The statistical physics of real-world networks


In the past 15 years, statistical physics has been successful as a framework for modelling complex networks. On the theoretical side, this approach has unveiled a variety of physical phenomena, such as the emergence of mixed distributions and ensemble non-equivalence, that are observed in heterogeneous networks but not in homogeneous systems. At the same time, thanks to the deep connection between the principle of maximum entropy and information theory, statistical physics has led to the definition of null models for networks that reproduce features of real-world systems but that are otherwise as random as possible. We review here the statistical physics approach and the null models for complex networks, focusing in particular on analytical frameworks that reproduce local network features. We show how these models have been used to detect statistically significant structural patterns in real-world networks and to reconstruct the network structure in cases of incomplete information. We further survey the statistical physics models that reproduce more complex, semilocal network features using Markov chain Monte Carlo sampling, as well as models of generalized network structures, such as multiplex networks, interacting networks and simplicial complexes.

Key points

  • Statistical physics is a powerful framework to explain properties of complex networks, modelled as systems of heterogeneous entities whose degrees of freedom are their interactions rather than their states.

  • The statistical physics of complex networks has brought theoretical insights into physical phenomena that are different in heterogeneous networks than in homogeneous systems.

  • From an applied perspective, statistical physics defines null models for real-world networks that reproduce local features but are otherwise as random as possible.

  • These models have been used, on the one hand, to detect statistically significant patterns in real-world networks and, on the other, to infer the network structure when information is incomplete.

  • These applications are particularly useful in the current information age to make consistent inference from huge streams of continuously produced, high-dimensional, noisy data.

  • The statistical mechanics approach has also been extended using numerical techniques to reproduce semilocal network features and, more recently, to encompass structures such as multilayer networks and simplicial complexes.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Construction of the microcanonical and canonical ensemble of networks from local constraints.
Fig. 2: One-mode projection of the network of countries and products they export and its statistical validation against a null hypothesis derived from the BiCM.
Fig. 3: Statistical reconstruction of an interbank network given by bilateral exchanges among banks.
Fig. 4: Generalized network structures.


  1. 1.

    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 

  2. 2.

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

    ADS  MathSciNet  MATH  Google Scholar 

  3. 3.

    Yook, S. H., Jeong, H., Barabási, A.-L. & Tu, Y. Weighted evolving networks. Phys. Rev. Lett. 86, 5835–5838 (2001).

    ADS  Google Scholar 

  4. 4.

    Barrat, A., Barthelemy, M. & Vespignani, A. Weighted evolving networks: coupling topology and weight dynamics. Phys. Rev. Lett. 92, 228701 (2004).

    ADS  Google Scholar 

  5. 5.

    Newman, M. E. J. & Girvan, M. Finding and evaluating community structure in networks. Phys. Rev. E 69, 026113 (2004).

    ADS  Google Scholar 

  6. 6.

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

    ADS  MathSciNet  Google Scholar 

  7. 7.

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

    ADS  MATH  Google Scholar 

  8. 8.

    Amaral, L. A. N., Scala, A., Barthélémy, M. & Stanley, H. E. Classes of small-world networks. Proc. Natl. Acad. Sci. U.S.A. 97, 11149–11152 (2000).

    ADS  Google Scholar 

  9. 9.

    Chung, F. & Lu, L. The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci. U.S.A. 99, 15879–15882 (2002).

    ADS  MathSciNet  MATH  Google Scholar 

  10. 10.

    Albert, R. & Barabási, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys0. 74, 47–97 (2002).

    ADS  MathSciNet  MATH  Google Scholar 

  11. 11.

    Newman, M. E. J. The structure and function of complex networks. SIAM Rev. Soc. Ind. Appl. Math. 45, 167–256 (2003).

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D.-U. Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006).

    ADS  MathSciNet  MATH  Google Scholar 

  13. 13.

    Bianconi, G. & Barabási, A. L. Bose-einstein condensation in complex network. Phys. Rev. Lett. 86, 5632–5635 (2001).

    ADS  Google Scholar 

  14. 14.

    Caldarelli, G., Capocci, A., De Los Rios, P. & Muñoz, M. A. Scale-free networks from varying vertex intrinsic fitness. Phys. Rev. Lett. 89, 258702 (2002).

    ADS  Google Scholar 

  15. 15.

    Dorogovtsev, S. N., Mendes, J. F. F. & Samukhin, A. N. Structure of growing networks with preferential linking. Phys. Rev. Lett. 85, 4633–4636 (2000).

    ADS  Google Scholar 

  16. 16.

    Medo, M., Cimini, G. & Gualdi, S. Temporal effects in the growth of networks. Phys. Rev. Lett. 107, 238701 (2011).

    ADS  Google Scholar 

  17. 17.

    Holland, P. W. & Leinhardt, S. An exponential family of probability distributions for directed graphs. J. Am. Stat. Assoc. 76, 33–50 (1981). This paper introduces ERGs as a formalism to define probability distributions for the structures of social networks.

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Frank, O. & Strauss, D. Markov graphs. J. Am. Stat. Assoc. 81, 832–842 (1986).

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Strauss, D. On a general class of models for interaction. SIAM Rev. Soc. Ind. Appl. Math. 28, 513–527 (1986).

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Wasserman, S. & Pattison, P. Logit models and logistic regressions for social networks: I. An introduction to markov graphs and p. Psychometrika 61, 401–425 (1996).

