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Network geometry

Nature Reviews Physics volume 3, pages 114–135 (2021)Cite this article

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Abstract

Networks are finite metric spaces, with distances defined by the shortest paths between nodes. However, this is not the only form of network geometry: two others are the geometry of latent spaces underlying many networks and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale invariance, self-similarity and other forms of fundamental symmetries in networks. Network geometry is also of great use in a variety of practical applications, from understanding how the brain works to routing in the Internet. We review the most important theoretical and practical developments dealing with these approaches to network geometry and offer perspectives on future research directions and challenges in this frontier in the study of complexity.

Key points

  • The fractal geometry of networks enables casting the self-similar symmetries underlying the organization of complex systems under the three pillars of scaling, universality and renormalization.

  • Latent metric spaces with hyperbolic geometry provide a natural explanation for the architecture of real complex networks, including small-worldness, degree heterogeneity, clustering, community structure, symmetries and navigability.

  • Multiscale unfolding of complex networks is possible by means of a geometric renormalization technique that uncovers self-similarity at different scales.

  • Network dynamical processes induce kinematic distances that characterize an effective geometry of a system’s function, which cannot be obtained by purely topological latent geometry.

  • Network geometry enhances our understanding of complex systems across their multiple scales of organization and of collective phenomena emerging from their information exchange.

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Fig. 1: Structural self-similarity, renormalization-group flow and universality.
Fig. 2: Network evolution and transport theory.
Fig. 3: Networks in latent geometry.
Fig. 4: Multiscale unfolding of network structure and function by geometric renormalization.
Fig. 5: Geometry induced by spreading dynamics and universal temporal distance.
Fig. 6: Diffusion geometry of complex networks and multilayer networks.

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References

  1. Cimini, G. et al. The statistical physics of real-world networks. Nat. Rev. Phys. 1, 58–71 (2019).

    Google Scholar 

  2. Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

    MATH  ADS  Google Scholar 

  3. Barabási, A.-L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999).

    MathSciNet  MATH  ADS  Google Scholar 

  4. Ravasz, E., Somera, A. L., Mongru, D. A., Oltvai, Z. N. & Barabási, A.-L. Hierarchical organization of modularity in metabolic networks. Science 297, 1551–1555 (2002).

    ADS  Google Scholar 

  5. Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008).

    ADS  Google Scholar 

  6. Gao, J., Buldyrev, S. V., Stanley, H. E. & Havlin, S. Networks formed from interdependent networks. Nat. Phys. 8, 40–48 (2012).

    Google Scholar 

  7. D’Souza, R. M. & Nagler, J. Anomalous critical and supercritical phenomena in explosive percolation. Nat. Phys. 11, 531–538 (2015).

    Google Scholar 

  8. Gao, J., Barzel, B. & Barabási, A.-L. Universal resilience patterns in complex networks. Nature 530, 307–312 (2016).

    ADS  Google Scholar 

  9. Bianconi, G. Interdisciplinary and physics challenges of network theory. Europhys. Lett. 111, 56001 (2015).

    ADS  Google Scholar 

  10. Estrada, E. The Structure of Complex Networks: Theory and Applications (Oxford Univ. Press, 2012).

  11. Garlaschelli, D., Capocci, A. & Caldarelli, G. Self-organized network evolution coupled to extremal dynamics. Nat. Phys. 3, 813–817 (2007).

    MATH  Google Scholar 

  12. Garlaschelli, D. & Loffredo, M. Generalized Bose–Fermi statistics and structural correlations in weighted networks. Phys. Rev. Lett. 102, 038701 (2009).

    ADS  Google Scholar 

  13. Kalinin, N. et al. Self-organized criticality and pattern emergence through the lens of tropical geometry. Proc. Natl Acad. Sci. USA 115, E8135–E8142 (2018).

    MathSciNet  MATH  Google Scholar 

  14. Song, C., Havlin, S. & Makse, H. A. Self-similarity of complex networks. Nature 433, 392–395 (2005).

    ADS  Google Scholar 

  15. Gallos, L. K., Song, C., Havlin, S. & Makse, H. A. Scaling theory of transport in complex biological networks. Proc. Natl Acad. Sci. USA 104, 7746–7751 (2007).

    ADS  Google Scholar 

  16. Condamin, S., Bénichou, O., Tejedor, V., Voituriez, R. & Klafter, J. First-passage times in complex scale-invariant media. Nature 450, 77–80 (2007).

    ADS  Google Scholar 

  17. Radicchi, F., Ramasco, J., Barrat, A. & Fortunato, S. Complex networks renormalization: flows and fixed points. Phys. Rev. Lett. 101, 148701 (2008).

    ADS  Google Scholar 

  18. Rozenfeld, H. D., Song, C. & Makse, H. A. Small-world to fractal transition in complex networks: a renormalization group approach. Phys. Rev. Lett. 104, 025701 (2010).

    ADS  Google Scholar 

  19. Song, C., Havlin, S. & Makse, H. A. Origins of fractality in the growth of complex networks. Nat. Phys. 2, 275–281 (2006).

    Google Scholar 

  20. Serrano, M. Á., Krioukov, D. & Boguñá, M. Self-similarity of complex networks and hidden metric spaces. Phys. Rev. Lett. 100, 078701 (2008).

    ADS  Google Scholar 

  21. Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. & Boguñá, M. Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106 (2010).

    MathSciNet  ADS  Google Scholar 

  22. Sorokin, A. P. Social Mobility (Harper, 1927).

  23. McFarland, D. D. & Brown, D. J. Social Distance as a Metric: A Systematic Introduction to Smallest Space Analysis 213–252 (John Wiley, 1973).

  24. Hoff, P. D., Raftery, A. E. & Handcock, M. S. Latent space approaches to social network analysis. J. Am. Stat. Assoc. 97, 1090–1098 (2002).

    MathSciNet  MATH  Google Scholar 

  25. Boguñá, M., Krioukov, D. & Claffy, K. C. Navigability of complex networks. Nat. Phys. 5, 74–80 (2009).

    Google Scholar 

  26. Gulyás, A., Bíró, J. J., Kőrösi, A., Rétvári, G. & Krioukov, D. Navigable networks as Nash equilibria of navigation games. Nat. Commun. 6, 7651 (2015).

    ADS  Google Scholar 

  27. Allard, A. & Serrano, M. Á. Navigable maps of structural brain networks across species. PLoS Comput. Biol. 16, e1007584 (2020).

    ADS  Google Scholar 

  28. Muscoloni, A. & Cannistraci, C. V. Navigability evaluation of complex networks by greedy routing efficiency. Proc. Natl Acad. Sci. USA 116, 1468–1469 (2019).

    Google Scholar 

  29. Zuev, K., Boguñá, M., Bianconi, G. & Krioukov, D. Emergence of soft communities from geometric preferential attachment. Sci. Rep. 5, 9421 (2015).

