Dynamics-based centrality for general directed networks

TL;DR

This paper introduces a new Laplacian-based centrality measure for general directed networks, extending existing methods to compare arbitrary nodes and interpreting it through dynamical processes like random walks.

Contribution

It extends Laplacian-based centrality to general directed networks and provides a dynamical interpretation in terms of random walk absorption probabilities.

Findings

Laplacian-based centrality is equivalent to absorption probability in random walks.

The measure can compare importance of arbitrary nodes in directed networks.

Numerical simulations demonstrate the applicability of the proposed centrality.

Abstract

Determining the relative importance of nodes in directed networks is important in, for example, ranking websites, publications, and sports teams, and for understanding signal flows in systems biology. A prevailing centrality measure in this respect is the PageRank. In this work, we focus on another class of centrality derived from the Laplacian of the network. We extend the Laplacian-based centrality, which has mainly been applied to strongly connected networks, to the case of general directed networks such that we can quantitatively compare arbitrary nodes. Toward this end, we adopt the idea used in the PageRank to introduce global connectivity between all the pairs of nodes with a certain strength. Numerical simulations are carried out on some networks. We also offer interpretations of the Laplacian-based centrality for general directed networks in terms of various dynamical and structural properties of networks. Importantly, the Laplacian-based centrality defined as the stationary density of the continuous-time random walk with random jumps is shown to be equivalent to the absorption probability of the random walk with sinks at each node but without random jumps. Similarly, the proposed centrality represents the importance of nodes in dynamics on the original network supplied with sinks but not with random jumps.