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Anderson, C. J., Wasserman, S. & Crouch, B. A p* primer: logit models for social networks. Soc. Networks 21, 37–66 (1999).

    Google Scholar 

  22. 22.

    Snijders, T. A. B., Pattison, P. E., Robins, G. L. & Handcock, M. S. New specifications for exponential random graph models. Sociol. Methodol. 36, 99–153 (2006).

    Google Scholar 

  23. 23.

    Robins, G., Pattison, P., Kalish, Y. & Lusher, D. An introduction to exponential random graph (p*) models for social networks. Soc. Networks 29, 173–191 (2007).

    Google Scholar 

  24. 24.

    Cranmer, S. J. & Desmarais, B. A. Inferential network analysis with exponential random graph models. Polit. Anal. 19, 6686 (2011).

    Google Scholar 

  25. 25.

    Snijders, T. A. B. Statistical models for social networks. Annu. Rev. Sociol. 37, 131–153 (2011).

    Google Scholar 

  26. 26.

    Park, J. & Newman, M. E. J. Statistical mechanics of networks. Phys. Rev. E 70, 066117 (2004). In this paper, ERGs are interpreted for the first time as the statistical physics framework for complex networks.

    ADS  MathSciNet  Google Scholar 

  27. 27.

    Jaynes, E. T. Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957). In this milestone paper, Jaynes shows that equilibrium statistical mechanics provides an unbiased prescription to make inferences from partial information.

    ADS  MathSciNet  MATH  Google Scholar 

  28. 28.

    Shore, J. & Johnson, R. Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans. Inf. Theory 26, 26–37 (1980).

    ADS  MathSciNet  MATH  Google Scholar 

  29. 29.

    Pressé, S., Ghosh, K., Lee, J. & Dill, K. A. Principles of maximum entropy and maximum caliber in statistical physics. Rev. Mod. Phys. 85, 1115–1141 (2013).

    ADS  Google Scholar 

  30. 30.

    Jaynes, E. T. On the rationale of maximum-entropy methods. Proc. IEEE 70, 939–952 (1982).

    ADS  Google Scholar 

  31. 31.

    Bianconi, G. The entropy of randomized network ensembles. Europhys. Lett. 81, 28005 (2008). This paper derives the Boltzmann entropy of a variety of network ensembles to assess the role of structural network properties.

    ADS  MathSciNet  Google Scholar 

  32. 32.

    Squartini, T., Mastrandrea, R. & Garlaschelli, D. Unbiased sampling of network ensembles. New J. Phys. 17, 023052 (2015).

    ADS  Google Scholar 

  33. 33.

    Anand, K. & Bianconi, G. Entropy measures for networks: toward an information theory of complex topologies. Phys. Rev. E 80, 045102 (2009).

    ADS  Google Scholar 

  34. 34.

    Squartini, T., de Mol, J., den Hollander, F. & Garlaschelli, D. Breaking of ensemble equivalence in networks. Phys. Rev. Lett. 115, 268701 (2015).

    ADS  Google Scholar 

  35. 35.

    Squartini, T. & Garlaschelli, D. Reconnecting statistical physics and combinatorics beyond ensemble equivalence. Preprint at (2018).

  36. 36.

    Garlaschelli, D. & Loffredo, M. I. Generalized bose-fermi statistics and structural correlations in weighted networks. Phys. Rev. Lett. 102, 038701 (2009). This paper develops the ERG approach for a general class of weighted networks.

    ADS  Google Scholar 

  37. 37.

    Garlaschelli, D. & Loffredo, M. I. Maximum likelihood: extracting unbiased information from complex networks. Phys. Rev. E 78, 015101(R) (2008).

    ADS  Google Scholar 

  38. 38.

    Squartini, T. & Garlaschelli, D. Analytical maximum-likelihood method to detect patterns in real networks. New J. Phys. 13, 083001 (2011). This paper turns ERGs into null models for empirically observed networks using the maximum likelihood principle.

    ADS  Google Scholar 

  39. 39.

    Erdos, P. & Rényi, A. On random graphs. Publ. Math. Debr. 6, 290–297 (1959). This paper introduces the first statistical ensemble of random graphs.

    MATH  Google Scholar 

  40. 40.

    Serrano, M. Á. & Boguñá, M. Weighted configuration model. AIP Conf. Proc. 776, 101–107 (2005).

    ADS  Google Scholar 

  41. 41.

    Mastrandrea, R., Squartini, T., Fagiolo, G. & Garlaschelli, D. Enhanced reconstruction of weighted networks from strengths and degrees. New J. Phys. 16, 043022 (2014).

    ADS  Google Scholar 

  42. 42.

    Maslov, S. & Sneppen, K. Specificity and stability in topology of protein networks. Science 296, 910–913 (2002). This paper introduces the local link rewiring method to build a null network model.

    ADS  Google Scholar 

  43. 43.

    Park, J. & Newman, M. E. J. Origin of degree correlations in the internet and other networks. Phys. Rev. E 68, 026112 (2003).

    ADS  Google Scholar 

  44. 44.

    Barrat, A., Barthelemy, M., Pastor-Satorras, R. & Vespignani, A. The architecture of complex weighted networks. Proc. Natl. Acad. Sci. U.S.A. 101, 3747–3752 (2004).

    ADS  Google Scholar 

  45. 45.