    ADS  Google Scholar 

  30. García-Pérez, G., Boguñá, M., Allard, A. & Serrano, M. Á. The hidden hyperbolic geometry of international trade: World Trade Atlas 1870–2013. Sci. Rep. 6, 33441 (2016).

    ADS  Google Scholar 

  31. Wang, Z., Li, Q., Jin, F., Xiong, W. & Wu, Y. Hyperbolic mapping of complex networks based on community information. Physica A 455, 104–119 (2016).

    MathSciNet  MATH  ADS  Google Scholar 

  32. García-Pérez, G., Serrano, M. Á. & Boguñá, M. Soft communities in similarity space. J. Stat. Phys. 173, 775–782 (2018).

    MathSciNet  MATH  ADS  Google Scholar 

  33. Muscoloni, A. & Cannistraci, C. Leveraging the nonuniform PSO network model as a benchmark for performance evaluation in community detection and link prediction. New J. Phys. 20, 063022 (2018).

    Google Scholar 

  34. García-Pérez, G., Boguñá, M. & Serrano, M. Á. Multiscale unfolding of real networks by geometric renormalization. Nat. Phys. 14, 583–589 (2018).

    Google Scholar 

  35. Papadopoulos, F., Kitsak, M., Serrano, M. Á., Boguñá, M. & Krioukov, D. Popularity versus similarity in growing networks. Nature 489, 537–540 (2012).

    ADS  Google Scholar 

  36. Zuev, K., Papadopoulos, F. & Krioukov, D. Hamiltonian dynamics of preferential attachment. J. Phys. A 49, 105001 (2016).

    MathSciNet  MATH  ADS  Google Scholar 

  37. Zheng, M., García-Pérez, G., Boguñá, M. & Serrano, M. A. Scaling up real networks by geometric branching growth. Preprint at https://arxiv.org/abs/1912.00704 (2019).

  38. Krioukov, D. & Ostilli, M. Duality between equilibrium and growing networks. Phys. Rev. E 88, 022808 (2013).

    ADS  Google Scholar 

  39. Coifman, R. R. et al. Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. Proc. Natl Acad. Sci. USA 102, 7426–7431 (2005).

    MATH  ADS  Google Scholar 

  40. Brockmann, D. & Helbing, D. The hidden geometry of complex, network-driven contagion phenomena. Science 342, 1337–1342 (2013).

    ADS  Google Scholar 

  41. Taylor, D. et al. Topological data analysis of contagion maps for examining spreading processes on networks. Nat. Commun. 6, 7723 (2015).

    ADS  Google Scholar 

  42. De Domenico, M. Diffusion geometry unravels the emergence of functional clusters in collective phenomena. Phys. Rev. Lett. 118, 168301 (2017).

    ADS  Google Scholar 

  43. Hens, C., Harush, U., Haber, S., Cohen, R. & Barzel, B. Spatiotemporal signal propagation in complex networks. Nat. Phys. 10, 403–412 (2019).

    Google Scholar 

  44. Feder, J. Fractals (Springer Science & Business Media, 2013).

  45. Frisch, U. & Kolmogorov, A. N. Turbulence: The Legacy of AN Kolmogorov (Cambridge Univ. Press, 1995).

  46. Cardy, J. Scaling and Renormalization in Statistical Physics Vol. 5 (Cambridge Univ. Press, 1996).

  47. Lesne, A. & Laguës, M. Scale Invariance: From Phase Transitions to Turbulence (Springer Science & Business Media, 2011).

  48. Bak, P. How Nature Works: The Science of Self-organized Criticality (Springer Science & Business Media, 2013).

  49. Carlson, J. M. & Doyle, J. Complexity and robustness. Proc. Natl Acad. Sci. USA 99, 2538–2545 (2002).

    ADS  Google Scholar 

  50. Bollobás, B. Modern Graph Theory Vol. 184 (Springer Science & Business Media, 2013).

  51. Chung, F. & Lu, L. The average distances in random graphs with given expected degrees. Proc. Natl Acad. Sci. USA 99, 15879–82 (2002).

    MathSciNet  MATH  ADS  Google Scholar 

  52. Cohen, R. & Havlin, S. Scale-free networks are ultrasmall. Phys. Rev. Lett. 90, 058701 (2003).

    ADS  Google Scholar 

  53. Dorogovtsev, S. N., Mendes, J. F. F. & Samukhin, A. N. Metric structure of random networks. Nucl. Phys. B 653, 307–422 (2003).

    MathSciNet  MATH  ADS  Google Scholar 

  54. Mandelbrot, B. B. The Fractal Geometry of Nature Vol. 2 (WH Freeman, 1982).

  55. Harary, F. Graph Theory (Addison-Wesley, 1994).

  56. Wiener, H. Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947).

    Google Scholar 

  57. Havlin, S., Trus, B. & Stanley, H. Cluster-growth model for branched polymers that are “chemically linear”. Phys. Rev. Lett. 53, 1288–1291 (1984).

    ADS  Google Scholar 

  58. Wilson, K. G. The renormalization group and critical phenomena. Rev. Mod. Phys. 55, 583–600 (1983).

    MathSciNet  ADS  Google Scholar 

  59. Efrati, E., Wang, Z., Kolan, A. & Kadanoff, L. P. Real-space renormalization in statistical mechanics. Rev. Mod. Phys. 86, 647 (2014).

    ADS  Google Scholar 

  60. Gallos, L. K., Song, C. & Makse, H. A. Scaling of degree correlations and its influence on diffusion in scale-free networks. Phys. Rev. Lett. 100, 248701 (2008).

    ADS  Google Scholar 

  61. Bunde, A. & Havlin, S. Fractals and Disordered Systems (Springer Science & Business Media, 2012).

  62. Goh, K.-I., Salvi, G., Kahng, B. & Kim, D. Skeleton and fractal scaling in complex networks. Phys. Rev. Lett. 96, 018701 (2006).

    ADS  Google Scholar 

  63. Kim, J. et al. Fractality in complex networks: critical and supercritical skeletons. Phys. Rev. E 75, 016110 (2007).

    ADS  Google Scholar 

  64. Yook, S.-H., Radicchi, F. & Meyer-Ortmanns, H. Self-similar scale-free networks and disassortativity. Phys. Rev. E 72, 045105 (2005).

    ADS  Google Scholar 

  65. Galvão, V. et al. Modularity map of the network of human cell differentiation. Proc. Natl Acad. Sci. USA 107, 5750–5755 (2010).

    ADS  Google Scholar 

  66. Newman, M. E. J. Modularity and community structure in networks. Proc. Natl Acad. Sci. USA 103, 8577–8582 (2006).

    ADS  Google Scholar 

  67. Fortunato, S. Community detection in graphs. Phys. Rep. 486, 75–174 (2010).

    MathSciNet  ADS  Google Scholar 

  68. Stanley, H. E. Scaling, universality, and renormalization: three pillars of modern critical phenomena. Rev. Mod. Phys. 71, S358–S366 (1999).

