  Maximum entropy network ensembles have been very successful in modelling
sparse network topologies and in solving challenging inference problems.
However the sparse maximum entropy network models proposed so far have fixed
number of nodes and are typically not exchangeable. Here we consider
hierarchical models for exchangeable networks in the sparse limit, i.e. with
the total number of links scaling linearly with the total number of nodes. The
approach is grand canonical, i.e. the number of nodes of the network is not
fixed a priori: it is finite but can be arbitrarily large. In this way the
grand canonical network ensembles circumvent the difficulties in treating
infinite sparse exchangeable networks which according to the Aldous-Hoover
theorem must vanish. The approach can treat networks with given degree
distribution or networks with given distribution of latent variables. When only
a subgraph induced by a subset of nodes is known, this model allows a Bayesian
estimation of the network size and the degree sequence (or the sequence of
latent variables) of the entire network which can be used for network
reconstruction.
