  Random graph mixture models are now very popular for modeling real data
networks. In these setups, parameter estimation procedures usually rely on
variational approximations, either combined with the expectation-maximisation
(\textsc{em}) algorithm or with Bayesian approaches. Despite good results on
synthetic data, the validity of the variational approximation is however not
established. Moreover, the behavior of the maximum likelihood or of the maximum
a posteriori estimators approximated by these procedures is not known in these
models, due to the dependency structure on the variables. In this work, we show
that in many different affiliation contexts (for binary or weighted graphs),
estimators based either on moment equations or on the maximization of some
composite likelihood are strongly consistent and $\sqrt{n}$-convergent, where
$n$ is the number of nodes. As a consequence, our result establishes that the
overall structure of an affiliation model can be caught by the description of
the network in terms of its number of triads (order 3 structures) and edges
(order 2 structures). We illustrate the efficiency of our method on simulated
data and compare its performances with other existing procedures. A data set of
cross-citations among economics journals is also analyzed.
