Abstract: This work extends the rank aggregation framework for the setting of discovering optimal course sequences at the university level, and contributes to the literature on educational applications of network analysis. Each student provides a partial ranking of the courses taken throughout her or his undergraduate career. We build a network of courses by computing pairwise rank comparisons between courses based on the order students typically take them, and aggregate the results over the entire student population, to obtain a proxy for the rank offset between pairs of courses. We extract a global ranking of the courses via several state-of-the art algorithms for ranking with pairwise noisy information, including SerialRank, Rank Centrality, and the recent SyncRank based on the group synchronization problem. We test this application of rank aggregation on 15 years of student data from the Department of Mathematics at the University of California, Los Angeles (UCLA). Furthermore, we experiment with the above approach on different subsets of the student population conditioned on final GPA, and highlight several differences in the obtained rankings that uncover potential hidden pre-requisites in the Mathematics curriculum.
Abstract: We introduce several novel and computationally efficient methods for detecting "core--periphery structure" in networks. Core--periphery structure is a type of mesoscale structure that includes densely-connected core vertices and sparsely-connected peripheral vertices. Core vertices tend to be well-connected both among themselves and to peripheral vertices, which tend not to be well-connected to other vertices. Our first method, which is based on transportation in networks, aggregates information from many geodesic paths in a network and yields a score for each vertex that reflects the likelihood that a vertex is a core vertex. Our second method is based on a low-rank approximation of a network's adjacency matrix, which can often be expressed as a tensor-product matrix. Our third approach uses the bottom eigenvector of the random-walk Laplacian to infer a coreness score and a classification into core and peripheral vertices. We also design an objective function to (1) help classify vertices into core or peripheral vertices and (2) provide a goodness-of-fit criterion for classifications into core versus peripheral vertices. To examine the performance of our methods, we apply our algorithms to both synthetically-generated networks and a variety of networks constructed from real-world data sets.