    Google Scholar 

  69. Radicchi, F., Barrat, A., Fortunato, S. & Ramasco, J. Renormalization flows in complex networks. Phys. Rev. E 79, 026104 (2009).

    MathSciNet  ADS  Google Scholar 

  70. Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena (Oxford Univ. Press, 1971).

  71. Kleinberg, J. Navigation in a small world. Nature 406, 845 (2000).

    ADS  Google Scholar 

  72. Viswanathan, G. M. et al. Optimizing the success of random searches. Nature 401, 911–914 (1999).

    ADS  Google Scholar 

  73. Strogatz, S. H. Complex systems: Romanesque networks. Nature 433, 365–366 (2005).

    ADS  Google Scholar 

  74. Dorogovtsev, S. N. & Mendes, J. F. F. Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford Univ. Press, 2003).

  75. Rozenfeld, H. D., Gallos, L. K., Song, C. & Makse, H. A. in Encyclopedia of Complexity and Systems Science (ed. Meyers R.) 3924–3943 (Springer, 2009).

  76. Kitano, H. Systems biology: a brief overview. Science 295, 1662–1664 (2002).

    ADS  Google Scholar 

  77. Jin, Y., Turaev, D., Weinmaier, T., Rattei, T. & Makse, H. A. The evolutionary dynamics of protein–protein interaction networks inferred from the reconstruction of ancient networks. PLoS ONE 8, e58134 (2013).

    ADS  Google Scholar 

  78. Villani, C. Optimal Transport: Old and New Vol. 338 (Springer Science & Business Media, 2008).

  79. Ben-Avraham, D. & Havlin, S. Diffusion and Reactions in Fractals and Disordered Systems (Cambridge Univ. Press, 2000).

  80. Haynes, C. P. & Roberts, A. P. Generalization of the fractal Einstein law relating conduction and diffusion on networks. Phys. Rev. Lett. 103, 020601 (2009).

    ADS  Google Scholar 

  81. Masuda, N., Porter, M. A. & Lambiotte, R. Random walks and diffusion on networks. Phys. Rep. 716, 1–58 (2017).

    MathSciNet  MATH  ADS  Google Scholar 

  82. Gallos, L. K., Potiguar, F. Q., Andrade Jr, J. S. & Makse, H. A. IMDb network revisited: unveiling fractal and modular properties from a typical small-world network. PLoS ONE 8, e66443 (2013).

    ADS  Google Scholar 

  83. Gallos, L. K., Makse, H. A. & Sigman, M. A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks. Proc. Natl Acad. Sci. USA 109, 2825–2830 (2012).

    ADS  Google Scholar 

  84. Avena-Koenigsberger, A., Misic, B. & Sporns, O. Communication dynamics in complex brain networks. Nat. Rev. Neurosci. 19, 17–33 (2017).

    Google Scholar 

  85. Gallos, L. K., Song, C. & Makse, H. A. A review of fractality and self-similarity in complex networks. Physica A 386, 686–691 (2007).

    ADS  Google Scholar 

  86. Bloch, A. On methods for the construction of networks dual to non-planar networks. Proc. Phys. Soc. 58, 677–694 (1946).

    MathSciNet  MATH  ADS  Google Scholar 

  87. Estrada, E. The communicability distance in graphs. Linear Algebra Appl. 436, 4317–4328 (2012).

    MathSciNet  MATH  Google Scholar 

  88. Gromov, M. in Essays in Group Theory (ed. Gersten S. M.) 75–263 (Mathematical Sciences Research Institute Publications Vol. 8, Springer, 1987).

  89. Nekrashevych, V. Hyperbolic spaces from self-similar group actions. Algebra Discret. Math. 2, 77–86 (2003).

    MathSciNet  MATH  Google Scholar 

  90. Nekrashevych, V. Self-Similar Groups (Mathematical Surveys and Monographs Vol. 117, American Mathematical Society, 2005).

  91. Furstenberg, H. Ergodic Theory and Fractal Geometry Vol. 120 (American Mathematical Society, 2014).

  92. Furstenberg, H. & Weiss, B. Markov processes and Ramsey theory for trees. Comb. Probab. Comput. 12, 547–563 (2003).

    MathSciNet  MATH  Google Scholar 

  93. Dorogovtsev, S. N. & Mendes, J. F. F. Evolution of networks. Adv. Phys. 51, 1079–1187 (2002).

    ADS  Google Scholar 

  94. Buyalo, S. & Schroeder, V. Elements of Asymptotic Geometry (European Mathematical Society Publishing House, 2007).

  95. Grigorchuk, R., Nekrashevych, V. & Šunić, Z. From Self-Similar Groups to Self-Similar Sets and Spectra 175–207 (Birkhäuser, 2015).

  96. Boguñá, M. & Krioukov, D. Navigating ultrasmall worlds in ultrashort time. Phys. Rev. Lett. 102, 058701 (2009).

    ADS  Google Scholar 

  97. Boguñá, M., Papadopoulos, F. & Krioukov, D. Sustaining the Internet with hyperbolic mapping. Nat. Commun. 1, 62 (2010).

    ADS  Google Scholar 

  98. Penrose, M. Random Geometric Graphs (Oxford Univ. Press, 2003).

  99. Boguñá, M., Krioukov, D., Almagro, P. & Serrano, M. Á. Small worlds and clustering in spatial networks. Phys. Rev. Res. 2, 023040 (2020).

    Google Scholar 

  100. van der Hoorn, P., Lippner, G. & Krioukov, D. Sparse maximum-entropy random graphs with a given power-law degree distribution. J. Stat. Phys. 173, 806–844 (2018).

    MathSciNet  MATH  ADS  Google Scholar 

  101. Voitalov, I., van der Hoorn, P., van der Hofstad, R. & Krioukov, D. Scale-free networks well done. Phys. Rev. Res. 1, 033034 (2019).

    Google Scholar 

  102. Cannon, J., Floyd, W., Kenyon, R. & Parry, W. Hyperbolic Geometry 59–116 (MSRI, 1997).

  103. Matoušek, J. in Lectures on Discrete Geometry. Graduate Texts in Mathematics Vol 212. (ed. Matoušek J.) Ch. 15 (Springer, 2002).

  104. Munzner, T. Exploring large graphs in 3D hyperbolic space. IEEE Comput. Graph. Appl. 18, 18–23 (1998).

    Google Scholar 

  105. Aste, T., Dimatteo, T. & Hyde, S. Complex networks on hyperbolic surfaces. Physica A 346, 20–26 (2005).

    ADS  Google Scholar 

  106. Lohsoonthorn, P. Hyperbolic Geometry of Networks. PhD thesis, USC (2003).

  107. Jonckheere, E., Lohsoonthorn, P. & Bonahon, F. Scaled Gromov hyperbolic graphs. J. Graph. Theory 57, 157–180 (2008).

    MathSciNet  MATH  Google Scholar 

  108. Kleinberg, R. Geographic routing using hyperbolic space. In IEEE INFOCOM 2007 - 26th IEEE Int. Conf. Comput. Commun. 1902–1909 (IEEE, 2007).

