P. & Xu, S. Spectral coarse grained controllability of complex networks. Physica A 478 (2017).
33. Aygün, E. & Erzan, A. Spectral renormalization group theory on networks. J. Phys. Conf. Ser. 319 (2011).
34. Tuncer, A. & Erzan, A. Spectral renormalization group for the Gaussian model and $Psi$4 theory on nonspatial networks. Phys. Rev. E 92 (2015).
35. Villegas, P., Gili, T., Caldarelli, G. & Gabrielli, A. Laplacian renormalization group for heterogeneous networks. Nat. Phys. 19 (2023).
36. Chen, H., Hou, Z., Xin, H. & Yan, Y. Statistically consistent coarse-grained simulations for critical phenomena in complex networks. Phys. Rev. E 82 (2010).
37. Long, Y.-S., Jia, Z. & Wang, Y.-Y. Coarse graining method based on generalized degree in complex network. Physica A 505 (2018).
38. Wang, Y. Coarse graining method based on node similarity in complex network. Commun. Netw. 10 (2018).
39. Lorrain, F. & White, H. C. Structural equivalence of individuals in social networks. J. Math. Sociol. 1 (1971).
40. Dunne, J. A. Food webs. In textit{Encyclopedia of Complexity and Systems Science (ed. Meyers, R.) 3661–3682 (Springer, 2009).
41. Itzkovitz, S. Coarse-graining and self-dissimilarity of complex networks. Phys. Rev. E 71 (2005).
42. Serrano, M. Á., Krioukov, D. & Boguñá, M. Self-similarity of complex networks and hidden metric spaces. Phys. Rev. Lett. 100 (2008).
43. Alvarez-Hamelin, J., Dall’Asta, L., Barrat, A. & Vespignani, A. K-core decomposition of Internet graphs: hierarchies, self-similarity and measurement biases. Netw. Heterog. Media 3 (2008).
44. Lambiotte, R. Multi-scale modularity in complex networks. In textit{8th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks 546–553 (IEEE, 2010).
45. Karypis, G. & Kumar, V. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20 (1998).
46. Abou-Rjeili, A. & Karypis, G. Multilevel algorithms for partitioning power-law graphs. In textit{Proc. 20th IEEE International Parallel & Distributed Processing Symposium (IEEE, 2006).
47. Leifer, I., Phillips, D., Sorrentino, F. & Makse, H. A. Symmetry-driven network reconstruction through pseudobalanced coloring optimization. J. Stat. Mech. 2022 (2022).
48. Boldi, P. & Vigna, S. Fibrations of graphs. Discrete Math. 243 (2002).
49. Cardelli, L., Tribastone, M., Tschaikowski, M. & Vandin, A. Maximal aggregation of polynomial dynamical systems. Proc. Natl Acad. Sci. USA 114 (2017).
50. Cardelli, L., Tribastone, M., Tschaikowski, M. & Vandin, A. Erode: a tool for the evaluation and reduction of ordinary differential equations. In textit{Proc. Tools and Algorithms for the Construction and Analysis of Systems: 23rd International Conference, TACAS 2017 Part II 23, 310–328 (Springer, 2017).
51. Argyris, G. A., Lluch Lafuente, A., Tribastone, M., Tschaikowski, M. & Vandin, A. Reducing Boolean networks with backward equivalence. BMC Bioinformatics 24 (2023).
52. Morone, F., Leifer, I. & Makse, H. A. Fibration symmetries uncover the building blocks of biological networks. Proc. Natl Acad. Sci. USA 117 (2020).
53. Gili, T. et al. Fibration symmetry-breaking supports functional transitions in a brain network engaged in language. Preprint at arXiv (2024).
54. Papadopoulos, F., Psounis, K. & Govindan, R. Performance preserving topological downscaling of internet-like networks. IEEE J. Sel. Areas Commun. 24 (2006).
55. Papadopoulos, F. & Psounis, K. Efficient identification of uncongested internet links for topology downscaling. SIGCOMM Comput. Commun. Rev. 37 (2007).
56. Di, X., Zhao, Y., Huang, S. & Liu, H. X. A similitude theory for modeling traffic flow dynamics. IEEE Trans. Intell. Transp. Syst. 20 (2019).