    Maslov, S., Sneppen, K. & Zaliznyak, A. Detection of topological patterns in complex networks: correlation profile of the internet. Phys. A Stat. Mech. Appl. 333, 529–540 (2004).

    Google Scholar 

  46. 46.

    Colizza, V., Flammini, A., Serrano, M. A. & Vespignani, A. Detecting rich-club ordering in complex networks. Nat. Phys. 2, 110 (2006).

    Google Scholar 

  47. 47.

    Serrano, M. Á., Boguñá, M. & Pastor-Satorras, R. Correlations in weighted networks. Phys. Rev. E 74, 055101 (2006).

    ADS  Google Scholar 

  48. 48.

    Guimera, R., Sales-Pardo, M. & Amaral, L. A. N. Classes of complex networks defined by role-to-role connectivity profiles. Nat. Phys. 3, 63 (2006).

    Google Scholar 

  49. 49.

    Bhattacharya, K., Mukherjee, G., Saramäki, J., Kaski, K. & Manna, S. S. The international trade network: weighted network analysis and modelling. J. Stat. Mech. Theory Exp. 2008, P02002 (2008).

    Google Scholar 

  50. 50.

    Opsahl, T., Colizza, V., Panzarasa, P. & Ramasco, J. J. Prominence and control: the weighted rich-club effect. Phys. Rev. Lett. 101, 168702 (2008).

    ADS  Google Scholar 

  51. 51.

    Serrano, M. Á. & Boguñá, M. Topology of the world trade web. Phys. Rev. E 68, 015101 (2003).

    ADS  Google Scholar 

  52. 52.

    Garlaschelli, D. & Loffredo, M. I. Fitness-dependent topological properties of the world trade web. Phys. Rev. Lett. 93, 188701 (2004).

    ADS  Google Scholar 

  53. 53.

    Garlaschelli, D. & Loffredo, M. I. Structure and evolution of the world trade network. Phys. A Stat. Mech. Appl. 355, 138–144 (2005).

    MathSciNet  Google Scholar 

  54. 54.

    Fagiolo, G., Reyes, J. & Schiavo, S. World trade web: topological properties, dynamics, and evolution. Phys. Rev. E 79, 036115 (2009).

    ADS  MathSciNet  Google Scholar 

  55. 55.

    Newman, M. E. J. Analysis of weighted networks. Phys. Rev. E 70, 056131 (2004).

    ADS  Google Scholar 

  56. 56.

    Ahnert, S. E., Garlaschelli, D., Fink, T. M. A. & Caldarelli, G. Ensemble approach to the analysis of weighted networks. Phys. Rev. E 76, 016101 (2007).

    ADS  Google Scholar 

  57. 57.

    Saramäki, J., Kivelä, M., Onnela, J.-P., Kaski, K. & Kertész, J. Generalizations of the clustering coefficient to weighted complex networks. Phys. Rev. E 75, 027105 (2007).

    ADS  Google Scholar 

  58. 58.

    Milo, R. et al. Network motifs: simple building blocks of complex networks. Science 298, 824–827 (2002).

    ADS  Google Scholar 

  59. 59.

    Shen-Orr, S. S., Milo, R., Mangan, S. & Alon, U. Network motifs in the transcriptional regulation network of escherichia coli. Nat. Genet. 31, 64 (2002).

    Google Scholar 

  60. 60.

    Garlaschelli, D. & Loffredo, M. I. Patterns of link reciprocity in directed networks. Phys. Rev. Lett. 93, 268701 (2004).

    ADS  Google Scholar 

  61. 61.

    Garlaschelli, D. & Loffredo, M. I. Multispecies grand-canonical models for networks with reciprocity. Phys. Rev. E 73, 015101 (2006).

    ADS  MathSciNet  Google Scholar 

  62. 62.

    Squartini, T. & Garlaschelli, D. in Self-Organizing Systems (eds Kuipers, F. A. & Heegaard, P. E.) 24–35 (Springer Berlin, Heidelberg, 2012).

  63. 63.

    Stouer, D. B., Camacho, J., Jiang, W. & Amaral, L. A. N. Evidence for the existence of a robust pattern of prey selection in food webs. Proc. R. Soc. Lond. B Biol. Sci. 274, 1931–1940 (2007).

    Google Scholar 

  64. 64.

    Squartini, T., van Lelyveld, I. & Garlaschelli, D. Early-warning signals of topological collapse in interbank networks. Sci. Rep. 3, 3357 (2013).

    Google Scholar 

  65. 65.

    Guimerà, R., Sales-Pardo, M. & Amaral, L. A. N. Modularity from uctuations in random graphs and complex networks. Phys. Rev. E 70, 025101 (2004).

    ADS  Google Scholar 

  66. 66.

    Reichardt, J. & Bornholdt, S. Partitioning and modularity of graphs with arbitrary degree distribution. Phys. Rev. E 76, 015102 (2007).

    ADS  Google Scholar 

  67. 67.

    Chung, F. & Lu, L. Connected components in random graphs with given expected degree sequences. Ann. Comb. 6, 125–145 (2002). This paper defines a very popular analytic model of networks with given degree sequence, admitting self-loops and multilinks.

    MathSciNet  MATH  Google Scholar 

  68. 68.

    Bargigli, L. & Gallegati, M. Random digraphs with given expected degree sequences: a model for economic networks. J. Econ. Behav. Organ. 78, 396–411 (2011).

