Extreme Fluctuations in Stochastic Network Coordination with Time Delays

TL;DR

This paper investigates how uniform time delays influence the maximum fluctuations in stochastic network synchronization, revealing different extreme value behaviors in complex versus low-dimensional networks and examining nonlinear effects.

Contribution

It provides a comprehensive analysis of extreme fluctuations in delayed stochastic networks, including scaling laws, distribution types, and the impact of nonlinear couplings.

Findings

In large complex networks, fluctuations decouple and follow Gumbel distribution.

In low-dimensional networks, fluctuations are correlated with Airy distribution.

Nonlinear couplings affect the stability and extremes of synchronization.

Abstract

We study the effects of uniform time delays on the extreme fluctuations in stochastic synchronization and coordination problems with linear couplings in complex networks. We obtain the average size of the fluctuations at the nodes from the behavior of the underlying modes of the network. We then obtain the scaling behavior of the extreme fluctuations with system size, as well as the distribution of the extremes on complex networks, and compare them to those on regular one-dimensional lattices. For large complex networks, when the delay is not too close to the critical one, fluctuations at the nodes effectively decouple, and the limit distributions converge to the Fisher-Tippett-Gumbel density. In contrast, fluctuations in low-dimensional spatial graphs are strongly correlated, and the limit distribution of the extremes is the Airy density. Finally, we also explore the effects of nonlinear couplings on the stability and on the extremes of the synchronization landscapes.