57. Chen, H., Perozzi, B., Hu, Y. & Skiena, S. HARP: hierarchical representation learning for networks. In textit{Proc. 32nd AAAI Conference on Artificial Intelligence/30th Innovative Applications of Artificial Intelligence Conference/8th AAAI Symposium on Educational Advances in Artificial Intelligence (AAAI’18/IAAI’18/EAAI’18) 2127–2134 (AAAI, 2018).
58. Loukas, A. Graph reduction with spectral and cut guarantees. J. Mach. Learn. Res. 20 (2019).
59. Jin, Y., Loukas, A. & JaJa, J. Graph coarsening with preserved spectral properties. In textit{Proc. International Conference on Artificial Intelligence and Statistics 4452–4462 (PMLR, 2020).
60. Fahrbach, M., Goranci, G., Peng, R., Sachdeva, S. & Wang, C. Faster graph embeddings via coarsening. In textit{Proc. International Conference on Machine Learning 2953–2963 (PMLR, 2020).
61. Liang, J., Gurukar, S. & Parthasarathy, S. MILE: A multi-level framework for scalable graph embedding. In textit{Proc. International AAAI Conference on Web and Social Media Vol. 15, 361–372 (AAAI, 2021).
62. Huang, Z., Zhang, S., Xi, C., Liu, T. & Zhou, M. Scaling up graph neural networks via graph coarsening. In textit{Proc. 27th ACM SIGKDD Conference on Knowledge Discovery and Data Mining 675–684 (ACM, 2021).
63. Zhang, Z., Ghavasieh, A., Zhang, J. & De Domenico, M. Coarse-graining network flow through statistical physics and machine learning. Nat. Commun. 16 (2025).
64. Yedidia, J. S. & Bouchaud, J.-P. Renormalization group approach to error-correcting codes. J. Phys. A 36 (2003).
65. Klein, B. & Hoel, E. The emergence of informative higher scales in complex networks. Complexity 2020 (2020).
66. Faccin, M., Schaub, M. T. & Delvenne, J.-C. Entrograms and coarse graining of dynamics on complex networks. J. Complex Netw. 6 (2017).
67. Boguñá, M. Network geometry. Nat. Rev. Phys. 3 (2021).
68. García-Pérez, G., Boguñá, M. & Serrano, M. Á. Multiscale unfolding of real networks by geometric renormalization. Nat. Phys. 14 (2018).
69. Zheng, M., García-Pérez, G., Boguñá, M. & Serrano, M. Á. Scaling up real networks by geometric branching growth. Proc. Natl Acad. Sci. USA (2021).
70. Zheng, M., García-Pérez, G., Boguñá, M. & Serrano, M. Á. Geometric renormalization of weighted networks. Commun. Phys. 7 (2024).
71. Kolk, J., Serrano, M. & Boguñá, M. Renormalization of networks with weak geometric coupling. Phys. Rev. E 110 (2024).
72. Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. & Boguñá, M. Hyperbolic geometry of complex networks. Phys. Rev. E 82 (2010).
73. Gilbert, E. N. Random plane networks. J. Soc. Indust. Appl. Math. 9 (1961).
74. Penrose, M. Random Geometric Graphs Vol. 5 (Oxford Univ. Press, 2003).
75. Hoff, P. D., Raftery, A. E. & Handcock, M. S. Latent space approaches to social network analysis. J. Am. Stat. Assoc. 97 (2002).
76. Boguñá, M., Krioukov, D., Almagro, P. & Serrano, M. Á. Small worlds and clustering in spatial networks. Phys. Rev. Res. 2 (2020).
77. Kolk, J., Serrano, M. Á. & Boguñá, M. An anomalous topological phase transition in spatial random graphs. Commun. Phys. 5 (2022).
78. Kolk, J., Serrano, M. Á. & Boguñá, M. Random graphs and real networks with weak geometric coupling. Phys. Rev. Res. 6 (2024).
79. García-Pérez, G., Allard, A., Serrano, M. Á. & Boguñá, M. Mercator: uncovering faithful hyperbolic embeddings of complex networks. New J. Phys. 21 (2019).