    Google Scholar 

  69. 69.

    Fronczak, P., Fronczak, A. & Bujok, M. Exponential random graph models for networks with community structure. Phys. Rev. E 88, 32810 (2013).

    ADS  Google Scholar 

  70. 70.

    Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. Phys. Rev. E 78, 046110 (2008).

    ADS  Google Scholar 

  71. 71.

    Karrer, B. & Newman, M. E. J. Stochastic blockmodels and community structure in networks. Phys. Rev. E 83, 016107 (2011).

    ADS  MathSciNet  Google Scholar 

  72. 72.

    Peixoto, T. P. Entropy of stochastic blockmodel ensembles. Phys. Rev. E 85, 056122 (2012).

    ADS  Google Scholar 

  73. 73.

    Holme, P., Liljeros, F., Edling, C. R. & Kim, B. J. Network bipartivity. Phys. Rev. E 68, 056107 (2003).

    ADS  Google Scholar 

  74. 74.

    Saracco, F., Di Clemente, R., Gabrielli, A. & Squartini, T. Randomizing bipartite networks: the case of the world trade web. Sci. Rep. 5, 10595 (2015).

    ADS  Google Scholar 

  75. 75.

    Tacchella, A., Cristelli, M., Caldarelli, G., Gabrielli, A. & Pietronero, L. A new metrics for countries’ fitness and products’ complexity. Sci. Rep. 2, 723 (2012).

    ADS  MATH  Google Scholar 

  76. 76.

    Caldarelli, G. et al. A network analysis of countries' export flows: firm grounds for the building blocks of the economy. PLoS ONE 7, e47278 (2012).

    ADS  Google Scholar 

  77. 77.

    Saracco, F., Di Clemente, R., Gabrielli, A. & Squartini, T. Detecting early signs of the 2007–2008 crisis in the world trade. Sci. Rep. 6, 30286 (2016).

    ADS  Google Scholar 

  78. 78.

    Payrató Borrás, C., Hernández, L. & Moreno, Y. Breaking the spell of nestedness. Preprint at (2017).

  79. 79.

    Zhou, T., Ren, J., Medo, M. & Zhang, Y.-C. Bipartite network projection and personal recommendation. Phys. Rev. E 76, 046115 (2007).

    ADS  Google Scholar 

  80. 80.

    Tumminello, M., Aste, T., Di Matteo, T. & Mantegna, R. N. A tool for filtering information in complex systems. Proc. Natl. Acad. Sci. U.S.A. 102, 10421–10426 (2005).

    ADS  Google Scholar 

  81. 81.

    Serrano, M. Á., Boguñá, M. & Vespignani, A. Extracting the multiscale backbone of complex weighted networks. Proc. Natl. Acad. Sci. U.S.A. 106, 6483–6488 (2009).

    ADS  Google Scholar 

  82. 82.

    Slater, P. B. A two-stage algorithm for extracting the multiscale backbone of complex weighted networks. Proc. Natl. Acad. Sci. U.S.A. 106, E66 (2009).

    ADS  Google Scholar 

  83. 83.

    Radicchi, F., Ramasco, J. J. & Fortunato, S. Information filtering in complex weighted networks. Phys. Rev. E 83, 046101 (2011).

    ADS  Google Scholar 

  84. 84.

    Goldberg, D. S. & Roth, F. P. Assessing experimentally derived interactions in a small world. Proc. Natl. Acad. Sci. U.S.A. 100, 4372–4376 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  85. 85.

    Latapy, M., Magnien, C. & Vecchio, N. D. Basic notions for the analysis of large two-mode networks. Soc. Networks 30, 31–48 (2008).

    Google Scholar 

  86. 86.

    Tumminello, M., Miccichè, S., Lillo, F., Piilo, J. & Mantegna, R. N. Statistically validated networks in bipartite complex systems. PLoS ONE 6, e17994 (2011).

    ADS  Google Scholar 

  87. 87.

    Tumminello, M., Lillo, F., Piilo, J. & Mantegna, R. N. Identification of clusters of investors from their real trading activity in a financial market. New J. Phys. 14, 013041 (2012).

    ADS  Google Scholar 

  88. 88.

    Neal, Z. Identifying statistically significant edges in one-mode projections. Soc. Netw. Anal. Min. 3, 915–924 (2013).

    Google Scholar 

  89. 89.

    Zweig, K. A. & Kaufmann, M. A systematic approach to the one-mode projection of bipartite graphs. Soc. Netw. Anal. Min. 1, 187–218 (2011).

    Google Scholar 

  90. 90.

    Horvát, E.-Á. & Zweig, K. A. A fixed degree sequence model for the one-mode projection of multiplex bipartite graphs. Soc. Netw. Anal. Min. 3, 1209–1224 (2013).

    Google Scholar 

  91. 91.

    Gionis, A., & Mannila, H., & Mielikäinen, T. & Tsaparas, P. Assessing data mining results via swap randomization. ACM Trans. Knowl. Discov. Data 1, 14 (2007).

    Google Scholar 

  92. 92.

    Neal, Z. The backbone of bipartite projections: inferring relationships from co-authorship, cosponsorship, co-attendance and other co-behaviors. Soc. Networks 39, 84–97 (2014).

    Google Scholar 

  93. 93.

