A random model that relies on maximal bicliques to preserve the overlaps in bipartite networks

Context. Many real-networks, also refered to as complex networks, lend themselves to the use of graphs in order to analyse their structure and model their properties. Since the seminal papers of Barabási and Watts, one usually considers that, whatever the context in which they emerge, all networks share non trivial properties such as a low density, a low average distance, an heterogeneous degree distribution, a high local density, etc. Such properties distinguish those networks from classic random graph models such as the ones generated by the Erdős-Rényi model which only reproduce the density of the networks. As a consequence, significant effort is dedicated to the elaboration of random models able to capture more intricate properties. Among them, one can cite the Barabási-Albert model which succeeds in producing a heterogeneous (scale-free) degree distribution but fail in generating graphs with a high local density, the Watts and Strogatz model which generates networks with the opposite features or the Configuration Model [3] which generate random graphs with a prescribed degree sequence but with a low local density. All in all, and despite the different attempts, generating a graph exhibiting all expected properties is still an open issue. The purpose of this study is to present a new step toward that goal by exploiting the bipartite version of the configuration model. Indeed, although useful, the representation of networks as unipartite graphs does not account for the inherent complexity induced by the hierachical structure observed in most real networks. This observation led the scientific community to turn to bipartite graphs to describe such complex structure when possible. This formalism allows to define explicitly two disjoint sets of nodes and the links only relate a node of one set to a node of the other set. The natural extension of the configuration model to bipartite graphs allows to preserve the degree of every nodes while shuffling the links, as depicted below: