Modelling Hierarchical Structures in Networks Using Graph Theory : With Application to Knowledge Networks in Graph Curricula

Abstract: Community detection is a topic in network theory that involves assigning labels to nodes based on some distance measure or centrality index. Detecting communities within a network can be useful to perform information condensation. In this thesis we explore how to use the approach for pedagogical purposes, and more precisely to condense and visualise the networks of facts, concepts and procedures (also called Knowledge Components (KCs)) that are offered in higher education programmes. In details, we consider one of the most common quantities used to evaluate the goodness of a community classification, which is the concept of modularity. Detecting communities by computing the maximum possible modularity indexes is indeed usually desired, but this approach is generally unavailable because the associated optimisation problem is NP-complete. This is why practitioners use other algorithms, that instead of computing the optimum they rely on various heuristics to find communities: some use modularity directly, some start from the entire graph and divide it repeatedly, and some contain random elements. This thesis investigates the trade-offs of using different community detection algorithms and variations of the concept of modularity first in general terms, and then for the purpose of identifying communities in knowledge graphs associated to higher education programmes, which can be modelled as directed graphs of KCs. We discover, tweaking and applying these algorithms both on synthetic but also field data that the Louvain algorithm is among the better algorithms of those that we considered, which is mostly thanks to its efficiency. It does not produce a full hierarchy, however, so we recommend Fast Newman if hierarchy is important.