  109. Ratcliffe, J. Foundations of Hyperbolic Manifolds (Springer, 2006).

  110. Dall, J. & Christensen, M. Random geometric graphs. Phys. Rev. E 66, 016121 (2002).

    MathSciNet  ADS  Google Scholar 

  111. Krioukov, D. et al. Network cosmology. Sci. Rep. 2, 793 (2012).

    Google Scholar 

  112. Allard, A., Serrano, M. Á., García-Pérez, G. & Boguñá, M. The geometric nature of weights in real complex networks. Nat. Commun. 8, 14103 (2017).

    ADS  Google Scholar 

  113. Kleineberg, K.-K., Boguñá, M., Ángeles Serrano, M. & Papadopoulos, F. Hidden geometric correlations in real multiplex networks. Nat. Phys. 12, 1076–1081 (2016).

    Google Scholar 

  114. Kleineberg, K.-K., Buzna, L., Papadopoulos, F., Boguñá, M. & Serrano, M. A. Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks. Phys. Rev. Lett. 118, 218301 (2017).

    ADS  Google Scholar 

  115. Muscoloni, A. & Cannistraci, C. V. A nonuniform popularity-similarity optimization (nPSO) model to efficiently generate realistic complex networks with communities. New J. Phys. 20, 052002 (2018).

    MathSciNet  ADS  Google Scholar 

  116. Maldacena, J. The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998).

    MathSciNet  MATH  ADS  Google Scholar 

  117. Gugelmann, L., Panagiotou, K. & Peter, U. Random Hyperbolic Graphs: Degree Sequence and Clustering 573–585 (Springer, 2012).

  118. Fountoulakis, N. On a geometrization of the Chung–Lu model for complex networks. J. Complex Netw. 3, 361–387 (2015).

    MathSciNet  MATH  Google Scholar 

  119. Candellero, E. & Fountoulakis, N. Clustering and the hyperbolic geometry of complex networks. Internet Math. 12, 2–53 (2016).

    MathSciNet  MATH  Google Scholar 

  120. Bode, M., Fountoulakis, N. & Müller, T. The probability of connectivity in a hyperbolic model of complex networks. Random Struct. Algorithms 49, 65–94 (2016).

    MathSciNet  MATH  Google Scholar 

  121. Bringmann, K., Keusch, R., Lengler, J., Maus, Y. & Molla, A. R. Greedy routing and the algorithmic small-world phenomenon. In Proc. ACM Symp. Princ. Distrib. Comput. - PODC’17 371–380 (ACM Press, 2017).

  122. Abdullah, M. A., Fountoulakis, N. & Bode, M. Typical distances in a geometric model for complex networks. Internet Math. https://doi.org/10.24166/im.13.2017 (2017).

    Article  MathSciNet  Google Scholar 

  123. Bläsius, T., Friedrich, T. & Krohmer, A. Cliques in hyperbolic random graphs. Algorithmica 80, 2324–2344 (2018).

    MathSciNet  MATH  Google Scholar 

  124. Fountoulakis, N. & Müller, T. Law of large numbers for the largest component in a hyperbolic model of complex networks. Ann. Appl. Probab. 28, 607–650 (2018).

    MathSciNet  MATH  Google Scholar 

  125. Friedrich, T. & Krohmer, A. On the diameter of hyperbolic random graphs. SIAM J. Discret. Math. 32, 1314–1334 (2018).

    MathSciNet  MATH  Google Scholar 

  126. Kiwi, M. & Mitsche, D. Spectral gap of random hyperbolic graphs and related parameters. Ann. Appl. Probab. 28, 941–989 (2018).

    MathSciNet  MATH  Google Scholar 

  127. Bringmann, K., Keusch, R. & Lengler, J. Geometric inhomogeneous random graphs. Theor. Comput. Sci. 760, 35–54 (2019).

    MathSciNet  MATH  Google Scholar 

  128. Müller, T. & Staps, M. The diameter of KPKVB random graphs. Adv. Appl. Probab. 51, 358–377 (2019).

    MathSciNet  MATH  Google Scholar 

  129. Papadopoulos, F., Psomas, C. & Krioukov, D. Network mapping by replaying hyperbolic growth. IEEE/ACM Trans. Netw. 23, 198–211 (2015).

    Google Scholar 

  130. Papadopoulos, F., Aldecoa, R. & Krioukov, D. Network geometry inference using common neighbors. Phys. Rev. E 92, 022807 (2015).

    ADS  Google Scholar 

  131. Blasius, T., Friedrich, T., Krohmer, A. & Laue, S. Efficient embedding of scale-free graphs in the hyperbolic plane. IEEE/ACM Trans. Netw. 26, 920–933 (2018).

    MATH  Google Scholar 

  132. Muscoloni, A., Thomas, J. M., Ciucci, S., Bianconi, G. & Cannistraci, C. V. Machine learning meets complex networks via coalescent embedding in the hyperbolic space. Nat. Commun. 8, 1615 (2017).

    ADS  Google Scholar 

  133. Alanis-Lobato, G., Mier, P. & Andrade-Navarro, M. A. Efficient embedding of complex networks to hyperbolic space via their Laplacian. Sci. Rep. 6, 30108 (2016).

    ADS  Google Scholar 

  134. Wang, Z., Wu, Y., Li, Q., Jin, F. & Xiong, W. Link prediction based on hyperbolic mapping with community structure for complex networks. Physica A 450, 609–623 (2016).

    ADS  Google Scholar 

  135. Muscoloni, A. & Cannistraci, C. V. Minimum curvilinear automata with similarity attachment for network embedding and link prediction in the hyperbolic space. Preprint at https://arxiv.org/abs/1802.01183 (2018).

  136. Alanis-Lobato, G., Mier, P. & Andrade-Navarro, M. A. Manifold learning and maximum likelihood estimation for hyperbolic network embedding. Appl. Netw. Sci. 1, 10 (2016).

    Google Scholar 

  137. Garcia-Perez, G., Allard, A., Serrano, M. A. & Boguna, M. Mercator: uncovering faithful hyperbolic embeddings of complex networks. New J. Phys. 21, 123033 (2019).

    MathSciNet  Google Scholar 

  138. Kitsak, M., Voitalov, I. & Krioukov, D. Link prediction with hyperbolic geometry. Phys. Rev. Res. 2, 043113 (2020).

    Google Scholar 

  139. Serrano, M. Á., Boguñá, M. & Sagués, F. Uncovering the hidden geometry behind metabolic networks. Mol. Biosyst. 8, 843–850 (2012).

    Google Scholar 

  140. Kleineberg, K.-K. Metric clusters in evolutionary games on scale-free networks. Nat. Commun. 8, 1888 (2017).

    ADS  Google Scholar 

  141. García-Pérez, G., Aliakbarisani, R., Ghasemi, A. & Serrano, M. Á. Precision as a measure of predictability of missing links in real networks. Phys. Rev. E 101, 052318 (2020).

