Detection of core–periphery structure in networks using spectral methods and geodesic paths

• Published in: European journal of applied mathematics Vol. 27; no. 6; pp. 846 - 887
• Main Authors: CUCURINGU, MIHAI, ROMBACH, PUCK, LEE, SANG HOON, PORTER, MASON A.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2016
• Subjects: Algorithms; Applied mathematics; Graphs; Matrix; Networks; low-rank matrix approximations; core–periphery structure; shortest-path algorithms; graph Laplacians
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S095679251600022X

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 consists of 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 that vertex is a core vertex. Our second method is based on a low-rank approximation of a network's adjacency matrix, which we express as a perturbation of 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.