80. Jankowski, R., Allard, A., Boguñá, M. & Serrano, M. Á. The D-Mercator method for the multidimensional hyperbolic embedding of real networks. Nat. Commun. 14 (2023).
81. Boguñá, M., Papadopoulos, F. & Krioukov, D. Sustaining the internet with hyperbolic mapping. Nat. Commun. 1 (2010).
82. Gugelmann, L., Panagiotou, K. & Peter, U. Random hyperbolic graphs: degree sequence and clustering. In textit{Proc. Automata, Languages and Programming, 39th International Colloquium (ICALP 2012) (eds Czumaj, A. et al.) 573–585 (Springer, 2012).
83. Candellero, E. & Fountoulakis, N. Clustering and the hyperbolic geometry of complex networks. Internet Math. 12 (2016).
84. Fountoulakis, N., Hoorn, P., Müller, T. & Schepers, M. Clustering in a hyperbolic model of complex networks. Electron. J. Probab. 26 (2021).
85. Abdullah, M. A., Fountoulakis, N. & Bode, M. Typical distances in a geometric model for complex networks. Internet Math. (2017).
86. Friedrich, T. & Krohmer, A. On the diameter of hyperbolic random graphs. SIAM J. Discrete Math. 32 (2018).
87. Müller, T. & Staps, M. The diameter of KPKVB random graphs. Adv. Appl. Probab. 51 (2019).
88. Serrano, M. Á., Krioukov, D. & Boguñá, M. Percolation in self-similar networks. Phys. Rev. Lett. 106 (2011).
89. Fountoulakis, N. & Müller, T. Law of large numbers for the largest component in a hyperbolic model of complex networks. Ann. Appl. Probab. 28 (2018).
90. Bianconi, G. & Ziff, R. Topological percolation on hyperbolic simplicial complexes. Phys. Rev. E 98 (2018).
91. Kiwi, M. & Mitsche, D. Spectral gap of random hyperbolic graphs and related parameters. Ann. Appl. Probab. 28 (2018).
92. Papadopoulos, F., Kitsak, M., Serrano, M. Á., Boguñá, M. & Krioukov, D. Popularity versus similarity in growing networks. Nature 489 (2012).
93. Allard, A., Serrano, M. Á., García-Pérez, G. & Boguñá, M. The geometric nature of weights in real complex networks. Nat. Commun. 8 (2017).
94. Serrano, M. Á., Boguñá, M. & Sagués, F. Uncovering the hidden geometry behind metabolic networks. Mol. Biosyst. 8 (2012).
95. Kitsak, M., Papadopoulos, F. & Krioukov, D. Latent geometry of bipartite networks. Phys. Rev. E 95 (2017).
96. Budel, G. & Kitsak, M. Complementarity in complex networks. Preprint at arXiv (2023).
97. Kleineberg, K.-K., Boguñá, M., Serrano, M. Á. & Papadopoulos, F. Hidden geometric correlations in real multiplex networks. Nat. Phys. 12 (2016).
98. Kleineberg, K.-K., Buzna, L., Papadopoulos, F., Boguñá, M. & Serrano, M. Á. Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks. Phys. Rev. Lett. 118 (2017).
99. Zuev, K., Boguñá, M., Bianconi, G. & Krioukov, D. Emergence of soft communities from geometric preferential attachment. Sci. Rep. 5 (2015).
100. García-Pérez, G., Serrano, M. Á. & Boguñá, M. Soft communities in similarity space. J. Stat. Phys. 173 (2018).
101. Muscoloni, A. & Cannistraci, C. V. A nonuniform popularity-similarity optimization (nPSO) model to efficiently generate realistic complex networks with communities. New J. Phys. 20 (2018).
102. Allard, A., Serrano, M. Á. & Boguñá, M. Geometric description of clustering in directed networks. Nat. Phys. 20 (2024).
103. Aliakbarisani, R., Ángeles Serrano, M. & Boguñá, M. Feature-enriched hyperbolic network geometry. Preprint at arXiv (2023).
104. Zheng, M., Allard, A., Hagmann, P., Alemán-Gómez, Y. & Serrano, M. Á. Geometric renormalization unravels self-similarity of the multiscale human connectome. Proc. Natl Acad. Sci. USA 117 (2020).