    Gualdi, S., Cimini, G., Primicerio, K., Di Clemente, R. & Challet, D. Statistically validated network of portfolio overlaps and systemic risk. Sci. Rep. 6, 39467 (2016).

    ADS  Google Scholar 

  94. 94.

    Saracco, F. et al. Inferring monopartite projections of bipartite networks: an entropy-based approach. New J. Phys. 19, 053022 (2017).

    ADS  Google Scholar 

  95. 95.

    Straka, M. J., Caldarelli, G. & Saracco, F. Grand canonical validation of the bipartite international trade network. Phys. Rev. E 96, 022306 (2017).

    ADS  Google Scholar 

  96. 96.

    Pugliese, E. et al. Unfolding the innovation system for the development of countries: co-evolution of science, technology and production. Preprint at (2017).

  97. 97.

    Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).

    ADS  MathSciNet  Google Scholar 

  98. 98.

    Wells, S. J. Financial interlinkages in the United Kingdom's interbank market and the risk of contagion. Bank of England Working Paper (2004).

  99. 99.

    Upper, C. Simulation methods to assess the danger of contagion in interbank markets. J. Financ. Stab. 7, 111–125 (2011).

    Google Scholar 

  100. 100.

    Anand, K. et al. The missing links: a global study on uncovering financial network structures from partial data. J. Financ. Stab. 35, 107–119 (2018).

    Google Scholar 

  101. 101.

    Kossinets, G. Effects of missing data in social networks. Soc. Networks 28, 247–268 (2006).

    Google Scholar 

  102. 102.

    Lynch, C. How do your data grow? Nature 455, 28 (2008).

    ADS  Google Scholar 

  103. 103.

    Amaral, L. A. N. A truer measure of our ignorance. Proc. Natl. Acad. Sci. U.S.A. 105, 6795–6796 (2008).

    ADS  Google Scholar 

  104. 104.

    Guimerá, R. & Sales-Pardo, M. Missing and spurious interactions and the reconstruction of complex networks. Proc. Natl. Acad. Sci. U.S.A. 106, 22073–22078 (2009).

    ADS  Google Scholar 

  105. 105.

    Lu, L. & Zhou, T. Link prediction in complex networks: a survey. Phys. A Stat. Mech. Appl. 390, 1150–1170 (2011).

    Google Scholar 

  106. 106.

    Squartini, T., Caldarelli, G., Cimini, G., Gabrielli, A. & Garlaschelli, D. Reconstruction methods for networks: the case of economic and financial systems. Phys. Rep. 757, 1–47 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  107. 107.

    Boguñá, M. & Pastor-Satorras, R. Class of correlated random networks with hidden variables. Phys. Rev. E 68, 036112 (2003).

    ADS  Google Scholar 

  108. 108.

    Garlaschelli, D., Battiston, S., Castri, M., Servedio, V. D. P. & Caldarelli, G. The scale-free topology of market investments. Phys. A Stat. Mech. Appl. 350, 491–499 (2005).

    MathSciNet  Google Scholar 

  109. 109.

    De Masi, G., Iori, G. & Caldarelli, G. Fitness model for the italian interbank money market. Phys. Rev. E 74, 066112 (2006).

    ADS  Google Scholar 

  110. 110.

    Musmeci, N., Battiston, S., Caldarelli, G., Puliga, M. & Gabrielli, A. Bootstrapping topological properties and systemic risk of complex networks using the fitness model. J. Stat. Phys. 151, 1–15 (2013).

    MathSciNet  MATH  Google Scholar 

  111. 111.

    Cimini, G., Squartini, T., Gabrielli, A. & Garlaschelli, D. Estimating topological properties of weighted networks from limited information. Phys. Rev. E 92, 040802 (2015).

    ADS  Google Scholar 

  112. 112.

    Cimini, G., Squartini, T., Garlaschelli, D. & Gabrielli, A. Systemic risk analysis on reconstructed economic and financial networks. Sci. Rep. 5, 15758 (2015). This paper uses ERGs in combination with the fitness model to reconstruct networks from partial information.

    ADS  Google Scholar 

  113. 113.

    Squartini, T., Cimini, G., Gabrielli, A. & Garlaschelli, D. Network reconstruction via density sampling. Appl. Netw. Sci. 2, 3 (2017).

    Google Scholar 

  114. 114.

    Squartini, T. et al. Enhanced capital-asset pricing model for the reconstruction of bipartite financial networks. Phys. Rev. E 96, 032315 (2017).

    ADS  Google Scholar 

  115. 115.

    Berg, J. & Lässig, M. Correlated random networks. Phys. Rev. Lett. 89, 228701 (2002).

    ADS  Google Scholar 

  116. 116.

    Park, M. E. J. & Newman, J. Solution of the two-star model of a network. Phys. Rev. E 70, 066146 (2004).

    ADS  MathSciNet  Google Scholar 

  117. 117.

    Yin, M. & Zhu, L. Reciprocity in directed networks. Phys. A Stat. Mech. Appl. 447, 71–84 (2016).

    MathSciNet  MATH  Google Scholar 

  118. 118.

    Park, J. & Newman, M. E. J. Solution for the properties of a clustered network. Phys. Rev. E 72, 026136 (2005).

    ADS  Google Scholar 

  119. 119.

    Fronczak, P., Fronczak, A. & Holyst, J. A. Phase transitions in social networks. Eur. Phys. J. B 59, 133–139 (2007).