    MathSciNet  ADS  Google Scholar 

  142. Kerrache, S., Alharbi, R. & Benhidour, H. A scalable similarity–popularity link prediction method. Sci. Rep. 10, 6394 (2020).

    ADS  Google Scholar 

  143. Voitalov, I., Aldecoa, R., Wang, L. & Krioukov, D. Geohyperbolic routing and addressing schemes. ACM SIGCOMM Comput. Commun. Rev 47, 11–18 (2017).

    Google Scholar 

  144. 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, 20244–20253 (2020).

    ADS  Google Scholar 

  145. Cacciola, A. et al. Coalescent embedding in the hyperbolic space unsupervisedly discloses the hidden geometry of the brain. Preprint at https://arxiv.org/abs/1705.04192 (2017).

  146. Faqeeh, A., Osat, S. & Radicchi, F. Characterizing the analogy between hyperbolic embedding and community structure of complex networks. Phys. Rev. Lett. 121, 098301 (2018).

    ADS  Google Scholar 

  147. Boguñá, M., Pastor-Satorras, R. & Vespignani, A. Cut-offs and finite size effects in scale-free networks. Eur Phys. J. B 38, 205–209 (2004).

    ADS  Google Scholar 

  148. Catanzaro, M., Boguñá, M. & Pastor-Satorras, R. Generation of uncorrelated random scale-free networks. Phys. Rev. E 71, 027103 (2005).

    ADS  Google Scholar 

  149. Korman, A. & Peleg, D. Dynamic Routing Schemes for General Graphs 619–630 (Lecture Notes in Computer Science Vol. 4051, Springer, 2006).

  150. Serrano, M. Á., Krioukov, D. & Boguñá, M. Percolation in self-similar networks. Phys. Rev. Lett. 106, 048701 (2011).

    ADS  Google Scholar 

  151. Cohen, R., Erez, K., Ben-Avraham, D. & Havlin, S. Resilience of the internet to random breakdowns. Phys. Rev. Lett. 85, 4626–4628 (2000).

    ADS  Google Scholar 

  152. Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001).

    ADS  Google Scholar 

  153. Bianconi, G. Mean field solution of the Ising model on a Barabási–Albert network. Phys. Lett. A 303, 166–168 (2002).

    MathSciNet  MATH  ADS  Google Scholar 

  154. Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Ising model on networks with an arbitrary distribution of connections. Phys. Rev. E 66, 016104 (2002).

    ADS  Google Scholar 

  155. Leone, M., Vázquez, A., Vespignani, A. & Zecchina, R. Ferromagnetic ordering in graphs with arbitrary degree distribution. Eur. Phys. J. B 28, 191–197 (2002).

    ADS  Google Scholar 

  156. Kadanoff, L. P. Statistical Physics: Statics, Dynamics and Renormalization (World Scientific, 2000).

  157. Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y. & Zhou, C. Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008).

    MathSciNet  ADS  Google Scholar 

  158. van den Heuvel, M. P., Kahn, R. S., Goni, J. & Sporns, O. High-cost, high-capacity backbone for global brain communication. Proc. Natl Acad. Sci. USA 109, 11372–11377 (2012).

    ADS  Google Scholar 

  159. Goñi, J. et al. Exploring the morphospace of communication efficiency in complex networks. PLoS ONE 8, e58070 (2013).

    ADS  Google Scholar 

  160. Fornito, A., Zalesky, A. & Breakspear, M. Graph analysis of the human connectome: promise, progress, and pitfalls. Neuroimage 80, 426–444 (2013).

    Google Scholar 

  161. Mišić, B., Sporns, O. & McIntosh, A. R. Communication efficiency and congestion of signal traffic in large-scale brain networks. PLoS Comput. Biol. 10, e1003427 (2014).

    ADS  Google Scholar 

  162. Roberts, J. A. et al. The contribution of geometry to the human connectome. Neuroimage 124, 379–393 (2016).

    Google Scholar 

  163. Avena-Koenigsberger, A. et al. Path ensembles and a tradeoff between communication efficiency and resilience in the human connectome. Brain Struct. Funct. 222, 603–618 (2017).

    Google Scholar 

  164. Seguin, C., van den Heuvel, M. P. & Zalesky, A. Navigation of brain networks. Proc. Natl Acad. Sci. USA 115, 6297–6302 (2018).

    Google Scholar 

  165. Tadić, B., Andjelković, M. & Šuvakov, M. Origin of hyperbolicity in brain-to-brain coordination networks. Front. Phys. 6, 7 (2018).

    Google Scholar 

  166. Avena-Koenigsberger, A. et al. A spectrum of routing strategies for brain networks. PLoS Comput. Biol. 15, e1006833 (2019).

    Google Scholar 

  167. Wang, X. et al. Synchronization lag in post stroke: relation to motor function and structural connectivity. Netw. Neurosci. 3, 1121–1140 (2019).

    Google Scholar 

  168. Betzel, R. F. et al. Generative models of the human connectome. Neuroimage, 124 1054–1064 (2015).

    Google Scholar 

  169. Horvát, S. et al. Spatial embedding and wiring cost constrain the functional layout of the cortical network of rodents and primates. PLoS Biol. 14, e1002512 (2016).

    Google Scholar 

  170. Stiso, J. & Bassett, D. S. Spatial embedding imposes constraints on neuronal network architectures. Trends Cogn. Sci. 22, 1127–1142 (2018).

    Google Scholar 

  171. Allen, Q. Y. et al. The intrinsic geometry of the human brain connectome. Brain Inform. 2, 197–210 (2015).

    Google Scholar 

  172. Farooq, H., Chen, Y., Georgiou, T. T., Tannenbaum, A. & Lenglet, C. Network curvature as a hallmark of brain structural connectivity. Nat. Commun. 10, 4937 (2019).

    ADS  Google Scholar 

  173. Assaf, Y., Bouznach, A., Zomet, O., Marom, A. & Yovel, Y. Conservation of brain connectivity and wiring across the mammalian class. Nat. Neurosci. 23, 805–808 (2020).

    Google Scholar 

  174. Nickel, M. & Kiela, D. Poincaré embeddings for learning hierarchical representations. Adv. Neural Inf. Process. Syst. (2017).

  175. Dhingra, B., Shallue, C. J., Norouzi, M., Dai, A. M. & Dahl, G. E. Embedding text in hyperbolic spaces. In Proc. Twelfth Work Graph-Based Methods Nat. Lang. Process 59–69 (Association for Computational Linguistics, 2018).

  176. Ganea, O.-E., Bécigneul, G. & Hofmann, T. Hyperbolic entailment cones for learning hierarchical embeddings. In Proc 35th Int. Conf. Mach. Learn. (eds Dy, J., & Krause, A.) 1646–1655 (PMLR, 2018).

