  An intense activity is nowadays devoted to the definition of models capturing
the properties of complex networks. Among the most promising approaches, it has
been proposed to model these graphs via their clique incidence bipartite
graphs. However, this approach has, until now, severe limitations resulting
from its incapacity to reproduce a key property of this object: the overlapping
nature of cliques in complex networks. In order to get rid of these limitations
we propose to encode the structure of clique overlaps in a network thanks to a
process consisting in iteratively factorising the maximal bicliques between the
upper level and the other levels of a multipartite graph. We show that the most
natural definition of this factorising process leads to infinite series for
some instances. Our main result is to design a restriction of this process that
terminates for any arbitrary graph. Moreover, we show that the resulting
multipartite graph has remarkable combinatorial properties and is closely
related to another fundamental combinatorial object. Finally, we show that, in
practice, this multipartite graph is computationally tractable and has a size
that makes it suitable for complex network modelling.
