  In the last 15 years, statistical physics has been a very successful
framework to model complex networks. On the theoretical side, this approach has
brought novel insights into a variety of physical phenomena, such as
self-organisation, scale invariance, emergence of mixed distributions and
ensemble non-equivalence, that display unconventional features on heterogeneous
networks. At the same time, thanks to their deep connection with information
theory, statistical physics and the principle of maximum entropy have led to
the definition of null models for networks reproducing some features of
real-world systems, but otherwise as random as possible. We review here the
statistical physics approach and the various null models for complex networks,
focusing in particular on the analytic frameworks reproducing the local network
features. We then show how these models have been used to detect statistically
significant and predictive structural patterns in real-world networks, as well
as to reconstruct the network structure in case of incomplete information. We
further survey the statistical physics models that reproduce more complex,
semi-local network features using Markov chain Monte Carlo sampling, as well as
the models of generalised network structures such as multiplex networks,
interacting networks and simplicial complexes.