  177. Ganea, O.-E., Bécigneul, G. & Hofmann, T. Hyperbolic neural networks. Adv. Neural Inf. Process. Syst. (2018).

  178. Nickel, M. & Kiela, D. Learning continuous hierarchies in the Lorentz model of hyperbolic geometry. In Int. Conf. Mach. Learn. (eds Dy, J., & Krause, A.) 3776–3785 (PMLR, 2018).

  179. Ovinnikov, I. Poincaré Wasserstein autoencoder. Preprint at https://arxiv.org/abs/1901.01427 (2019).

  180. Sala, F. et al. Representation tradeoffs for hyperbolic embeddings. In Int. Conf. Mach. Learn. (eds Dy, J., & Krause, A.) 4460–4469 (PMLR, 2018).

  181. Gulcehre, C. et al. Hyperbolic attention networks. In Int. Conf. Learn. Represent ICLR 2019 (OpenReview.net, 2019).

  182. Chami, I., Ying, R., Ré, C. & Leskovec, J. Hyperbolic graph convolutional neural networks. In Adv. Neural Inf. Process. Syst. (Wallach, H. et al) 4868–4879 (2019).

  183. Liu, Q., Nickel, M. & Kiela, D. Hyperbolic graph neural networks. In Adv. Neural Inf. Process. Syst. (Wallach, H. et al) 8230–8241 (2019).

  184. Suzuki, A., Wang, J., Tian, F., Nitanda, A. & Yamanishi, K. Hyperbolic ordinal embedding. In Proc. Elev. Asian Conf. Mach. Learn (Lee, W. S., & Suzuki, T.) 1065–1080 (PMLR, 2019).

  185. Tifrea, A., Bécigneul, G. & Ganea, O.-E. Poincaré GloVe: hyperbolic word embeddings. In Int. Conf. Learn. Represent (OpenReview.net, 2019).

  186. Orsini, C. et al. Quantifying randomness in real networks. Nat. Commun. 6, 8627 (2015).

    ADS  Google Scholar 

  187. Gu, W., Tandon, A., Ahn, Y.-Y. & Radicchi, F. Defining and identifying the optimal embedding dimension of networks. Preprint at https://arxiv.org/abs/2004.09928 (2020).

  188. Chaudhari, P. et al. Entropy-SGD: biasing gradient descent into wide valleys. J. Stat. Mech. Theor. Exp. 2019, 124018 (2019).

    MathSciNet  Google Scholar 

  189. Krioukov, D. Clustering implies geometry in networks. Phys. Rev. Lett. 116, 208302 (2016).

    MathSciNet  ADS  Google Scholar 

  190. Holme, P. & Saramäki, J. Temporal networks. Phy. Rep. 519, 97–125 (2012).

    ADS  Google Scholar 

  191. Kallenberg, O. Probabilistic Symmetries and Invariance Principles (Springer, 2005).

  192. Lovász, L. Large Networks and Graph Limits (American Mathematical Society, 2012).

  193. Orbanz, P. & Roy, D. M. Bayesian models of graphs, arrays and other exchangeable random structures. IEEE Trans. Pattern Anal. Mach. Intell. 37, 437–461 (2015).

    Google Scholar 

  194. Aldous, D. J. Representations for partially exchangeable arrays of random variables. J. Multivar. Anal. 11, 581–598 (1981).

    MathSciNet  MATH  Google Scholar 

  195. Hoover, D. N. Relations on Probability Spaces and Arrays of Random Variables Technical Report (Institute for Adanced Study, 1979).

  196. Caron, F. & Fox, E. B. Sparse graphs using exchangeable random measures. J. R. Stat. Soc. Ser. B 79, 1295–1366 (2017).

    MathSciNet  MATH  Google Scholar 

  197. Veitch, V. & Roy, D. M. Sampling and estimation for (sparse) exchangeable graphs. Ann. Stat. 47, 3274–3299 (2019).

    MathSciNet  MATH  Google Scholar 

  198. Borgs, C., Chayes, J. T., Cohn, H. & Zhao, Y. An Lp theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions. Trans. Am. Math. Soc. 372, 3019–3062 (2019).

    MATH  Google Scholar 

  199. Janson, S. On edge exchangeable random graphs. J. Stat. Phys. 173, 448–484 (2018).

    MathSciNet  MATH  ADS  Google Scholar 

  200. Muscoloni, A. & Cannistraci, C. V. Local-ring network automata and the impact of hyperbolic geometry in complex network link-prediction. Preprint at https://arxiv.org/abs/1707.09496 (2017).

  201. Klein, D. J. & Randić, M. Resistance distance. J. Math. Chem. 12, 81–95 (1993).

    MathSciNet  Google Scholar 

  202. Babić, D., Klein, D., Lukovits, I., Nikolić, S. & Trinajstić, N. Resistance-distance matrix: a computational algorithm and its application. Int. J. Quantum Chem. 90, 166–176 (2002).

    Google Scholar 

  203. McRae, B. H. Isolation by resistance. Evolution 60, 1551–1561 (2006).

    Google Scholar 

  204. Luxburg, U. V., Radl, A. & Hein, M. Getting lost in space: large sample analysis of the resistance distance. Adv. Neural Inf. Process. Syst. 23, 2622–2630 (2010).

    Google Scholar 

  205. Estrada, E. Complex networks in the euclidean space of communicability distances. Phys. Rev. E 85, 066122 (2012).

    ADS  Google Scholar 

  206. Estrada, E., Sánchez-Lirola, M. & De La Peña, J. A. Hyperspherical embedding of graphs and networks in communicability spaces. Discret. Appl. Math. 176, 53–77 (2014).

    MathSciNet  MATH  Google Scholar 

  207. Estrada, E. & Hatano, N. Communicability angle and the spatial efficiency of networks. SIAM Rev. 58, 692–715 (2016).

    MathSciNet  MATH  Google Scholar 

  208. Akbarzadeh, M. & Estrada, E. Communicability geometry captures traffic flows in cities. Nat. Hum. Behav. 2, 645–652 (2018).

    Google Scholar 

  209. Estrada, E. Communicability geometry of multiplexes. New J. Phys. 21, 015004 (2018).

    Google Scholar 

  210. De Domenico, M., Granell, C., Porter, M. A. & Arenas, A. The physics of spreading processes in multilayer networks. Nat. Phys. 12, 901–906 (2016).

    Google Scholar 

  211. Estrada, E., Hatano, N. & Benzi, M. The physics of communicability in complex networks. Phys. Rep. 514, 89–119 (2012).

    MathSciNet  ADS  Google Scholar 

  212. Chebotarev, P. A class of graph–geodetic distances generalizing the shortest-path and the resistance distances. Discret. Appl. Math. 159, 295–302 (2011).

    MathSciNet  MATH  Google Scholar 

  213. Chebotarev, P. The walk distances in graphs. Discret. Appl. Math. 160, 1484–1500 (2012).

    MathSciNet  MATH  Google Scholar 

  214. Iannelli, F., Koher, A., Brockmann, D., Hoevel, P. & Sokolov, I. M. Effective distances for epidemics spreading on complex networks. Phys. Rev. E 95, 012313 (2016).