105. Barrat, A., Barthelemy, M., Pastor-Satorras, R. & Vespignani, A. The architecture of complex weighted networks. Proc. Natl Acad. Sci. USA 101 (2004).
106. Serrano, M. Á., Boguñá, M. & Pastor-Satorras, R. Correlations in weighted networks. Phys. Rev. E 74 (2006).
107. Nolan, J. P. Univariate Stable Distributions: Models for Heavy Tailed Data (Springer Nature, 2019).
108. Caldarelli, G., Gabrielli, A., Gili, T. & Villegas, P. Laplacian renormalization group: an introduction to heterogeneous coarse-graining. J. Stat. Mech. 2024 (2024).
109. Domenico, M. & Biamonte, J. Spectral entropies as information-theoretic tools for complex network comparison. Phys. Rev. X 6 (2016).
110. Masuda, N., Porter, M. A. & Lambiotte, R. Random walks and diffusion on networks. Phys. Rep. 716 (2017).
111. Kadanoff, L. P. Notes on Migdal’s recursion formulas. Ann. Phys. 100 (1976).
112. Migdal, A. A. Phase transitions in gauge and spin-lattice systems. Zh. Eksp. Teor. Fiz. 69 (1975).
113. Wilson, K. G. & Kogut, J. The renormalization group and the $$varepsilo$$ expansion. Phys. Rep. 12 (1974).
114. Newman, M. E. J. Networks: An Introduction (Oxford Univ. Press, 2010).
115. Moretti, P. & Zaiser, M. Network analysis predicts failure of materials and structures. Proc. Natl Acad. Sci. USA 116 (2019).
116. Burioni, R. & Cassi, D. Random walks on graphs: ideas, techniques and results. J. Phys. A 38 (2005).
117. Villegas, P., Gabrielli, A., Poggialini, A. & Gili, T. Multi-scale Laplacian community detection in heterogeneous networks. Phys. Rev. Res. 7 (2025).
118. Kadanoff, L. P. Scaling laws for Ising models near Tc. Phys. Phys. Fiz. 2 (1966).
119. Wilson, K. G. Problems in physics with many scales of length. Sci. Am. 241 (1979).
120. Hinrichsen, H. Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49 (2000).
121. Dornic, I., Chaté, H. & Muñoz, M. A. Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions. Phys. Rev. Lett. 94 (2005).
122. Burioni, R. & Cassi, D. Geometrical universality in vibrational dynamics. Mod. Phys. Lett. B 11 (1997).
123. Cassi, D. Phase transitions and random walks on graphs: a generalization of the Mermin–Wagner theorem to disordered lattices, fractals, and other discrete structures. Phys. Rev. Lett. 68 (1992).
124. Reitz, M. & Bianconi, G. The higher-order spectrum of simplicial complexes: a renormalization group approach. J. Phys. A 53 (2020).
125. Bianconi, G. & Dorogovstev, S. N. The spectral dimension of simplicial complexes: a renormalization group theory. J. Stat. Mech. 2020 (2020).
126. Nurisso, M. et al. Higher-order Laplacian renormalization. Nat. Phys. (2025).
127. Cheng, A., Xu, Y., Sun, P. & Tian, Y. A simplex path integral and a simplex renormalization group for high-order interactions. Rep. Prog. Phys. 87 (2024).
128. Ghavasieh, A., Nicolini, C. & Domenico, M. Statistical physics of complex information dynamics. Phys. Rev. E 102 (2020).
129. Villegas, P., Gabrielli, A., Santucci, F., Caldarelli, G. & Gili, T. Laplacian paths in complex networks: information core emerges from entropic transitions. Phys. Rev. Res. 4 (2022).
130. Binney, J. J., Dowrick, N. J., Fisher , A. J. & Newman, M. E. The Theory of Critical Phenomena: An Introduction to the Renormalization Group (Oxford Univ. Press, 1992).
131. Pathria, R. K. & Beale, P. D. Statistical Mechanics (Elsevier/Academic, 2011).
132. Holland, P. W., Laskey, K. B. & Leinhardt, S. Stochastic blockmodels: first steps. Soc. Networks 5 (1983).
133. Burioni, R. & Cassi, D. Universal properties of spectral dimension. Phys. Rev. Lett. 76 (1996).
134. Moretti, P. & Muñoz, M. A. Griffiths phases and the stretching of criticality in brain networks. Nat. Commun. 4 (2013).
