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.
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.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
A positive statistical benchmark to assess network agreement
Nature Communications Open Access 24 May 2023
Sustainable development goals as unifying narratives in large UK firms’ Twitter discussions
Scientific Reports Open Access 29 April 2023
Supercritical fluids behave as complex networks
Nature Communications Open Access 10 April 2023
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 digital issues and online access to articles
$99.00 per year
only $8.25 per issue
Rent or buy this article
Get just this article for as long as you need it
Prices may be subject to local taxes which are calculated during checkout
Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008).
Barabási, A.-L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999).
Yook, S. H., Jeong, H., Barabási, A.-L. & Tu, Y. Weighted evolving networks. Phys. Rev. Lett. 86, 5835–5838 (2001).
Barrat, A., Barthelemy, M. & Vespignani, A. Weighted evolving networks: coupling topology and weight dynamics. Phys. Rev. Lett. 92, 228701 (2004).
Newman, M. E. J. & Girvan, M. Finding and evaluating community structure in networks. Phys. Rev. E 69, 026113 (2004).
Fortunato, S. Community detection in graphs. Phys. Rep. 486, 75–174 (2010).
Watts, D. J. & Strogatz, S. H. Collective dynamics of small-world networks. Nature 393, 440–442 (1998).
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).
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).
Albert, R. & Barabási, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys0. 74, 47–97 (2002).
Newman, M. E. J. The structure and function of complex networks. SIAM Rev. Soc. Ind. Appl. Math. 45, 167–256 (2003).
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. & Hwang, D.-U. Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006).
Bianconi, G. & Barabási, A. L. Bose-einstein condensation in complex network. Phys. Rev. Lett. 86, 5632–5635 (2001).
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).
Dorogovtsev, S. N., Mendes, J. F. F. & Samukhin, A. N. Structure of growing networks with preferential linking. Phys. Rev. Lett. 85, 4633–4636 (2000).
Medo, M., Cimini, G. & Gualdi, S. Temporal effects in the growth of networks. Phys. Rev. Lett. 107, 238701 (2011).
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.
Frank, O. & Strauss, D. Markov graphs. J. Am. Stat. Assoc. 81, 832–842 (1986).
Strauss, D. On a general class of models for interaction. SIAM Rev. Soc. Ind. Appl. Math. 28, 513–527 (1986).
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).
Anderson, C. J., Wasserman, S. & Crouch, B. A p* primer: logit models for social networks. Soc. Networks 21, 37–66 (1999).
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).
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).
Cranmer, S. J. & Desmarais, B. A. Inferential network analysis with exponential random graph models. Polit. Anal. 19, 6686 (2011).
Snijders, T. A. B. Statistical models for social networks. Annu. Rev. Sociol. 37, 131–153 (2011).
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.
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.
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).
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).
Jaynes, E. T. On the rationale of maximum-entropy methods. Proc. IEEE 70, 939–952 (1982).
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.
Squartini, T., Mastrandrea, R. & Garlaschelli, D. Unbiased sampling of network ensembles. New J. Phys. 17, 023052 (2015).
Anand, K. & Bianconi, G. Entropy measures for networks: toward an information theory of complex topologies. Phys. Rev. E 80, 045102 (2009).
Squartini, T., de Mol, J., den Hollander, F. & Garlaschelli, D. Breaking of ensemble equivalence in networks. Phys. Rev. Lett. 115, 268701 (2015).
Squartini, T. & Garlaschelli, D. Reconnecting statistical physics and combinatorics beyond ensemble equivalence. Preprint at https://arxiv.org/abs/1710.11422 (2018).
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.
Garlaschelli, D. & Loffredo, M. I. Maximum likelihood: extracting unbiased information from complex networks. Phys. Rev. E 78, 015101(R) (2008).
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.
Erdos, P. & Rényi, A. On random graphs. Publ. Math. Debr. 6, 290–297 (1959). This paper introduces the first statistical ensemble of random graphs.
