Tripartite Networks: a first exploratory step towards the understanding of multipartite networks

In tripartite networks nodes are divided in three different sets and edges are allowed only between couple of nodes belonging to different families. In other words the graph is defined by G = {V, E} where V = V1 ∪ V2 ∪ V3 and Vi ∩ Vj = ∅ for i = j, and E = {(u, v) : u ∈ Vi , v ∈ Vj and i = j}. Tripartite graphs are the natural way to describe a number of real world structures. Recently, we found them a useful way to describe the two following systems: Settembre Musica 1 gathering data of more than 30 years of organization of the prestigious international music festival hold in Torino. The tripartite graph in this case emerges considering as node classes the artists, the concerts or music pieces, and the directors. Edges therefore represent single concerts played in different editions of the festival. We-Sport which collects data of the vertical social-network http://www.we-sport.com. The main aim of this project is to connect people with sports avoiding one of the main problem of practicing sports: the lack of partner with which to practice one’s favorite sports. Considering as nodes athletes, chosen sports, and places where those are played, the tripartite structures is readily apparent. However, many other example of tripartite, or even multipartite, networks could be found in real world. In eco-epidemiological frameworks a tripartite network could be useful to describe the transmission cycle of Taenia Solium between raw pork meat, humans and, swine. Another interesting example can be found in the ubiquitination cycle, the main processes used by cells to mark proteins in order to degrade them. In fact the process requires three kind of enzymes and the marking of a particular protein is mediated by particular enzymes. Therefore, the understanding of the network behind it, of its modular structure, and of its principal features could be of great importance for public health, not last the fight against cancer. Working on those examples of tripartite networks we found that some of them show a particular structure other than the general tripartite one. For example, there exist tripartite networks in which a connection between a node u ∈ V1 and a node v ∈ V2 exists only if u and v are both tied whit w ∈ V3 . Another common and interesting case of tripartite network is where nodes of V1 are only connected with nodes of V2 which, in turn, are connected only with nodes belonging to V3 . Given these differences, and since the research questions on them could be very different, an extra care must be taken in approaching them. In this work we try to detect the most efficient projection strategies according to the particular structure and to the questions made on it, in order to avoid the intrinsic loss of crucial information due to their bipartite or unipartite projection.