In a bipartite rating network, edges represent reviews of products purchased by consumers and are weighted by the numerical score received (i.e. Amazon review system). We provide theoretical tools for their analysis extending the procedure in [1, 2, 3]. Given an observed rating network, its random counterpart is given by a gran canonical ensemble of graphs, which probability distribution is defined maximising the entropy over a set of constraints per node, i.e. the observed degree for each possible score (Bipartite Score Configuration Model, BiSCM). With this method we are able to analytically obtain the probability to observe a specific score for each pair of nodes in the system. Since only a finite set of discrete and ordinal scores are possible, we are able to distinguish positive from negative reviews and this allows to define the “signed” version of many topological quantities, such as positive/negative degree and average nearest neighbour degree. We then analyse correlation and assortativity patterns on the Amazon Digital Music network [4], comparing the observed number of topological quantities with their expected counterpart. Left panel of Figure 1 shows that BiSCM is perfectly able to detect the observed assortativity trend, meaning that this higher order property can be directly explained by the imposed topological structure. Finally, the network has been projected onto the products layer. The projection has been validated using the method in [5] and the Louvain community detection algorithm has been applied to the validated graph. Right panel of Figure 1 shows that this procedure provides a very sharp division in communities, revealing strongly connected groups of users with similar musical tastes and purchases.
