Graph conductance, synchronization, and a new bottleneck measure

TL;DR

The paper addresses the challenge of predicting and ensuring global synchronization in diffusively coupled networks. It analyzes the limitations of traditional isoperimetric measures—graph conductance $\Phi_G$ and Cheeger constant $h_G$—for global synchronization and introduces a new bottleneck measure $\Xi(G)= \min_{S\subseteq V} \frac{|\partial S|^2}{vol(\delta \overline{S})}$ that targets the worst-case flow across a two-component cut. The authors illustrate Xi on a weighted path graph to show it identifies the actual bottleneck and can guide decentralized edge-weight adjustments to enhance synchronization for dynamics with Type I MSF. The results clarify that while Xi controls stability for Type I MSF, it does not suffice for Type II MSF, highlighting the need for further links between network bottlenecks, MSF types, and diameter effects. The practical impact is a strategy for improving global synchronizability by strengthening the critical cut identified by Xi, with potential applications in power grids and distributed systems.

Abstract

As a quantification of the main bottleneck to flow over a graph, the network property of conductance plays an important role in the process of synchronization of network-coupled dynamical systems. Diffusive coupling terms serve not only to exchange information between nodes within a networked system, but ultimately to dissipate the entropy of the collective dynamic state down toward that which can be associated with a single dynamic node when the synchronization manifold is stable. While the graph conductance can characterize the coupling strength that is required to maintain widespread synchronization across a majority of the nodes in such a system, it offers no guarantee for a stable synchronization manifold, which involves all nodes in the system. We define a new measure called the synchronization bottleneck of a graph, which we denote by $Ξ$; this new network property provides a quantification of the limiting bottleneck of the flow between any subset of nodes (regardless of its order) and the rest of the networked system. This quantity does control the coupling strength required for a stable synchronization manifold for a large class of dynamical systems. Solving for this quantity is combinatorial, as is the case with conductance, but heuristics based on this optimization problem can guide decentralized strategies for improving global synchronizability.

Figures (1)

• Figure 1: A weighted path graph on five vertices, which can represent the bottlenecks for dynamics coupled on many graphs when $w_l>w_c>>1>w_r$.