    MathSciNet  ADS  Google Scholar 

  215. Barthélemy, M. Spatial networks. Phys. Rep. 499, 1–101 (2011).

    MathSciNet  ADS  Google Scholar 

  216. Bonamassa, I., Gross, B., Danziger, M. M. & Havlin, S. Critical stretching of mean-field regimes in spatial networks. Phys. Rev.Lett. 123, 088301 (2019).

    MathSciNet  ADS  Google Scholar 

  217. Bertagnolli, G., Agostinelli, C. & Domenico, M. D. Network depth: identifying median and contours in complex networks. J. Complex Netw. 8, cnz041 (2019).

    MathSciNet  Google Scholar 

  218. Bertagnolli, G. & De Domenico, M. Diffusion geometry of multiplex and interdependent systems. Preprint at https://arxiv.org/abs/2006.13032 (2020).

  219. De Domenico, M. et al. Mathematical formulation of multi-layer networks. Phys. Rev. X 3, 041022 (2013).

    Google Scholar 

  220. De Domenico, M., Solé-Ribalta, A., Gómez, S. & Arenas, A. Navigability of interconnected networks under random failures. Proc. Natl Acad. Sci. USA 111, 8351–8356 (2014).

    MathSciNet  MATH  ADS  Google Scholar 

  221. Ravasz, E. & Barabási, A.-L. Hierarchical organization in complex networks. Phys. Rev. E 67, 026112 (2003).

    MATH  ADS  Google Scholar 

  222. Corominas-Murtra, B., Goñi, J., Solé, R. V. & Rodríguez-Caso, C. On the origins of hierarchy in complex networks. Proc. Natl Acad. Sci. USA 110, 13316–13321 (2013).

    MathSciNet  MATH  ADS  Google Scholar 

  223. Newman, M. E. J., Strogatz, S. H. & Watts, D. J. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 26118 (2001).

    ADS  Google Scholar 

  224. Dorogovtsev, S. N., Mendes, J. F. F. & Samukhin, A. N. Giant strongly connected component of directed networks. Phys. Rev. E 64, 025101 (2001).

    ADS  Google Scholar 

  225. Seidman, S. B. Network structure and minimum degree. Soc. Netw. 5, 269–287 (1983).

    MathSciNet  Google Scholar 

  226. Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. k-core organization of complex networks. Phys. Rev. Lett. 96, 040601 (2006).

    ADS  Google Scholar 

  227. Holme, P. Core–periphery organization of complex networks. Phys. Rev. E 72, 046111 (2005).

    ADS  Google Scholar 

  228. Rombach, M. P., Porter, M. A., Fowler, J. H. & Mucha, P. J. Core–periphery structure in networks. SIAM J. Appl. Math. 74, 167–190 (2014).

    MathSciNet  MATH  Google Scholar 

  229. Papadopoulos, F., Krioukov, D., Boguna, M. & Vahdat, A. Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces. In 2010 Proc IEEE INFOCOM 1–9 (IEEE, 2010).

  230. Cannistraci, C. V. & Muscoloni, A. Geometrical congruence and efficient greedy navigability of complex networks. Preprint at https://arxiv.org/abs/2005.13255 (2020).

  231. Wu, Z., Menichetti, G., Rahmede, C. & Bianconi, G. Emergent complex network geometry. Sci. Rep. 5, 10073 (2015).

    ADS  Google Scholar 

  232. Mulder, D. & Bianconi, G. Network geometry and complexity. J. Stat. Phys. 173, 783–805 (2018).

    MathSciNet  MATH  ADS  Google Scholar 

  233. Rideout, D. & Wallden, P. Spacelike distance from discrete causal order. Class. Quantum Gravity 26, 155013 (2009).

    MATH  ADS  Google Scholar 

  234. Dowker, F. Spacetime discreteness, Lorentz invariance and locality. J. Phys. Conf. Ser. 306, 012016 (2011).

    Google Scholar 

  235. Konopka, T., Markopoulou, F. & Severini, S. Quantum graphity: a model of emergent locality. Phys. Rev. D 77, 104029 (2008).

    MathSciNet  ADS  Google Scholar 

  236. Ambjørn, J., Jurkiewicz, J. & Loll, R. The spectral dimension of the universe is scale dependent. Phys. Rev. Lett. 95, 171301 (2005).

    ADS  Google Scholar 

  237. Rovelli, C. & Speziale, S. On the geometry of loop quantum gravity on a graph. Phys. Rev. D 82, 44018 (2010).

    ADS  Google Scholar 

  238. Grüber, D., Sahlmann, H. & Zilker, T. Geometry and entanglement entropy of surfaces in loop quantum gravity. Phys. Rev. D 98, 066009 (2018).

    MathSciNet  ADS  Google Scholar 

  239. Ambjørn, J., Jordan, S., Jurkiewicz, J. & Loll, R. Second-order phase transition in causal dynamical triangulations. Phys. Rev. Lett. 107, 211303 (2011).

    ADS  Google Scholar 

  240. Sotiriou, T. P., Visser, M. & Weinfurtner, S. Spectral dimension as a probe of the ultraviolet continuum regime of causal dynamical triangulations. Phys. Rev. Lett. 107, 131303 (2011).

    ADS  Google Scholar 

  241. Loll, R. Quantum gravity from causal dynamical triangulations: a review. Class. Quantum Grav. 37, 013002 (2020).

    MathSciNet  ADS  Google Scholar 

  242. Kelly, C., Trugenberger, C. A. & Biancalana, F. Self-assembly of geometric space from random graphs. Class. Quantum Gravity 36, 125012 (2019).

    MathSciNet  ADS  Google Scholar 

  243. Bianconi, G. & Rahmede, C. Complex quantum network manifolds in dimension d > 2 are scale-free. Sci. Rep. 5, 13979 (2015).

    ADS  Google Scholar 

  244. Bianconi, G. & Rahmede, C. Network geometry with flavor: from complexity to quantum geometry. Phys. Rev. E 93, 032315 (2016).

    MathSciNet  ADS  Google Scholar 

  245. Bianconi, G. & Rahmede, C. Emergent hyperbolic network geometry. Sci. Rep. 7, 41974 (2017).

    ADS  Google Scholar 

  246. Higuchi, Y. Combinatorial curvature for planar graphs. J. Graph. Theor. 38, 220–229 (2001).

    MathSciNet  MATH  Google Scholar 

  247. Ollivier, Y. Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256, 810–864 (2009).

    MathSciNet  MATH  Google Scholar 

  248. Lin, Y., Lu, L. & Yau, S.-T. Ricci curvature of graphs. Tohoku Math. J. Sec. Ser. 63, 605–627 (2011).

    MathSciNet  MATH  Google Scholar 

  249. Sreejith, R., Mohanraj, K., Jost, J., Saucan, E. & Samal, A. Forman curvature for complex networks. J. Stat. Mech. Theor. Exp. 2016, 063206 (2016).