    ADS  MATH  Google Scholar 

  120. 120.

    Bianconi, G., Coolen, A. C. C. & Perez Vicente, C. J. Entropies of complex networks with hierarchically constrained topologies. Phys. Rev. E 78, 016114 (2008).

    ADS  MathSciNet  Google Scholar 

  121. 121.

    Bianconi, G. Entropy of network ensembles. Phys. Rev. E 79, 036114 (2009).

    ADS  MathSciNet  Google Scholar 

  122. 122.

    Mondragón, R. J. Network null-model based on maximal entropy and the rich-club. J. Complex Netw. 2, 288–298 (2014).

    Google Scholar 

  123. 123.

    Annibale, A., Coolen, A. C. C., Fernandes, L. P., Fraternali, F. & Kleinjung, J. Tailored graph ensembles as proxies or null models for real networks I: tools for quantifying structure. J. Phys. A Math. Theor. 42, 485001 (2009).

    MathSciNet  MATH  Google Scholar 

  124. 124.

    Roberts, E. S., Schlitt, T. & Coolen, A. C. C. Tailored graph ensembles as proxies or null models for real networks II: results on directed graphs. J. Phys. A Math. Theor. 44, 275002 (2011).

    ADS  MathSciNet  MATH  Google Scholar 

  125. 125.

    Roberts, E. S. & Coolen, A. C. C. Entropies of tailored random graph ensembles: bipartite graphs, generalized degrees, and node neighbourhoods. J. Phys. A Math. Theor. 47, 435101 (2014).

    ADS  MathSciNet  MATH  Google Scholar 

  126. 126.

    Artzy-Randrup, Y. & Stone, L. Generating uniformly distributed random networks. Phys. Rev. E 72, 056708 (2005).

    ADS  MathSciNet  Google Scholar 

  127. 127.

    Coolen, A. C. C., De Martino, A. & Annibale, A. Constrained markovian dynamics of random graphs. J. Stat. Phys. 136, 1035–1067 (2009). This paper introduces Monte Carlo processes for uniform sampling of network ensembles.

    ADS  MathSciNet  MATH  Google Scholar 

  128. 128.

    Roberts, E. S. & Coolen, A. C. C. Unbiased degree-preserving randomization of directed binary networks. Phys. Rev. E 85, 046103 (2012).

    ADS  Google Scholar 

  129. 129.

    Strauss, D. & Ikeda, M. Pseudolikelihood estimation for social networks. J. Am. Stat. Assoc. 85, 204–212 (1990).

    MathSciNet  Google Scholar 

  130. 130.

    van Duijn, M. A. J., Gile, K. J. & Handcock, M. S. A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models. Soc. Networks 31, 52–62 (2009).

    Google Scholar 

  131. 131.

    Snijders, T. A. B., Koskinen, J. & Schweinberger, M. Maximum likelihood estimation for social network dynamics. Ann. Appl. Stat. 4, 567–588 (2010).

    MathSciNet  MATH  Google Scholar 

  132. 132.

    Schweinberger, M. Instability, sensitivity, and degeneracy of discrete exponential families. J. Am. Stat. Assoc. 106, 1361–1370 (2011).

    MathSciNet  MATH  Google Scholar 

  133. 133.

    Desmarais, B. A. & Cranmer, S. J. Statistical mechanics of networks: estimation and uncertainty. Phys. A Stat. Mech. Appl. 391, 1865–1876 (2012).

    Google Scholar 

  134. 134.

    Chatterjee, S. & Diaconis, P. Estimating and understanding exponential random graph models. Ann. Stat. 41, 2428–2461 (2013).

    MathSciNet  MATH  Google Scholar 

  135. 135.

    Horvát, S., Czabarka, É. & Toroczkai, Z. Reducing degeneracy in maximum entropy models of networks. Phys. Rev. Lett. 114, 158701 (2015).

    ADS  Google Scholar 

  136. 136.

    Hastings, W. K. Monte carlo sampling methods using markov chains and their applications. Biometrika 57, 97–109 (1970).

    MathSciNet  MATH  Google Scholar 

  137. 137.

    Mahadevan, P., Krioukov, D., Fall, K. & Vahdat, A. Systematic topology analysis and generation using degree correlations. SIGCOMM Comput. Commun. Rev. 36, 135–146 (2006).

    Google Scholar 

  138. 138.

    Orsini, C. et al. Quantifying randomness in real networks. Nat. Commun. 6, 8627 (2015). This paper uses the dk -series approach to show that degree distributions, degree correlations and clustering often represent sufficient statistics to describe a network.

    Google Scholar 

  139. 139.

    Foster, D., Foster, J., Paczuski, M. & Grassberger, P. Communities, clustering phase transitions, and hysteresis: pitfalls in constructing network ensembles. Phys. Rev. E 81, 046115 (2010).

    ADS  MathSciNet  Google Scholar 

  140. 140.

    Fischer, R., Leitão, J. C., Peixoto, T. P. & Altmann, E. G. Sampling motif-constrained ensembles of networks. Phys. Rev. Lett. 115, 188701 (2015).

    ADS  Google Scholar 

  141. 141.

    Fugao Wang & Landau, D. P. Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett. 86, 2050–2053 (2001).

    ADS  Google Scholar 

  142. 142.

    Kivelä, M. et al. Multilayer networks. J. Complex Netw. 2, 203–271 (2014).

    Google Scholar 

  143. 143.

