  This paper introduces a new methodology to analyse bipartite and unipartite
networks with nonnegative edge values. The proposed approach combines and
adapts a number of ideas from the literature on latent variable network models.
The resulting framework is a new type of latent position model which exhibits
great flexibility, and is able to capture important features that are generally
exhibited by observed networks, such as sparsity and heavy tailed degree
distributions. A crucial advantage of the proposed method is that the number of
latent dimensions is automatically deduced from the data in one single
algorithmic framework. In addition, the model attaches a weight to each of the
latent dimensions, hence providing a measure of their relative importance. A
fast variational Bayesian algorithm is proposed to estimate the parameters of
the model. Finally, applications of the proposed methodology are illustrated on
both artificial and real datasets, and comparisons with other existing
procedures are provided.
