- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Geometric criticality in scale-invariant networks

Lorenzo Lucarini1,2, Giulio Cimini1,2, and Pablo Villegas2,3,*

  • *Contact author: pablo.villegas@cref.it
PDF

Phys. Rev. Research 8, 013330 – Published 26 March, 2026

DOI: https://doi.org/10.1103/75gt-z6x6

Abstract

Dimension in physical systems determines universal properties at criticality. Yet, the impact of structural perturbations on dimensionality remains largely unexplored. Here, we characterize the attraction basins of structural fixed points in scale-invariant networks from a renormalization group perspective, demonstrating that basin stability connects to a structural phase transition. This topology-dependent effect, which we term geometric criticality, triggers a geometric breakdown hitherto unknown, which induces nontrivial fractal dimensions and unveils hidden Laplacian renormalization group flows toward unstable structural fixed points. Our systematic study of how networks and lattices respond to disorder paves the way for future analysis of nonergodic behavior induced by quenched disorder.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (47)

  1. R. F. Betzel and D. S. Bassett, Specificity and robustness of long-distance connections in weighted, interareal connectomes, Proc. Natl. Acad. Sci. USA 115, E4880 (2018).
  2. P. Martin, A. McGovern, G. Orozco, et al., Capture Hi-C reveals novel candidate genes and complex long-range interactions with related autoimmune risk loci, Nat. Commun. 6, 10069 (2015).
  3. A. Simonov and A. L. Goodwin, Designing disorder into crystalline materials, Nat. Rev. Chem. 4, 657 (2020).
  4. O. Sporns, Networks of the Brain (MIT Press, Cambridge, MA, 2010).
  5. M. Kaiser and C. C. Hilgetag, Nonoptimal component placement, but short processing paths, due to long-distance projections in neural systems, PLoS Comput. Biol. 2, e95 (2006).
  6. E. Bullmore and O. Sporns, Complex brain networks: Graph theoretical analysis of structural and functional systems, Nat. Rev. Neurosci. 10, 186 (2009).
  7. R. F. Betzel and D. S. Bassett, Multi-scale brain networks, NeuroImage 160, 73 (2017).
  8. F. Schein and P. C. Aichelburg, Traversable wormholes in geometries of charged shells, Phys. Rev. Lett. 77, 4130 (1996).
  9. Y.-C. Chen, Z.-G. Huang, and Y.-C. Lai, Controlling extreme events on complex networks, Sci. Rep. 4, 6121 (2014).
  10. R. E. Morris and J. Čejka, Exploiting chemically selective weakness in solids as a route to new porous materials, Nat. Chem. 7, 381 (2015).
  11. W. J. Roth, et al., A family of zeolites with controlled pore size prepared using a top-down method, Nat. Chem. 5, 628 (2013).
  12. Z. Zhang, F. Zuo, C. Wan, A. Dutta, J. Kim, J. Rensberg, R. Nawrodt, H. H. Park, T. J. Larrabee, X. Guan, Y. Zhou, S. M. Prokes, C. Ronning, V. M. Shalaev, A. Boltasseva, M. A. Kats, and S. Ramanathan, Evolution of metallicity in vanadium dioxide by creation of oxygen vacancies, Phys. Rev. Appl. 7, 034008 (2017).
  13. J. J. Brown, Y. Shao, Z. Ke, and A. J. Page, Anion vacancy effects on structural stability in oxynitrides, Mater. Adv. 2, 2398 (2021).
  14. J. Zhou, Vacancy defects in skyrmion crystals: Structural and magnetic effects, J. Magn. Magn. Mater. 562, 170147 (2023).
  15. D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature (London) 393, 440 (1998).
  16. M. Newman and D. Watts, Renormalization group analysis of the small-world network model, Phys. Lett. A 263, 341 (1999).
  17. M. E. J. Newman, Models of the small world, J. Stat. Phys. 101, 819 (2000).
  18. A. P. Millán, G. Gori, F. Battiston, T. Enss, and N. Defenu, Complex networks with tuneable spectral dimension as a universality playground, Phys. Rev. Res. 3, 023015 (2021).
  19. T. Vojta, Rare region effects at classical, quantum and nonequilibrium phase transitions, J. Phys. A: Math. Gen. 39, R143 (2006).
  20. R. B. Griffiths, Nonanalytic behavior above the critical point in a random Ising ferromagnet, Phys. Rev. Lett. 23, 17 (1969).
