Combinatorics of the paths towards synchronization

TL;DR

This paper introduces a combinatorial framework for analyzing the paths toward synchronization in network flows, focusing on the transition diagram for Laplacian and Kuramoto flows over complete graphs.

Contribution

It develops a novel combinatorial structure called the transition diagram to study synchronization paths in network flows, specifically for Laplacian and Kuramoto models.

Findings

Transition diagram for Laplacian flow over complete graphs

Application to Kuramoto flow near the diagonal

Results on flows over complete bipartite graphs

Abstract

In this paper, we introduce a codification of the paths towards synchronization for synchronizing flows defined over a network. The collection of paths toward synchronization defines a combinatorial structure: the transition diagram. We describe the transition diagram corresponding to the Laplacian flow over the completely connected graph. This applies to the Kuramoto flow over the same graph when initial conditions close to the diagonal are considered. We present as well some results concerning the Laplacian and Kuramoto flows over the complete bipartite graph.