Coarse Graining for Synchronization in Directed Networks

TL;DR

This paper introduces a topology-aware coarse graining method for directed networks, effectively preserving synchronization properties during network simplification, validated through stability analysis and numerical simulations on various network types.

Contribution

The paper presents a novel coarse graining approach specifically designed for directed networks, addressing a gap in existing methods that mainly focus on undirected networks.

Findings

The TCG method preserves network synchronizability effectively.

Numerical simulations confirm the method's robustness across different network types.

Linear stability analysis supports the method's theoretical foundation.

Abstract

Coarse graining model is a promising way to analyze and visualize large-scale networks. The coarse-grained networks are required to preserve the same statistical properties as well as the dynamic behaviors as the initial networks. Some methods have been proposed and found effective in undirected networks, while the study on coarse graining in directed networks lacks of consideration. In this paper, we proposed a Topology-aware Coarse Graining (TCG) method to coarse grain the directed networks. Performing the linear stability analysis of synchronization and numerical simulation of the Kuramoto model on four kinds of directed networks, including tree-like networks and variants of Barab\'{a}si-Albert networks, Watts-Strogatz networks and Erd\"{o}s-R\'{e}nyi networks, we find our method can effectively preserve the network synchronizability.