Serrano, M. Á. & Boguñá, M. Weighted configuration model. AIP Conf. Proc. 776, 101–107 (2005).
Mastrandrea, R., Squartini, T., Fagiolo, G. & Garlaschelli, D. Enhanced reconstruction of weighted networks from strengths and degrees. New J. Phys. 16, 043022 (2014).
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.
Park, J. & Newman, M. E. J. Origin of degree correlations in the internet and other networks. Phys. Rev. E 68, 026112 (2003).
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).
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).
Colizza, V., Flammini, A., Serrano, M. A. & Vespignani, A. Detecting rich-club ordering in complex networks. Nat. Phys. 2, 110 (2006).
Serrano, M. Á., Boguñá, M. & Pastor-Satorras, R. Correlations in weighted networks. Phys. Rev. E 74, 055101 (2006).
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).
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).
Opsahl, T., Colizza, V., Panzarasa, P. & Ramasco, J. J. Prominence and control: the weighted rich-club effect. Phys. Rev. Lett. 101, 168702 (2008).
Serrano, M. Á. & Boguñá, M. Topology of the world trade web. Phys. Rev. E 68, 015101 (2003).
Garlaschelli, D. & Loffredo, M. I. Fitness-dependent topological properties of the world trade web. Phys. Rev. Lett. 93, 188701 (2004).
Garlaschelli, D. & Loffredo, M. I. Structure and evolution of the world trade network. Phys. A Stat. Mech. Appl. 355, 138–144 (2005).
Fagiolo, G., Reyes, J. & Schiavo, S. World trade web: topological properties, dynamics, and evolution. Phys. Rev. E 79, 036115 (2009).
Newman, M. E. J. Analysis of weighted networks. Phys. Rev. E 70, 056131 (2004).
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).
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).
Milo, R. et al. Network motifs: simple building blocks of complex networks. Science 298, 824–827 (2002).
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).
Garlaschelli, D. & Loffredo, M. I. Patterns of link reciprocity in directed networks. Phys. Rev. Lett. 93, 268701 (2004).
Garlaschelli, D. & Loffredo, M. I. Multispecies grand-canonical models for networks with reciprocity. Phys. Rev. E 73, 015101 (2006).
Squartini, T. & Garlaschelli, D. in Self-Organizing Systems (eds Kuipers, F. A. & Heegaard, P. E.) 24–35 (Springer Berlin, Heidelberg, 2012).
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).
Squartini, T., van Lelyveld, I. & Garlaschelli, D. Early-warning signals of topological collapse in interbank networks. Sci. Rep. 3, 3357 (2013).
Guimerà, R., Sales-Pardo, M. & Amaral, L. A. N. Modularity from uctuations in random graphs and complex networks. Phys. Rev. E 70, 025101 (2004).
Reichardt, J. & Bornholdt, S. Partitioning and modularity of graphs with arbitrary degree distribution. Phys. Rev. E 76, 015102 (2007).
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.
Bargigli, L. & Gallegati, M. Random digraphs with given expected degree sequences: a model for economic networks. J. Econ. Behav. Organ. 78, 396–411 (2011).
Fronczak, P., Fronczak, A. & Bujok, M. Exponential random graph models for networks with community structure. Phys. Rev. E 88, 32810 (2013).
Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. Phys. Rev. E 78, 046110 (2008).
Karrer, B. & Newman, M. E. J. Stochastic blockmodels and community structure in networks. Phys. Rev. E 83, 016107 (2011).
Peixoto, T. P. Entropy of stochastic blockmodel ensembles. Phys. Rev. E 85, 056122 (2012).
Holme, P., Liljeros, F., Edling, C. R. & Kim, B. J. Network bipartivity. Phys. Rev. E 68, 056107 (2003).
Saracco, F., Di Clemente, R., Gabrielli, A. & Squartini, T. Randomizing bipartite networks: the case of the world trade web. Sci. Rep. 5, 10595 (2015).
