Tunable eigenvector-based centralities for multiplex and temporal networks

Characterizing the importances (i.e., centralities) of nodes in social, biological, and technological networks is a core topic in both network analysis and data science. We present a linear-algebraic framework that generalizes eigenvector-based centralities, including PageRank and hub/authority scores, to provide a common framework for two popular classes of multilayer networks: multiplex networks (which have layers that encode different types of relationships) and temporal networks (in which relationships change over time). Our approach involves the study of joint, marginal, and conditional “supracentralities” that one can calculate from the dominant eigenvector of a supracentrality matrix [Taylor et al., Multiscale Model. Simul., 15 (2017), pp. 537-574; [110] in this paper], which couples centrality matrices that are associated with individual network layers. We extend this prior work (which was restricted to temporal networks with layers that are coupled by adjacent-in-time coupling) by allowing the layers to be coupled through a (possibly asymmetric) interlayer-adjacency matrix Ã, where the entry Ã_tt0 ≥ 0 encodes the coupling between layers t and t⁰. Our framework provides a unifying foundation for centrality analysis of multiplex and temporal networks, and it also illustrates a complicated dependency of the supracentralities on the topology and weights of interlayer coupling. By scaling à by an interlayer-coupling strength ω ≥ 0 and developing a singular perturbation theory for the limits of weak (ω → 0⁺) and strong (ω → ∞) coupling, we also reveal an interesting dependence of supracentralities on the right and left dominant eigenvectors of Ã. We provide additional theoretical and practical insights by applying our framework to two empirical data sets: a multiplex network of airline transportation in Europe and a temporal network that encodes the graduation and hiring of mathematical scientists at United States universities.