Principal components-based quantification of hierarchical k-core assortativity

Hierarchical networks typically get rated to be not assortative on the basis of the degrees of the end vertices of the edges. However, degree centrality is not cognizant of the location of the nodes in the hierarchy. Rather, we propose the use of hierarchy-aware metrics (such as the k-core index of a node: the largest value of k for which the node would be part of a k-core and not part of any k′-core, where k′ > k and the number of k-core levels for which there exists a non-empty k-core for the network) for assortativity calculations. We first determine the assortativity index ( \(AssI_{kcore}\) $Ass{I}_{\mathit{kcore}}$ ) of a network on the basis of the k-core indexes of the vertices. We then conduct principal component analysis (PCA) on a dataset of the \(AssI_{kcore}\) $Ass{I}_{\mathit{kcore}}$ and the number of k-core levels for a suite of 77 real-world networks. We quantify the hierarchical k-core assortativity of a network as the weighted sum of the entries in the two principal components of this dataset.