Tacchella, A., Cristelli, M., Caldarelli, G., Gabrielli, A. & Pietronero, L. A new metrics for countries’ fitness and products’ complexity. Sci. Rep. 2, 723 (2012).
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).
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).
Payrató Borrás, C., Hernández, L. & Moreno, Y. Breaking the spell of nestedness. Preprint at https://arxiv.org/abs/1711.03134 (2017).
Zhou, T., Ren, J., Medo, M. & Zhang, Y.-C. Bipartite network projection and personal recommendation. Phys. Rev. E 76, 046115 (2007).
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).
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).
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).
Radicchi, F., Ramasco, J. J. & Fortunato, S. Information filtering in complex weighted networks. Phys. Rev. E 83, 046101 (2011).
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).
Latapy, M., Magnien, C. & Vecchio, N. D. Basic notions for the analysis of large two-mode networks. Soc. Networks 30, 31–48 (2008).
Tumminello, M., Miccichè, S., Lillo, F., Piilo, J. & Mantegna, R. N. Statistically validated networks in bipartite complex systems. PLoS ONE 6, e17994 (2011).
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).
Neal, Z. Identifying statistically significant edges in one-mode projections. Soc. Netw. Anal. Min. 3, 915–924 (2013).
Zweig, K. A. & Kaufmann, M. A systematic approach to the one-mode projection of bipartite graphs. Soc. Netw. Anal. Min. 1, 187–218 (2011).
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).
Gionis, A., & Mannila, H., & Mielikäinen, T. & Tsaparas, P. Assessing data mining results via swap randomization. ACM Trans. Knowl. Discov. Data 1, 14 (2007).
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).
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).
Saracco, F. et al. Inferring monopartite projections of bipartite networks: an entropy-based approach. New J. Phys. 19, 053022 (2017).
Straka, M. J., Caldarelli, G. & Saracco, F. Grand canonical validation of the bipartite international trade network. Phys. Rev. E 96, 022306 (2017).
Pugliese, E. et al. Unfolding the innovation system for the development of countries: co-evolution of science, technology and production. Preprint at https://arxiv.org/abs/1707.05146 (2017).
Pastor-Satorras, R., Castellano, C., Van Mieghem, P. & Vespignani, A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).
Wells, S. J. Financial interlinkages in the United Kingdom's interbank market and the risk of contagion. Bank of England Working Paper https://doi.org/10.2139/ssrn.641288 (2004).
Upper, C. Simulation methods to assess the danger of contagion in interbank markets. J. Financ. Stab. 7, 111–125 (2011).
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).
Kossinets, G. Effects of missing data in social networks. Soc. Networks 28, 247–268 (2006).
Lynch, C. How do your data grow? Nature 455, 28 (2008).
Amaral, L. A. N. A truer measure of our ignorance. Proc. Natl. Acad. Sci. U.S.A. 105, 6795–6796 (2008).
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).
Lu, L. & Zhou, T. Link prediction in complex networks: a survey. Phys. A Stat. Mech. Appl. 390, 1150–1170 (2011).
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).
Boguñá, M. & Pastor-Satorras, R. Class of correlated random networks with hidden variables. Phys. Rev. E 68, 036112 (2003).
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).
De Masi, G., Iori, G. & Caldarelli, G. Fitness model for the italian interbank money market. Phys. Rev. E 74, 066112 (2006).
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).
Cimini, G., Squartini, T., Gabrielli, A. & Garlaschelli, D. Estimating topological properties of weighted networks from limited information. Phys. Rev. E 92, 040802 (2015).
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.
Squartini, T., Cimini, G., Gabrielli, A. & Garlaschelli, D. Network reconstruction via density sampling. Appl. Netw. Sci. 2, 3 (2017).
Squartini, T. et al. Enhanced capital-asset pricing model for the reconstruction of bipartite financial networks. Phys. Rev. E 96, 032315 (2017).
