Tunable Eigenvector-Based Centralities for Multiplex and Temporal Networks

Characterizing the importances of nodes in social, biological, information and technological networks is a core topic for the network-science and data-science communities. We present a linear-algebraic framework that generalizes eigenvector-based centralities---including PageRank---to provide a common framework for two popular classes of multilayer networks: multiplex networks with layers encoding different types of relationships and temporal networks in which the relationships change in time. Our approach involves the study of joint, marginal, and conditional supracentralities that can be obtained from the dominant eigenvector of a supracentrality matrix [Taylor et al., 2017], which couples centrality matrices that are associated with the individual network layers. We extend this prior work (which was restricted to temporal networks with layers that are coupled by undirected, adjacent-in-time coupling) by allowing the layers to be coupled through a (possibly asymmetric) interlayer adjacency matrix $\tilde{{\bf A}}$ in which each entry gives the coupling between layers $t$ and $t'$. Our framework provides a unifying foundation for centrality analysis of multiplex and/or temporal networks and reveals a complicated dependency of the supracentralities on the topology of interlayer coupling. We scale $\tilde{{\bf A}}$ by a coupling strength $\omega\ge0$ and develop singular perturbation theory for the limits of weak ($\omega\to0^+$) and strong coupling ($\omega\to\infty$), revealing an interesting dependence on the dominant eigenvectors of $\tilde{{\bf A}}$. We apply the framework to two empirical data sets: a multiplex network representation of airline transportation in Europe, and a temporal network encoding the graduation and hiring of mathematicians at U.S. colleges and universities.