Tweaking Synchronisation by Link Addition

TL;DR

This paper investigates how adding links to tree-like networks affects their synchronization, showing that cycle length and link placement can significantly alter stability and eigenvalue spectra.

Contribution

It introduces two methods of link addition and analyzes their impact on synchronization, highlighting the importance of cycle length and link position.

Findings

Short cycles can significantly change the Laplacian spectrum.

Adding links can enhance synchronization stability.

Position of added links is crucial for network dynamics.

Abstract

Natural and man-made networks often possess locally tree-like sub-structures. Taking such tree networks as our starting point, we show how the addition of links changes the synchronization properties of the network. We focus on two different methods of link addition. The first method adds single links that create cycles of a well-defined length. Following a topological approach we introduce cycles of varying length and analyze how this feature, as well as the position in the network, alters the synchronous behaviour. We show that in particular short cycles can lead to a maximum change of the Laplacian's eigenvalue spectrum, dictating the synchronization properties of such networks. The second method connects a certain proportion of the initially unconnected nodes. We simulate dynamical systems on these network topologies, with the nodes' local dynamics being either a discrete or continuous. Here our main result is that a certain amount of additional links, with the relative position in the network being crucial, can be beneficial to ensure stable synchronization.