# Indirect Long-range Interactions and Network Synchronization

## TL;DR

This paper investigates how long-range interactions in networks influence synchronization, showing they generally enhance synchronizability and that their effects depend on network structure, with implications for network robustness.

## Contribution

The study introduces a model incorporating long-range interactions via $d$-path Laplacian matrices and analyzes their impact on network synchronization both theoretically and numerically.

## Key findings

- Long-range interactions improve network synchronizability.
- The impact of long-range interactions depends on network diameter.
- Edge removal in networks with long-range interactions is more critical.

## Abstract

The dynamical behavior of networked complex systems is shaped not only by the direct links among the units, but also by the long-range interactions occurring through the many existing paths connecting the network nodes. In this work, we study how synchronization dynamics is influenced by these long-range interactions, formulating a model of coupled oscillators that incorporates this type of interactions through the use of $d-$path Laplacian matrices. We study synchronizability of these networks by the analysis of the Laplacian spectra, both theoretically and numerically, for real-world networks and artificial models. Our analysis reveal that in all networks long-range interactions improve network synchronizability with an impact that depends on the original structure, for instance it is greater for graphs having a larger diameter. We also investigate the effects of edge removal in graphs with long-range interactions and, as a major result, find that the removal process becomes more critical, since also the long-range influence of the removed link disappears.