135. Poggialini, A., Villegas, P., Muñoz, M. A. & Gabrielli, A. Networks with many structural scales: a renormalization group perspective. Phys. Rev. Lett. 134 (2024).
136. de C. Loures, M., Piovesana, A. A. & Brum, J. A. Laplacian coarse graining in complex networks. Preprint at arXiv (2023).
137. Garuccio, E., Lalli, M. & Garlaschelli, D. Multiscale network renormalization: scale-invariance without geometry. Phys. Rev. Res. 5 (2023).
138. Avena, L., Garlaschelli, D., Hazra, R. S. & Lalli, M. Inhomogeneous random graphs with infinite-mean fitness variables. Preprint at arXiv (2022).
139. Lalli, M. & Garlaschelli, D. Geometry-free renormalization of directed networks: scale-invariance and reciprocity. Preprint at arXiv (2024).
140. Uchaikin, V. V. & Zolotarev, V. M. Chance and Stability: Stable Distributions and their Applications (Walter de Gruyter, 2011).
141. Samorodnitsky, G. & Taqqu, M. S. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Chapman & Hall, 1994).
142. Lévy, P. L’addition des variables aléatoires définies sur une circonférence. Bull. Soc. Math. Fr. 67 (1939).
143. Balkema, A. A. & Resnick, S. I. Max-infinite divisibility. J. Appl. Probab. 14 (1977).
144. Giné, E., Hahn, M. G. & Vatan, P. Max-infinitely divisible and max-stable sample continuous processes. Probab. Theory Relat. Fields 87 (1990).
145. Verteletskyi, V. Renormalization of Networks with Weighted Links. MSc thesis, Leiden Univ. (2022).
146. Garlaschelli, D. & Loffredo, M. I. Patterns of link reciprocity in directed networks. Phys. Rev. Lett. 93 (2004).
147. Garlaschelli, D. & Loffredo, M. I. Multispecies grand-canonical models for networks with reciprocity. Phys. Rev. E 73 (2006).
148. Squartini, T., Picciolo, F., Ruzzenenti, F. & Garlaschelli, D. Reciprocity of weighted networks. Sci. Rep. 3 (2013).
149. Gallo, A., Saracco, F., Lambiotte, R., Garlaschelli, D. & Squartini, T. Patterns of link reciprocity in directed, signed networks. Phys. Rev. E 111 (2025).
150. Ialongo, L. N., Bangma, S., Jansen, F. & Garlaschelli, D. Multi-scale reconstruction of large supply networks. Preprint at arXiv (2024).
151. Milocco, R., Jansen, F. & Garlaschelli, D. Multi-scale node embeddings for graph modeling and generation. Preprint at arXiv (2024).
152. Caldarelli, G., Capocci, A., Los Rios, P. & Munoz, M. A. Scale-free networks from varying vertex intrinsic fitness. Phys. Rev. Lett. 89 (2002).
153. Dehghan-Kooshkghazi, A., Kamiński, B., Kraiński, Ł., Prałat, P. & Théberge, F. Evaluating node embeddings of complex networks. J. Complex Netw. 10 (2022).
154. van der Hofstad, R. Random Graphs and Complex Networks (Cambridge Univ. Press, 2024).
155. Hattori, K., Hattori, T. & Watanabe, H. Gaussian field theories on general networks and the spectral dimensions. Prog. Theor. Phys. Suppl. 92 (1987).
156. Burioni, R. & Cassi, D. Random walks on graphs: ideas, techniques and results. J. Phys. A 38 (2005).
157. Boettcher, S. & Brunson, C. T. Renormalization group for critical phenomena in complex networks. Front. Physiol. 2 (2011).
158. Dorogovtsev, S. N. Renormalization group for evolving networks. Phys. Rev. E 67 (2003).
159. Meshulam, L., Gauthier, J. L., Brody, C. D., Tank, D. W. & Bialek, W. Coarse graining, fixed points, and scaling in a large population of neurons. Phys. Rev. Lett. 123 (2019).
160. Griffiths, R. B. Nonanalytic behavior above the critical point in a random Ising ferromagnet. Phys. Rev. Lett. 23 (1969).
