A rank-3 network representation for single-affiliation systems

Single-affiliation systems are observed across nature and society. Examples include collaboration, organisational affiliations, and trade-blocs. The study of such systems is commonly approached through network analysis. Multilayer networks extend the representation of network analysis to include more information through increased dimensionality. Thus, they are able to more accurately represent the systems they are modelling. However, multilayer networks are often represented by rank-4 adjacency tensors, resulting in a N2M2 solution space. Single-affiliation systems are unable to occupy the full extent of this space leading to sparse data where it is difficult to attain statistical confidence through subsequent analysis. To overcome these limitations, this paper presents a rank-3 tensor representation for single-affiliation systems. The representations is able to maintain full information of single-affiliation networks in directionless networks, maintain near full information in directed networks, reduce the solution space it resides in (N2M) leading to statistically significant findings, and maintain the analytical capability of multilayer approaches. This is shown through a comparison of the rank-3 and rank-4 representations which is performed on two datasets: the University of Bath departmental journal co-authorship 2000-2017 and an Erdos-Renyi network with random single-affiliation. The results demonstrate that the structure of the network is maintained through both representations, while the rank-3 representation provides greater statistical confidence in node-based measures, and can readily show inter- and intra-affiliation dynamics.