Analysis of relative influence of nodes in directed networks

TL;DR

This paper investigates the influence centrality measure in directed networks, analytically and numerically demonstrating its significance in various network dynamics and its relation to PageRank and entrainment thresholds.

Contribution

The paper provides analytical calculations of influence in diverse directed networks and links influence to network hierarchy, shortcuts, and dynamics, extending understanding of node importance.

Findings

Influence correlates with network hierarchy and shortcuts.

Threshold for entrainment inversely related to influence.

Analytical influence calculation applicable to PageRank.

Abstract

Many complex networks are described by directed links; in such networks, a link represents, for example, the control of one node over the other node or unidirectional information flows. Some centrality measures are used to determine the relative importance of nodes specifically in directed networks. We analyze such a centrality measure called the influence. The influence represents the importance of nodes in various dynamics such as synchronization, evolutionary dynamics, random walk, and social dynamics. We analytically calculate the influence in various networks, including directed multipartite networks and a directed version of the Watts-Strogatz small-world network. The global properties of networks such as hierarchy and position of shortcuts, rather than local properties of the nodes, such as the degree, are shown to be the chief determinants of the influence of nodes in many cases. The developed method is also applicable to the calculation of the PageRank. We also numerically show that in a coupled oscillator system, the threshold for entrainment by a pacemaker is low when the pacemaker is placed on influential nodes. For a type of random network, the analytically derived threshold is approximately equal to the inverse of the influence. We numerically show that this relationship also holds true in a random scale-free network and a neural network.