    Boccaletti, S. et al. The structure and dynamics of multilayer networks. Phys. Rep. 544, 1–122 (2014).

    ADS  MathSciNet  Google Scholar 

  144. 144.

    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 

  145. 145.

    Bianconi, G. Statistical mechanics of multiplex networks: entropy and overlap. Phys. Rev. E 87, 062806 (2013). This paper develops the ERG framework for multiplex networks.

    ADS  Google Scholar 

  146. 146.

    Gemmetto, V. & Garlaschelli, D. Multiplexity versus correlation: the role of local constraints in real multiplexes. Sci. Rep. 5, 9120 (2015).

    ADS  Google Scholar 

  147. 147.

    Menichetti, G., Remondini, D., Panzarasa, P., Mondragón, R. J. & Bianconi, G. Weighted multiplex networks. PLoS ONE 9, e97857 (2014).

    ADS  Google Scholar 

  148. 148.

    Menichetti, G., Remondini, D. & Bianconi, G. Correlations between weights and overlap in ensembles of weighted multiplex networks. Phys. Rev. E 90, 062817 (2014).

    ADS  Google Scholar 

  149. 149.

    Sagarra, O., Pérez Vicente, C. J. & Díaz-Guilera, A. Statistical mechanics of multiedge networks. Phys. Rev. E 88, 062806 (2013).

    ADS  Google Scholar 

  150. 150.

    Sagarra, O., Font-Clos, F., Péerez-Vicente, C. J. & Díaz-Guilera, A. The configuration multiedge model: assessing the effect of fixing node strengths on weighted network magnitudes. Europhys. Lett. 107, 38002 (2014).

    ADS  Google Scholar 

  151. 151.

    Sagarra, O., Pérez Vicente, C. J. & Díaz-Guilera, A. Role of adjacency-matrix degeneracy in maximum-entropy-weighted network models. Phys. Rev. E 92, 052816 (2015).

    ADS  Google Scholar 

  152. 152.

    Mastrandrea, R., Squartini, T., Fagiolo, G. & Garlaschelli, D. Reconstructing the world trade multiplex: the role of intensive and extensive biases. Phys. Rev. E 90, 062804 (2014).

    ADS  Google Scholar 

  153. 153.

    Zuev, K., Eisenberg, O. & Krioukov, D. Exponential random simplicial complexes. J. Phys. A Math. Theor. 48, 465002 (2015).

    ADS  MathSciNet  MATH  Google Scholar 

  154. 154.

    Courtney, O. T. & Bianconi, G. Generalized network structures: the configuration model and the canonical ensemble of simplicial complexes. Phys. Rev. E 93, 062311 (2016).

    ADS  Google Scholar 

  155. 155.

    Young, J.-G., Petri, G., Vaccarino, F. & Patania, A. Construction of and efficient sampling from the simplicial configuration model. Phys. Rev. E 96, 032312 (2017).

    ADS  Google Scholar 

  156. 156.

    Dixit, P. D. et al. Perspective: maximum caliber is a general variational principle for dynamical systems. J. Chem. Phys. 148, 010901 (2018).

    ADS  Google Scholar 

  157. 157.

    Newman, M. E. J., Strogatz, S. H. & Watts, D. J. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118 (2001).

    ADS  Google Scholar 

  158. 158.

    Itzkovitz, S., Milo, R., Kashtan, N., Newman, M. E. J. & Alon, U. Reply to comment on ‘subgraphs in random networks’. Phys. Rev. E 70, 058102 (2004).

    ADS  MathSciNet  Google Scholar 

  159. 159.

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

    ADS  Google Scholar 

  160. 160.

    Zamora-Lopez, G., Zlatic, V., Zhou, C., Stefancic, H. & Kurths, J. Reciprocity of networks with degree correlations and arbitrary degree sequences. Phys. Rev. E 77, 016106 (2008).

    ADS  Google Scholar 

  161. 161.

    Zlatic, V. et al. On the rich-club effect in dense and weighted networks. Eur. Phys. J. B 67, 271–275 (2009).

    ADS  Google Scholar 

  162. 162.

    Tabourier, L., Roth, C. & Cointet, J.-P. Generating constrained random graphs using multiple edge switches. J. Exp. Algorithm. 16, 1.1–1.15 (2011).

    MathSciNet  MATH  Google Scholar 

  163. 163.

    Carstens, C. J. & Horadam, K. J. Switching edges to randomize networks: what goes wrong and how to fix it. J. Complex Netw. 5, 337–351 (2017).

    MathSciNet  Google Scholar 

  164. 164.

    Del Genio, C. I., Kim, H., Toroczkai, Z. & Bassler, K. E. Efficient and exact sampling of simple graphs with given arbitrary degree sequence. PLoS ONE 5, e10012 (2010).

    Google Scholar 

  165. 165.

    Blitzstein, J. & Diaconis, P. A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math. 6, 489–522 (2011).

    MathSciNet  MATH  Google Scholar 

  166. 166.

    Kim, H., Del Genio, C. I., Bassler, K. E. & Toroczkai, Z. Constructing and sampling directed graphs with given degree sequences. New J. Phys. 14, 023012 (2012).

    ADS  Google Scholar 

  167. 167.

    Newman, M. E. J. Random graphs with clustering. Phys. Rev. Lett. 103, 058701 (2009).

    ADS  Google Scholar 

  168. 168.

