Statistics

Statistical model

Artificial intelligence

Complex network

Information theory

Physics

Bayesian probability

Mathematics

Evolving networks

Complex system

Computer science

Markov chain Monte Carlo

Statistical physics

World Wide Web

Theoretical computer science

Principle of maximum entropy

Statistical mechanics

Nature Reviews Physics

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. This Review describes advances in the statistical physics of complex networks and provides a reference for the state of the art in theoretical network modelling and applications to real-world systems for pattern detection and network reconstruction.

The statistical physics of real-world networks