# Stabilization Time in Weighted Minority Processes

## TL;DR

This paper proves that the stabilization time for weighted minority processes in graphs can be exponentially long, even under strict conditions and across various models, highlighting fundamental complexity in dynamic graph coloring.

## Contribution

The paper introduces the first exponential lower bound on stabilization time for weighted minority processes, applicable across multiple models and graph types.

## Key findings

- Exponential lower bound on stabilization time established
- Bound holds in adversarial, benevolent, and concurrent models
- Results apply to weighted and sparse graphs

## Abstract

A minority process in a weighted graph is a dynamically changing coloring. Each node repeatedly changes its color in order to minimize the sum of weighted conflicts with its neighbors. We study the number of steps until such a process stabilizes. Our main contribution is an exponential lower bound on stabilization time. We first present a construction showing this bound in the adversarial sequential model, and then we show how to extend the construction to establish the same bound in the benevolent sequential model, as well as in any reasonable concurrent model. Furthermore, we show that the stabilization time of our construction remains exponential even for very strict switching conditions, namely, if a node only changes color when almost all (i.e., any specific fraction) of its neighbors have the same color. Our lower bound works in a wide range of settings, both for node-weighted and edge-weighted graphs, or if we restrict minority processes to the class of sparse graphs.