Cycle-Star Motifs: Network Response to Link Modifications

TL;DR

This paper analyzes how link modifications in a specific network topology affect its spectral properties and synchronization, providing criteria to predict whether such changes enhance or hinder collective dynamics.

Contribution

It introduces criteria based on network structure and weights to predict the impact of link modifications on spectral gap and synchronization in directed networks.

Findings

Link additions can hinder collective dynamics, contrary to common assumptions.

Criteria depend on sizes, coupling structure, and weights of network components.

Application to chaotic systems demonstrates practical implications for network control.

Abstract

Understanding efficient modifications to improve network functionality is a fundamental problem of scientific and industrial interest. We study the response of network dynamics against link modifications on a weakly connected directed graph consisting of two strongly connected components: an undirected star and an undirected cycle. We assume that there are directed edges starting from the cycle and ending at the star (master-slave formalism). We modify the graph by adding directed edges of arbitrarily large weights starting from the star and ending at the cycle (opposite direction of the cutset). We provide criteria (based on the sizes of the star and cycle, the coupling structure, and the weights of cutset and modification edges) that determine how the modification affects the spectral gap of the Laplacian matrix. We apply our approach to understand the modifications that either enhance or hinder synchronization in networks of chaotic Lorenz systems as well as R\"ossler. Our results show that the hindrance of collective dynamics due to link additions is not atypical as previously anticipated by modification analysis and thus allows for better control of collective properties.