    MathSciNet  MATH  Google Scholar 

  250. Sreejith, R., Jost, J., Saucan, E. & Samal, A. Systematic evaluation of a new combinatorial curvature for complex networks. Chaos Solitons Fractals 101, 50–67 (2017).

    MathSciNet  MATH  ADS  Google Scholar 

  251. Samal, A. et al. Comparative analysis of two discretizations of Ricci curvature for complex networks. Sci. Rep. 8, 8650 (2018).

    ADS  Google Scholar 

  252. Prokhorenkova, L., Samosvat, E. & van der Hoorn, P. Global Graph Curvature Vol. 2, 16–35 (Springer International Publishing, 2020).

  253. van der Hoorn, P., Cunningham, W. J., Lippner, G., Trugenberger, C. & Krioukov, D. Ollivier–Ricci curvature convergence in random geometric graphs. Preprint at https://arxiv.org/abs/2008.01209 (2020).

  254. Sandhu, R. et al. Graph curvature for differentiating cancer networks. Sci. Rep. 5, 12323 (2015).

    ADS  Google Scholar 

  255. Sia, J., Jonckheere, E. & Bogdan, P. Ollivier–ricci curvature-based method to community detection in complex networks. Sci. Rep. 9, 9800 (2019).

    ADS  Google Scholar 

  256. Ni, C.-C., Lin, Y.-Y., Luo, F. & Gao, J. Community detection on networks with Ricci flow. Sci. Rep. 9, 1–12 (2019).

    Google Scholar 

  257. Robles-Kelly, A. & Hancock, E. R. A Riemannian approach to graph embedding. Pattern Recognit. 40, 1042–1056 (2007).

    MATH  Google Scholar 

  258. Majid, S. Noncommutative Riemannian geometry on graphs. J. Geom. Phys. 69, 74–93 (2013).

    MathSciNet  MATH  ADS  Google Scholar 

  259. Franzosi, R., Felice, D., Mancini, S. & Pettini, M. Riemannian-geometric entropy for measuring network complexity. Phys. Rev. E 93, 062317 (2016).

    MathSciNet  ADS  Google Scholar 

  260. Narayan, O. & Saniee, I. Large-scale curvature of networks. Phys. Rev. E 84, 066108 (2011).

    ADS  Google Scholar 

  261. Albert, R., DasGupta, B. & Mobasheri, N. Topological implications of negative curvature for biological and social networks. Phys. Rev. E 89, 032811 (2014).

    ADS  Google Scholar 

  262. Borassi, M., Chessa, A. & Caldarelli, G. Hyperbolicity measures democracy in real-world networks. Phys. Rev. E 92, 032812 (2015).

    ADS  Google Scholar 

  263. Tadić, B., Andjelković, M. & Melnik, R. Functional geometry of human connectomes. Sci. Rep. 9, 12060 (2019).

    ADS  Google Scholar 

  264. Chatterjee, S. & Sloman, L. Average Gromov hyperbolicity and the Parisi ansatz. Adv. Math 376, 107417 (2019).

    MathSciNet  MATH  Google Scholar 

  265. Shang, Y. Lack of Gromov-hyperbolicity in small-world networks. Cent. Eur. J. Math. 10, 1152–1158 (2012).

    MathSciNet  MATH  Google Scholar 

  266. Shang, Y. Non-Hyperbolicity of random graphs with given expected degrees. Stoch. Model. 29, 451–462 (2013).

    MathSciNet  MATH  Google Scholar 

  267. Aktas, M. E., Akbas, E. & El Fatmaoui, A. Persistence homology of networks: methods and applications. Appl. Netw. Sci. 4, 61 (2019).

    Google Scholar 

  268. Patania, A., Vaccarino, F. & Petri, G. Topological analysis of data. EPJ Data Sci. 6, 7 (2017).

    MATH  Google Scholar 

  269. Giusti, C., Pastalkova, E., Curto, C. & Itskov, V. Clique topology reveals intrinsic geometric structure in neural correlations. Proc. Natl Acad. Sci. USA 112, 13455–13460 (2015).

    MathSciNet  MATH  ADS  Google Scholar 

  270. Otter, N., Porter, M. A., Tillmann, U., Grindrod, P. & Harrington, H. A. A roadmap for the computation of persistent homology. EPJ Data Sci. 6, 17 (2017).

    Google Scholar 

  271. Battiston, F. et al. Networks beyond pairwise interactions: structure and dynamics. Phys. Rep. 874, 1–92 (2020).

    MathSciNet  ADS  Google Scholar 

  272. Song, C., Gallos, L. K., Havlin, S. & Makse, H. A. How to calculate the fractal dimension of a complex network: the box covering algorithm. J. Stat. Mech. Theor. Exp. 2007, P03006 (2007).

    MATH  Google Scholar 

  273. Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. J. Stat. Mech. Theor. Exp. 2008, P10008 (2008).

    MATH  Google Scholar 

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Acknowledgements

S.H. thanks the Israel Science Foundation, ONR, the BIU Center for Research in Applied Cryptography and Cyber Security, NSF-BSF grant number 2019740, and DTRA grant number HDTRA-1-19-1-0016 for financial support. M.B. and M.A.S. acknowledge support from: a James S. McDonnell Foundation Scholar Award in Complex Systems; the ICREA Academia award, funded by the Generalitat de Catalunya; Agencia estatal de investigación project number PID2019-106290GB-C22/AEI/10.13039/501100011033; the Spanish Ministerio de Ciencia, Innovación y Universidades project number FIS2016-76830-C2-2-P (AEI/FEDER, UE); project Mapping Big Data Systems: embedding large complex networks in low-dimensional hidden metric spaces, Ayudas Fundación BBVA a Equipos de Investigación Científica 2017, and Generalitat de Catalunya grant number 2017SGR1064. D.K. acknowledges support from the NSF grant number IIS-1741355, and the ARO grant numbers W911NF-16-1-0391 and W911NF-17-1-0491.

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Authors and Affiliations

  1. Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Spain

    Marián Boguñá & M. Ángeles Serrano

  2. Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Spain

    Marián Boguñá & M. Ángeles Serrano

  3. Department of Physics, Bar-Ilan University, Ramat-Gan, Israel

    Ivan Bonamassa & Shlomo Havlin

  4. CoMuNe Lab, Fondazione Bruno Kessler, Povo, Italy

    Manlio De Domenico

  5. Network Science Institute, Northeastern University, Boston, MA, USA

    Dmitri Krioukov

  6. Department of Physics, Northeastern University, Boston, MA, USA

    Dmitri Krioukov

  7. Department of Mathematics, Northeastern University, Boston, MA, USA

    Dmitri Krioukov

  8. Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, USA

    Dmitri Krioukov

  9. ICREA, Barcelona, Spain

    M. Ángeles Serrano

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  1. Marián Boguñá
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Boguñá, M., Bonamassa, I., De Domenico, M. et al. Network geometry. Nat Rev Phys 3, 114–135 (2021). https://doi.org/10.1038/s42254-020-00264-4

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