Efficient computation of \lowercase{$f$}-centralities and nonbacktracking centrality for temporal networks

TL;DR

This paper introduces a more efficient node-level formula for computing $f$-centralities and nonbacktracking centralities in temporal networks, improving tractability for dense networks and analyzing the effects of network extensions.

Contribution

It presents a novel node-level formula for $f$-centralities, demonstrating improved computational efficiency over existing edge-level methods for dense, time-evolving networks.

Findings

Node-level formula is more efficient for dense networks.

Adding a final time frame affects nonbacktracking Katz centrality.

Spectral theory of vector-valued matrices developed.

Abstract

We discuss efficient computation of $f$-centralities and nonbacktracking centralities for time-evolving networks with nonnegative weights. We present a node-level formula for its combinatorially exact computation which proves to be more tractable than previously existing formulae at edge-level for dense networks. Additionally, we investigate the impact of the addition of a final time frame to such a time-evolving network, analyzing its effect on the resulting nonbacktracking Katz centrality. Finally, we demonstrate by means of computational experiments that the node-level formula presented is much more efficient for dense networks than the previously known edge-level formula. As a tool for our goals, in an appendix of the paper, we develop a spectral theory of matrices whose elements are vectors.