  Exchangeability is a desired statistical property of network ensembles
requiring their invariance upon relabelling of the nodes. However combining
sparsity of network ensembles with exchangeability is challenging. Here we
propose a statistical physics framework and a Metropolis-Hastings algorithm
defining exchangeable sparse network ensembles. The model generates networks
with heterogeneous degree distributions by enforcing only global constraints
while existing (non exchangeable) exponential random graphs enforce an
extensive number of local constraints. This very general theoretical framework
to describe exchangeable networks is here first formulated for uncorrelated
simple networks and then it is extended to treat simple networks with degree
correlations, directed networks, bipartite networks and generalized network
structures including multiplex networks and simplicial complexes. In particular
here we formulate and treat both uncorrelated and correlated exchangeable
ensembles of simplicial complexes using statistical mechanics approaches.
