Extreme Fluctuations in Small-Worlds with Relaxational Dynamics

TL;DR

This paper investigates how adding random links to a one-dimensional lattice affects extreme height fluctuations in small-world networks, showing that fluctuations become finite and follow a Gumbel distribution, promoting synchronization.

Contribution

It demonstrates that small-world modifications lead to finite average fluctuations and Gumbel-distributed extremes, advancing understanding of synchronization in such systems.

Findings

Average fluctuations become finite with added links

Extreme heights follow Gumbel distribution

Synchronization is practically achievable in small-world systems

Abstract

We study the distribution and scaling of the extreme height fluctuations for Edwards-Wilkinson-type relaxation on small-world substrates. When random links are added to a one-dimensional lattice, the average size of the fluctuations becomes finite (synchronized state) and the extreme height diverges only logarithmically in the large system-size limit. This latter property ensures synchronization in a practical sense in small-world coupled multi-component autonomous systems. The statistics of the extreme heights is governed by the Fisher-Tippett-Gumbel distribution.