Prototyping and comparing networks through Archetypal Analysis

In recent years, in different fields, it has become possibile to observe large collections of networks referring to the same phenomenon, e.g. sets of collaboration networks, each describing different scientific field; sets of ego networks, where egos belong to the same category; sets of governance networks; sets of brain networks. Given such sets of networks, it could be of interest the comparison of net- works among each other. At the same time it could be relevant the detection of a small number of representative networks that can serve as a condensed view of the entire collection of networks. In this paper we focus on this latter aim which amounts, in a statistical perspective, in finding what would be called prototypical networks able to typify the network structures starting from the observed ones. To this aim, we adopt the approach proposed in Ragozini et. al, (2016) in the framework the analysis of N statistical statistical units, described by p variables, synthesized by a set m prototypes. The procedure we propose goes through 3 steps: i) describe a network through a mixture of features referring to local, global, and intermediate-scale (meso-scale) network structure ii) find in the space of descriptors a set of prototypes by applying the procedure proposed in Ragozini et. al, (2016); iii) find in the original space of the networks, on the base of results of the previous steps, the prototypical networks and profile them. This procedure allows us to typify the most characteristic network structures in the observed set of networks, and to have prototypical networks that are characterized by clear and interpretable profiles in terms of their most relevant features and their specificity in contrast to the others. We demonstrate via a simulation study how the proposed procedure is able to discriminate and describe different types of networks derived from several generative models.