    Melnik, S., Hackett, A., Porter, M. A., Mucha, P. J. & Gleeson, J. P. The unreasonable effectiveness of tree-based theory for networks with clustering. Phys. Rev. E 83, 036112 (2011).

    ADS  MathSciNet  Google Scholar 

  169. 169.

    Burda, Z. & Krzywicki, A. Uncorrelated random networks. Phys. Rev. E 67, 046118 (2003).

    ADS  Google Scholar 

  170. 170.

    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 

  171. 171.

    Neyman, J. & Pearson, E. S. On the problem of the most efficient tests of statistical hypotheses. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 231, 289–337 (1933).

    ADS  MATH  Google Scholar 

  172. 172.

    Burnham, K. P. & Anderson, D. R. (eds) Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (Springer-Verlag, New York, 2002).

  173. 173.

    Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control 19, 716–723 (1974).

    ADS  MathSciNet  MATH  Google Scholar 

  174. 174.

    Wagenmakers, E.-J. & Farrell, S. Aic model selection using akaike weights. Psychon. Bull. Rev. 11, 192–196 (2004).

    Google Scholar 

  175. 175.

    Burnham, K. P. & Anderson, D. R. Multimodel inference: understanding aic and bic in model selection. Sociol. Methods Res. 33, 261–304 (2004).

    MathSciNet  Google Scholar 

  176. 176.

    Braunstein, S. L., Ghosh, S. & Severini, S. The laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states. Ann. Comb. 10, 291–317 (2006).

    MathSciNet  MATH  Google Scholar 

  177. 177.

    Anand, K., Bianconi, G. & Severini, S. Shannon and von neumann entropy of random networks with heterogeneous expected degree. Phys. Rev. E 83, 036109 (2011).

    ADS  MathSciNet  Google Scholar 

  178. 178.

    Anand, K., Krioukov, D. & Bianconi, G. Entropy distribution and condensation in random networks with a given degree distribution. Phys. Rev. E 89, 062807 (2014).

    ADS  Google Scholar 

  179. 179.

    De Domenico, M. & Biamonte, J. Spectral entropies as information-theoretic tools for complex network comparison. Phys. Rev. X 6, 041062 (2016).

    Google Scholar 

  180. 180.

    Delvenne, J.-C., Lambiotte, R. & Rocha, L. E. C. Diffusion on networked systems is a question of time or structure. Nat. Commun. 6, 7366 (2015).

    Google Scholar 

  181. 181.

    Masuda, N., Porter, M. A. & Lambiotte, R. Random walks and diusion on networks. Phys. Rep. 716-717, 1–58 (2017).

    ADS  MATH  Google Scholar 

  182. 182.

    Demetrius, L. & Manke, T. Robustness and network evolution-an entropic principle. Phys. A Stat. Mech. Appl. 346, 682–696 (2005).

    Google Scholar 

  183. 183.

    Lott, J. & Villani, C. Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169, 903–991 (2009).

    MathSciNet  MATH  Google Scholar 

  184. 184.

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

    ADS  Google Scholar 

  185. 185.

    Sandhu, R. S., Georgiou, T. T. & Tannenbaum, A. R. Ricci curvature: an economic indicator for market fragility and systemic risk. Sci. Adv. 2, e1501495 (2016).

    ADS  Google Scholar 

Download references


G. Cimini, T.S., F.S. and G. Caldarelli acknowledge support from the EU projects CoeGSS (grant no. 676547), Openmaker (grant no. 687941), SoBigData (grant no. 654024) and DOLFINS (grant no. 640772). D.G. acknowledges support from the Dutch Econophysics Foundation (Stichting Econophysics, Leiden, Netherlands). A.G. acknowledges support from the CNR PNR Project CRISISLAB funded by the Italian government. G. Caldarelli also acknowledges the Israeli–Italian project MAC2MIC financed by Italian MAECI.

Author information




All authors contributed to all aspects of manuscript preparation, revision and editing.

Corresponding author

Correspondence to Guido Caldarelli.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s noteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.



Also known as vertices. Basic elements in the network or graph under consideration.


Also known as edges. Connections or interactions between two nodes or vertices of a network or graph, representing the fundamental degrees of freedom of the system.


A type of network for which every link is bidirectional, such as a network of colleagues (Alice works with Bob implies that Bob works with Alice).


A type of network for which links have a direction, such as an ecological network in which links represent predation (lions eat antelopes, but antelopes do not eat lions).


A type of network for which links are unweighted, that is, they can be described by either a 1 (the link exists) or a 0 (it does not).


A type of network for which links have weights, which represent, for example, carrying capacities or interaction strengths.


The tendency of node triples to be connected together, that is, to form triangles.


The mathematical abstraction of a network comprising a set of N vertices and a set of E edges, each associated with two nodes.


The fraction of possible connections that are actually realized in a network. Real-world networks are typically sparse, as their density is much smaller than 1.

ErdösRényi model

The random graph model in which a link between any two nodes exists with constant probability p, independent of all other links.


The tendency of nodes in a directed network to be mutually linked.


The tendency of nodes to be linked to other nodes with similar degrees. Conversely, disassortativity is the tendency of nodes to be linked to other nodes with dissimilar degrees.


The pattern in which the interactions of nodes with low degree are a subset of the interactions of nodes with high degree.


The core component of the network that is extracted by filtering redundant information.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Further reading