# Rapid Mixing of Local Graph Dynamics

## TL;DR

This paper introduces a local graph rewiring rule for peer-to-peer networks that guarantees rapid convergence to a well-connected equilibrium state within poly-logarithmic time, improving understanding of dynamic network behavior.

## Contribution

The paper presents a novel local rewiring rule that ensures fast, poly-logarithmic convergence to equilibrium in graph dynamics, addressing an open problem in network theory.

## Key findings

- Rewiring rule achieves poly-logarithmic convergence time
- Ensures high connectivity and expansion in equilibrium
- Proof involves isoperimetric profile analysis

## Abstract

Graph dynamics arise naturally in many contexts. For instance in peer-to-peer networks, a participating peer may replace an existing connection with one neighbour by a new connection with a neighbour's neighbour. Several such local rewiring rules have been proposed to ensure that peer-to-peer networks achieve good connectivity properties (e.g. high expansion) in equilibrium. However it has remained an open question whether there existed such rules that also led to fast convergence to equilibrium. In this work we provide an affirmative answer: We exhibit a local rewiring rule that converges to equilibrium after each participating node has undergone only a number of rewirings that is poly-logarithmic in the system size. The proof involves consideration of the whole isoperimetric profile of the graph, and may be of independent interest.