  21. V. Westphal, W. Kleemann, and M. D. Glinchuk, Diffuse phase transitions and random-field-induced domain states of the “relaxor” ferroelectric PbMg1/3Nb2/3O3, Phys. Rev. Lett. 68, 847 (1992).
  22. P. Timonin, Griffiths’ phase in dilute ferroelectrics, Ferroelectrics 199, 69 (1997).
  23. P. Villegas, T. Gili, G. Caldarelli, and A. Gabrielli, Laplacian renormalization group for heterogeneous networks, Nat. Phys. 19, 445 (2023).
  24. A. Poggialini, P. Villegas, M. A. Muñoz, and A. Gabrielli, Networks with many structural scales: A renormalization group perspective, Phys. Rev. Lett. 134, 057401 (2025).
  25. D. Cassi, Local vs average behavior on inhomogeneous structures: Recurrence on the average and a further extension of Mermin-Wagner theorem on graphs, Phys. Rev. Lett. 76, 2941 (1996).
  26. D. Cassi, 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, 3631 (1992).
  27. P. Villegas, Strange attractors in complex networks, Phys. Rev. E 111, L042301 (2025).
  28. A. Gabrielli, D. Garlaschelli, S. Patil, et al., Network renormalization, Nat. Rev. Phys. 7, 203 (2025).
  29. M. De Domenico and J. Biamonte, Spectral entropies as information-theoretic tools for complex network comparison, Phys. Rev. X 6, 041062 (2016).
  30. P. Villegas, A. Gabrielli, F. Santucci, G. Caldarelli, and T. Gili, Laplacian paths in complex networks: Information core emerges from entropic transitions, Phys. Rev. Res. 4, 033196 (2022).
  31. L. Falsi, P. Villegas, T. Gili, A. J. Agranat, and E. DelRe, Topological protection breakdown: A route to frustrated ferroelectricity, Phys. Rev. Res. 7, 043038 (2025).
  32. L. Donetti and C. Destri, The statistical geometry of scale-free random trees, J. Phys. A: Math. Gen. 37, 6003 (2004).
  33. H. D. Rozenfeld, S. Havlin, and D. Ben-Avraham, Fractal and transfractal recursive scale-free nets, New J. Phys. 9, 175 (2007).
  34. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Pseudofractal scale-free web, Phys. Rev. E 65, 066122 (2002).
  35. P. Holme and B. J. Kim, Growing scale-free networks with tunable clustering, Phys. Rev. E 65, 026107 (2002).
  36. P. Moretti and M. A. Muñoz, Griffiths phases and the stretching of criticality in brain networks, Nat. Commun. 4, 2521 (2013).
  37. See Supplemental Material at http://link.aps.org/supplemental/10.1103/75gt-z6x6 for videos supporting the findings in the main text.
  38. S. Alexander and R. Orbach, Density of states on fractals: ‘Fractons’, J. Physique Lett. 43, 625 (1982).
  39. P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Physica D 9, 189 (1983).
  40. P. Villegas, T. Gili, G. Caldarelli, and A. Gabrielli, Evidence of scale-free clusters of vegetation in tropical rainforests, Phys. Rev. E 109, L042402 (2024).
  41. B. B. Mandelbrot, The Fractal Geometry of Nature, revised and enlarged ed. (W. H. Freeman and Co., New York, 1983), p. 495.
  42. A. Gabrielli, F. S. Labini, M. Joyce, and L. Pietronero, Statistical Physics for Cosmic Structures (Springer Science & Business Media, New York, 2006).
  43. Y. Gefen, Y. Meir, B. B. Mandelbrot, and A. Aharony, Geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices, Phys. Rev. Lett. 50, 145 (1983).
  44. J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. Newman, The Theory of Critical Phenomena: An Introduction to the Renormalization Group (Oxford University Press, Oxford, 1992).
  45. D. J. Amit and V. Martin-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena, 3rd ed. (World Scientific, Singapore, 2005).
  46. L. P. Kadanoff, in Proceedings of the Enrico Fermi Summer School of Physics, Varenna 1970, edited by M. S. Green (Academic Press, London/New York, 1971), chap. 2, pp. 100–117.
  47. Y. Gefen, B. B. Mandelbrot, and A. Aharony, Critical phenomena on fractal lattices, Phys. Rev. Lett. 45, 855 (1980).

Outline

Information

Phys. Rev. Research 8, 013330– Published 26 March, 2026
  • Received 15 July 2025
  • Accepted 26 January 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation