Supracentrality Analysis of Temporal Networks with Directed Interlayer Coupling

We describe centralities in temporal networks using a supracentrality framework to study centrality trajectories, which characterize how the importances of nodes change in time. We study supracentrality generalizations of eigenvector-based centralities, a family of centrality measures for time-independent networks that includes PageRank, hub and authority scores, and eigenvector centrality. We start with a sequence of adjacency matrices, each of which represents a time layer of a network at a different point or interval of time. Coupling centrality matrices across time layers with weighted interlayer edges yields a \emph{supracentrality matrix} $\mathbb{C}(\omega)$, where $\omega$ controls the extent to which centrality trajectories change over time. We can flexibly tune the weight and topology of the interlayer coupling to cater to different scientific applications. The entries of the dominant eigenvector of $\mathbb{C}(\omega)$ represent \emph{joint centralities}, which simultaneously quantify the importance of every node in every time layer. Inspired by probability theory, we also compute \emph{marginal} and \emph{conditional centralities}. We illustrate how to adjust the coupling between time layers to tune the extent to which nodes' centrality trajectories are influenced by the oldest and newest time layers. We support our findings by analysis in the limits of small and large $\omega$.