Berg, J. & Lässig, M. Correlated random networks. Phys. Rev. Lett. 89, 228701 (2002).
Park, M. E. J. & Newman, J. Solution of the two-star model of a network. Phys. Rev. E 70, 066146 (2004).
Yin, M. & Zhu, L. Reciprocity in directed networks. Phys. A Stat. Mech. Appl. 447, 71–84 (2016).
Park, J. & Newman, M. E. J. Solution for the properties of a clustered network. Phys. Rev. E 72, 026136 (2005).
Fronczak, P., Fronczak, A. & Holyst, J. A. Phase transitions in social networks. Eur. Phys. J. B 59, 133–139 (2007).
Bianconi, G., Coolen, A. C. C. & Perez Vicente, C. J. Entropies of complex networks with hierarchically constrained topologies. Phys. Rev. E 78, 016114 (2008).
Bianconi, G. Entropy of network ensembles. Phys. Rev. E 79, 036114 (2009).
Mondragón, R. J. Network null-model based on maximal entropy and the rich-club. J. Complex Netw. 2, 288–298 (2014).
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).
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).
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).
Artzy-Randrup, Y. & Stone, L. Generating uniformly distributed random networks. Phys. Rev. E 72, 056708 (2005).
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.
Roberts, E. S. & Coolen, A. C. C. Unbiased degree-preserving randomization of directed binary networks. Phys. Rev. E 85, 046103 (2012).
Strauss, D. & Ikeda, M. Pseudolikelihood estimation for social networks. J. Am. Stat. Assoc. 85, 204–212 (1990).
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).
Snijders, T. A. B., Koskinen, J. & Schweinberger, M. Maximum likelihood estimation for social network dynamics. Ann. Appl. Stat. 4, 567–588 (2010).
Schweinberger, M. Instability, sensitivity, and degeneracy of discrete exponential families. J. Am. Stat. Assoc. 106, 1361–1370 (2011).
Desmarais, B. A. & Cranmer, S. J. Statistical mechanics of networks: estimation and uncertainty. Phys. A Stat. Mech. Appl. 391, 1865–1876 (2012).
Chatterjee, S. & Diaconis, P. Estimating and understanding exponential random graph models. Ann. Stat. 41, 2428–2461 (2013).
Horvát, S., Czabarka, É. & Toroczkai, Z. Reducing degeneracy in maximum entropy models of networks. Phys. Rev. Lett. 114, 158701 (2015).
Hastings, W. K. Monte carlo sampling methods using markov chains and their applications. Biometrika 57, 97–109 (1970).
Mahadevan, P., Krioukov, D., Fall, K. & Vahdat, A. Systematic topology analysis and generation using degree correlations. SIGCOMM Comput. Commun. Rev. 36, 135–146 (2006).
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.
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).
Fischer, R., Leitão, J. C., Peixoto, T. P. & Altmann, E. G. Sampling motif-constrained ensembles of networks. Phys. Rev. Lett. 115, 188701 (2015).
Fugao Wang & Landau, D. P. Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett. 86, 2050–2053 (2001).
Kivelä, M. et al. Multilayer networks. J. Complex Netw. 2, 203–271 (2014).
Boccaletti, S. et al. The structure and dynamics of multilayer networks. Phys. Rep. 544, 1–122 (2014).
De Domenico, M., Granell, C., Porter, M. A. & Arenas, A. The physics of spreading processes in multilayer networks. Nat. Phys. 12, 901–906 (2016).
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.
Gemmetto, V. & Garlaschelli, D. Multiplexity versus correlation: the role of local constraints in real multiplexes. Sci. Rep. 5, 9120 (2015).
Menichetti, G., Remondini, D., Panzarasa, P., Mondragón, R. J. & Bianconi, G. Weighted multiplex networks. PLoS ONE 9, e97857 (2014).
