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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.

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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.


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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.

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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.

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

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