  Multiplex networks describe a large number of systems ranging from social
networks to the brain. These multilayer structure encode information in their
structure. This information can be extracted by measuring the correlations
present in the multiplex networks structure, such as the overlap of the links
in different layers. Many multiplex networks are also weighted, and the weights
of the links can be strongly correlated with the structural properties of the
multiplex network. For example in multiplex network formed by the citation and
collaboration networks between PRE scientists it was found that the statistical
properties of citations to co-authors are different from the one of citations
to non-co-authors, i.e. the weights depend on the overlap of the links. Here we
present a theoretical framework for modelling multiplex weighted networks with
different types of correlations between weights and overlap. To this end, we
use the framework of canonical network ensembles, and the recently introduced
concept of multilinks, showing that null models of a large variety of network
structures can be constructed in this way. In order to provide a concrete
example of how this framework apply to real data we consider a multiplex
constructed from gene expression data of healthy and cancer tissues.