161. Munoz, M. A., Juhász, R., Castellano, C. & Ódor, G. Griffiths phases on complex networks. Phys. Rev. Lett. 105 (2010).
162. Amari, S.-i. Information Geometry and Its Applications (Springer, 2016).
163. Berman, D. S., Klinger, M. S. & Stapleton, A. G. Bayesian renormalization. Mach. Learn. Sci. Tech. 4 (2023).
164. Raju, A., Machta, B. B. & Sethna, J. P. Information loss under coarse graining: a geometric approach. Phys. Rev. E 98 (2018).
165. Machta, B. B., Chachra, R., Transtrum, M. K. & Sethna, J. P. Parameter space compression underlies emergent theories and predictive models. Science 342 (2013).
166. Har-Shemesh, O., Quax, R., Hoekstra, A. G. & Sloot, P. M. A. Information geometric analysis of phase transitions in complex patterns: the case of the Gray-Scott reaction–diffusion model. J. Stat. Mech. 4 (2016).
167. Ehrenfest, P. & Ehrenfest, T. Begriffliche Grundlagen der statistischen Auffassung in der Mechanik (Springer, 1907).
168. Niemeijer, T. & Leeuwen, J. M. J. Wilson theory for spin systems on a triangular lattice. Phys. Rev. Lett. 31 (1973).
169. Kadanoff, L. P. Variational principles and approximate renormalization group calculations. Phys. Rev. Lett. 34 (1975).
170. Southern, B. W. Kadanoff’s variational renormalisation group method: the Ising model on the square and triangular lattices. J. Phys. A 11 (1978).
171. Jan, N. & Glazer, A. Kadanoff’s approximate renormalization group transformation applied to the triangular Ising lattice. Physica A 91 (1978).
172. Nijs, M. & Knops, H. Variational renormalization method and the Potts model. Physica A 93 (1978).
173. Castro, C. & Jona-Lasinio, G. On the microscopic foundation of scaling laws. Phys. Lett. A 29 (1969).
174. Gell-Mann, M. & Low, F. E. Quantum electrodynamics at small distances. Phys. Rev. 95 (1954).
175. Zinn-Justin, J. Quantum Field Theory and Critical Phenomena (Oxford Univ. Press, 2021).
176. Amit, D. J. & Martin-Mayor, V. Field Theory, the Renormalization Group and Critical Phenomena (World Scientific, 2005).
177. Täuber, U. C. Renormalization group: applications in statistical physics. Nucl. Phys. B 228 (2012).
178. Pelissetto, A. & Vicari, E. Critical phenomena and renormalization-group theory. Phys. Rep. 368 (2002).
179. Efrati, E., Wang, Z., Kolan, A. & Kadanoff, L. P. Real-space renormalization in statistical mechanics. Rev. Mod. Phys. 86 (2014).
180. Gefen, Y., Mandelbrot, B. B. & Aharony, A. Critical phenomena on fractal lattices. Phys. Rev. Lett. 45 (1980).
181. Gefen, Y., Aharony, A., Mandelbrot, B. B. & Kirkpatrick, S. Solvable fractal family, and its possible relation to the backbone at percolation. Phys. Rev. Lett. 47 (1981).
182. Phani, M. K. & Dhar, D. Real-space renormalisation group: application to directed percolation. J. Phys. C 15 (1982).
183. Gefen, Y., Aharony, A. & Mandelbrot, B. B. Phase transitions on fractals. I. Quasi-linear lattices. J. Phys. A 16 (1983).
184. Gefen, Y., Aharony, A., Shapir, Y. & Mandelbrot, B. B. Phase transitions on fractals. II. Sierpinski gaskets. J. Phys. A 17 (1984).
185. Gefen, Y., Aharony, A. & Mandelbrot, B. B. Phase transitions on fractals. III. Infinitely ramified lattices. J. Phys. A 17 (1984).
186. Das, D., Dey, S., Jacobsen, J. L. & Dhar, D. Critical behavior of loops and biconnected clusters on fractals of dimension d < 2. J. Phys. A 41 (2008).
187. Della Morte, M. & Sannino, F. Renormalization group approach to pandemics as a time-dependent SIR model. Front. Phys. (2021).
