A notion of partial order in the Choose the Leader model

TL;DR

This paper introduces the concept of partial order as a form of asymptotic correlation in the Choose the Leader model, providing a new perspective on alignment phenomena in complex systems.

Contribution

It defines and analyzes the emergence and propagation of partial order in the CL model under critical scaling, expanding understanding of asymptotic correlations.

Findings

Partial order emerges in the CL model at critical scaling.

Quantitative estimates of convergence to partial order are provided.

Partial order allows for deviations from total adherence, offering a more realistic model of alignment.

Abstract

In this work we continue the study of non-chaotic asymptotic correlations in many element systems and discuss the emergence of a new notion of asymptotic correlation -- partial order -- in the Choose the Leader (CL) system. Similarly to the newly defined notion of order, partial order refers to alignment of the elements in the system -- though it allows for deviation from total adherence. Our presented work revolves around the definition of partial order and shows its emergence in the CL model in its original critical scaling. Furthermore, we discuss the propagation of partial order in the CL model and give a quantitative estimate to the convergence to this state. This new notion (as well as that of order) opens the door to exploring old and new (probabilistic) models of biological and societal nature in a more realistic way.