Menichetti, G., Remondini, D. & Bianconi, G. Correlations between weights and overlap in ensembles of weighted multiplex networks. Phys. Rev. E 90, 062817 (2014).
Sagarra, O., Pérez Vicente, C. J. & Díaz-Guilera, A. Statistical mechanics of multiedge networks. Phys. Rev. E 88, 062806 (2013).
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).
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).
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).
Zuev, K., Eisenberg, O. & Krioukov, D. Exponential random simplicial complexes. J. Phys. A Math. Theor. 48, 465002 (2015).
Courtney, O. T. & Bianconi, G. Generalized network structures: the configuration model and the canonical ensemble of simplicial complexes. Phys. Rev. E 93, 062311 (2016).
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).
Dixit, P. D. et al. Perspective: maximum caliber is a general variational principle for dynamical systems. J. Chem. Phys. 148, 010901 (2018).
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).
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).
Catanzaro, M., Boguñá, M. & Pastor-Satorras, R. Generation of uncorrelated random scalefree networks. Phys. Rev. E 71, 027103 (2005).
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).
Zlatic, V. et al. On the rich-club effect in dense and weighted networks. Eur. Phys. J. B 67, 271–275 (2009).
Tabourier, L., Roth, C. & Cointet, J.-P. Generating constrained random graphs using multiple edge switches. J. Exp. Algorithm. 16, 1.1–1.15 (2011).
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).
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).
Blitzstein, J. & Diaconis, P. A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math. 6, 489–522 (2011).
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).
Newman, M. E. J. Random graphs with clustering. Phys. Rev. Lett. 103, 058701 (2009).
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).
Burda, Z. & Krzywicki, A. Uncorrelated random networks. Phys. Rev. E 67, 046118 (2003).
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).
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).
Burnham, K. P. & Anderson, D. R. (eds) Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (Springer-Verlag, New York, 2002).
Akaike, H. A new look at the statistical model identification. IEEE Trans. Autom. Control 19, 716–723 (1974).
Wagenmakers, E.-J. & Farrell, S. Aic model selection using akaike weights. Psychon. Bull. Rev. 11, 192–196 (2004).
Burnham, K. P. & Anderson, D. R. Multimodel inference: understanding aic and bic in model selection. Sociol. Methods Res. 33, 261–304 (2004).
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).
Anand, K., Bianconi, G. & Severini, S. Shannon and von neumann entropy of random networks with heterogeneous expected degree. Phys. Rev. E 83, 036109 (2011).
Anand, K., Krioukov, D. & Bianconi, G. Entropy distribution and condensation in random networks with a given degree distribution. Phys. Rev. E 89, 062807 (2014).
De Domenico, M. & Biamonte, J. Spectral entropies as information-theoretic tools for complex network comparison. Phys. Rev. X 6, 041062 (2016).
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).
Masuda, N., Porter, M. A. & Lambiotte, R. Random walks and diusion on networks. Phys. Rep. 716-717, 1–58 (2017).
Demetrius, L. & Manke, T. Robustness and network evolution-an entropic principle. Phys. A Stat. Mech. Appl. 346, 682–696 (2005).
Lott, J. & Villani, C. Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169, 903–991 (2009).
Sandhu, R. et al. Graph curvature for differentiating cancer networks. Sci. Rep. 5, 12323 (2015).
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).
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.
The authors declare no competing interests.
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.
- Erdös–Rényi 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
About this article
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). https://doi.org/10.1038/s42254-018-0002-6
This article is cited by
Sustainable development goals as unifying narratives in large UK firms’ Twitter discussions
Scientific Reports (2023)
A positive statistical benchmark to assess network agreement
Nature Communications (2023)
The role of complexity for digital twins of cities
Nature Computational Science (2023)
Characterization of interactions’ persistence in time-varying networks
Scientific Reports (2023)
ERnet: a tool for the semantic segmentation and quantitative analysis of endoplasmic reticulum topology
Nature Methods (2023)