Percolation and connection times in multi-scale dynamic networks

TL;DR

This paper investigates how mobility influences percolation and connection times in multi-scale dynamic networks, revealing persistent randomness in long-term behavior and proposing a unified analytical framework.

Contribution

It introduces a general framework for analyzing limits in dynamic spatial networks, accounting for persistent randomness due to mobility effects.

Findings

Long-time averages may not approximate expected values in multi-scale models.

Strong random effects can persist in the limit, modeled by birth-death processes or Brownian motion.

A unified approach can handle diverse structures in dynamic spatial networks.

Abstract

We study the effects of mobility on two crucial characteristics in multi-scale dynamic networks: percolation and connection times. Our analysis provides insights into the question, to what extent long-time averages are well-approximated by the expected values of the corresponding quantities, i.e., the percolation and connection probabilities. In particular, we show that in multi-scale models, strong random effects may persist in the limit. Depending on the precise model choice, these may take the form of a spatial birth-death process or a Brownian motion. Despite the variety of structures that appear in the limit, we show that they can be tackled in a common framework with the potential to be applicable more generally in order to identify limits in dynamic spatial network models going beyond the examples considered in the present work.