  We define a new graph operator, called the weak-factor graph, which comes
from the context of complex network modelling. The weak-factor operator is
close to the well-known clique-graph operator but it rather operates in terms
of bicliques in a multipartite graph. We address the problem of the termination
of the series of graphs obtained by iteratively applying the weak-factor
operator starting from a given input graph. As for the clique-graph operator,
it turns out that some graphs give rise to series that do not terminate.
Therefore, we design a slight variation of the weak-factor operator, called
clean-factor, and prove that its associated series terminates for all input
graphs. In addition, we show that the multipartite graph on which the series
terminates has a very nice combinatorial structure: we exhibit a bijection
between its vertices and the chains of the inclusion order on the intersections
of the maximal cliques of the input graph.
