Majority dynamics and the median process: connections, convergence and some new conjectures

TL;DR

This paper explores the median dynamics process on graphs, analyzing its properties, connections to majority dynamics, and proposing new conjectures, with proofs on specific graph classes.

Contribution

It introduces a continuous median dynamics model, links it to majority dynamics, and formulates new conjectures about their behavior on general graphs.

Findings

Median dynamics provides insights into majority dynamics.

Conjectures are proved on certain graph classes.

Connections between median and majority dynamics are established.

Abstract

We consider the median dynamics process in general graphs. In this model, each vertex has an independent initial opinion uniformly distributed in the interval [0,1] and, with rate one, updates its opinion to coincide with the median of its neighbors. This process provides a continuous analog of binary majority dynamics. We deduce properties of median dynamics through this connection and raise new conjectures regarding the behavior of majority dynamics on general graphs. We also prove these conjectures on some graphs where majority dynamics has a simple description.