Collective frequency variation in network synchronization and reverse PageRank

TL;DR

This paper reveals that in directed networks, the collective frequency of synchronized systems depends on a reverse PageRank centrality measure, linking network structure to dynamical behavior and demonstrating this in power grid networks.

Contribution

It uncovers the relationship between collective frequency and reverse PageRank centrality in directed networks, a novel connection between network structure and synchronization dynamics.

Findings

Collective frequency differs from the mean natural frequency in directed networks.

The collective frequency is a weighted average based on reverse PageRank centrality.

Real-world power grids exhibit the predicted collective frequency variation.

Abstract

A wide range of natural and engineered phenomena rely on large networks of interacting units to reach a dynamical consensus state where the system collectively operates. Here we study the dynamics of self-organizing systems and show that for generic directed networks the collective frequency of the ensemble is {\it not} the same as the mean of the individuals' natural frequencies. Specifically, we show that the collective frequency equals a weighted average of the natural frequencies, where the weights are given by an out-flow centrality measure that is equivalent to a reverse PageRank centrality. Our findings uncover an intricate dependence of the collective frequency on both the structural directedness and dynamical heterogeneity of the network, and also reveal an unexplored connection between synchronization and PageRank, which opens the possibility of applying PageRank optimization to synchronization. Finally, we demonstrate the presence of collective frequency variation in real-world networks by considering the UK and Scandinavian power grids.