188. Cavagna, A. Dynamic scaling in natural swarms. Nat. Phys. 13 (2017).
189. Cavagna, A. Dynamical renormalization group approach to the collective behavior of swarms. Phys. Rev. Lett. (2019).
190. Cavagna, A. Natural swarms in 3.99 dimensions. Nat. Phys. 19 (2023).
191. Koch-Janusz, M. & Ringel, Z. Mutual information, neural networks and the renormalization group. Nat. Phys. 14 (2018).
192. Li, S.-H. & Wang, L. Neural network renormalization group. Phys. Rev. Lett. (2018).
193. Hu, H.-Y., Li, S.-H., Wang, L. & You, Y.-Z. Machine learning holographic mapping by neural network renormalization group. Phys. Rev. Res. 2 (2020).
194. Caso, F., Trappolini, G., Bacciu, A., Liò, P. & Silvestri, F. Renormalized graph neural networks. Preprint at arXiv (2023).
195. Young, A. P. & Stinchcombe, R. B. A renormalization group theory for percolation problems. J. Phys. C 8 (1975).
196. Reynolds, P. J., Stanley, H. E. & Klein, W. A real-space renormalization group for site and bond percolation. J. Phys. C 10 (1977).
197. Galam, S. Real space renormalization group and totalitarian paradox of majority rule voting. Physica A 285 (2000).
198. Kogan, O., Rogers, J. L., Cross, M. C. & Refael, G. Renormalization group approach to oscillator synchronization. Phys. Rev. E 80 (2009).
199. Östborn, P. Renormalization of oscillator lattices with disorder. Phys. Rev. E 79 (2009).
200. Garlaschelli, D., Hollander, F., Meylahn, J. & Zeegers, B. Synchronization of phase oscillators on the hierarchical lattice. J. Stat. Phys. 174 (2019).
201. Levin, M. & Nave, C. P. Tensor renormalization group approach to two-dimensional classical lattice models. Phys. Rev. Lett. (2007).
202. Evenbly, G. & Vidal, G. Tensor network renormalization. Phys. Rev. Lett. (2015).
203. Bal, M., Mariën, M., Haegeman, J. & Verstraete, F. Renormalization group flows of Hamiltonians using tensor networks. Phys. Rev. Lett. (2017).
204. Lenggenhager, P. M., Gökmen, D. E., Ringel, Z., Huber, S. D. & Koch-Janusz, M. Optimal renormalization group transformation from information theory. Phys. Rev. X 10 (2020).
205. Gökmen, D. E., Ringel, Z., Huber, S. D. & Koch-Janusz, M. Statistical physics through the lens of real-space mutual information. Phys. Rev. Lett. 127 (2021).
206. Gordon, A., Banerjee, A., Koch-Janusz, M. & Ringel, Z. Relevance in the renormalization group and in information theory. Phys. Rev. Lett. 126 (2021).
207. Sarra, L., Aiello, A. & Marquardt, F. Renormalized mutual information for artificial scientific discovery. Phys. Rev. Lett. 126 (2021).
208. Gökmen, D. E., Ringel, Z., Huber, S. D. & Koch-Janusz, M. Symmetries and phase diagrams with real-space mutual information neural estimation. Phys. Rev. E 104 (2021).
209. Canet, L., Delamotte, B., Deloubrière, O. & Wschebor, N. Nonperturbative renormalization-group study of reaction–diffusion processes. Phys. Rev. Lett. 92 (2004).
210. Canet, L., Chaté, H. & Delamotte, B. Quantitative phase diagrams of branching and annihilating random walks. Phys. Rev. Lett. 92 (2004).
211. Canet, L., Chaté, H., Delamotte, B., Dornic, I. & Muñoz, M. A. Nonperturbative fixed point in a nonequilibrium phase transition. Phys. Rev. Lett. 95 (2005).
212. Canet, L., Chaté, H. & Delamotte, B. General framework of the non-perturbative renormalization group for non-equilibrium steady states. J. Phys. A 44 (2011).
213. Tarpin, M., Benitez, F., Canet, L. & Wschebor, N. Nonperturbative renormalization group for the diffusive epidemic process. Phys. Rev. E 96 (2017).