# Stabilization Time in Minority Processes

## TL;DR

This paper investigates the stabilization time of minority processes in graphs, establishing new lower bounds that are nearly quadratic, applicable in various models including sequential and concurrent, revealing fundamental limits of such dynamics.

## Contribution

The paper introduces a novel graph construction that proves a nearly quadratic lower bound on stabilization time for minority processes, extending previous bounds to more general models.

## Key findings

- Establishes an $oldsymbol{ ext{Ω}(n^2)}$ lower bound in the sequential adversarial model.
- Proves a $oldsymbol{ ext{Ω}(n^{2-oldsymbol{ ext{ε}}})}$ lower bound for any $ ext{ε}>0$.
- Bounds hold even in benevolent and concurrent models.

## Abstract

We analyze the stabilization time of minority processes in graphs. A minority process is a dynamically changing coloring, where each node repeatedly changes its color to the color which is least frequent in its neighborhood. First, we present a simple $\Omega(n^2)$ stabilization time lower bound in the sequential adversarial model. Our main contribution is a graph construction which proves a ${\Omega}(n^{2-\epsilon})$ stabilization time lower bound for any $\epsilon>0$. This lower bound holds even if the order of nodes is chosen benevolently, not only in the sequential model, but also in any reasonable concurrent model of the process.