Statistical Modelling of One-Mode and Two-Mode Networks : Simultaneous Analysis of Graphs and Bipartite Graphs

A bipartite graph, in which the nodes (or actors in a social network) are partitioned into two sets, can be studied using recent statistical models for dyadic interactions. These models, which are loglinear for the probabilities of dyadic choices or interactions, allow not only arcs or relationships to exist between the sets but also within the sets. Thus, the methods described here are applicable not only to bipartite graphs, consisting of arcs existing between nodes in different sets, but also to directed graphs that are defined within the two sets of nodes. Data on both types of graphs can be analysed simultaneously.A bipartite graph has an adjacency matrix (or sociomatrix) with two 'modes.' The set of nodes in the row mode differs from the set of nodes in the column mode. For example, in marketing, one could study the dyadic relations in a 'buyers-by-sellers' network. Generally, the relations observed in a one-mode network, which has a square sociomatrix (row mode =column mode) are bidirectional — the actors indexing the columns may also 'relate to' the actors indexing the rows. The relations observed in a two-mode network are generally unidirectional — the row actors relate to or choose the column actors, but the column actors do not relate to the row actors. Referring to our example, a buyer might pay a seller for some item, but a seller would not pay a buyer.Statistical models for the separate analysis of these one-mode and two-mode matrices are extended in this paper to the simultaneous analysis of both types of networks. A superordinate one-mode sociomatrix is created in which the rows and columns consist of all actors (that is, all buyers and sellers). This larger matrix contains both the one-mode matrices and the two-mode matrices. Multivariate analysis of unidirectional and bidirectional relations in social networks and complex directed graphs becomes possible with this simultaneous consideration of both types of matrices