Majority Dynamics and Internal Partitions of Random Regular Graphs: Experimental Results

TL;DR

This paper investigates Majority Dynamics on random regular graphs, revealing that about half of the vertices oscillate, and proposes a modified dynamics to better analyze internal cuts and cores.

Contribution

It provides empirical insights into vertex behavior in Majority Dynamics on random regular graphs and introduces a modified process to better identify internal graph structures.

Findings

Approximately half of the vertices oscillate in odd-regular graphs.

Standard Majority Dynamics does not produce desired cores.

Modified dynamics can generate parts with desired cores.

Abstract

This paper focuses on Majority Dynamics in sparse graphs, in particular, as a tool to study internal cuts. It is known that, in Majority Dynamics on a finite graph, each vertex eventually either comes to a fixed state, or oscillates with period two. The empirical evidence acquired by simulations suggests that for random odd-regular graphs, approximately half of the vertices end up oscillating with high probability. We notice a local symmetry between oscillating and non-oscillating vertices, that potentially can explain why the fraction of the oscillating vertices is concentrated around $\frac{1}{2}$. In our simulations, we observe that the parts of random odd-regular graph under Majority Dynamics with high probability do not contain $\lceil \frac{d}{2} \rceil$-cores at any timestep, and thus, one cannot use Majority Dynamics to prove that internal cuts exist in odd-regular graphs almost surely. However, we suggest a modification of Majority Dynamics, that yields parts with desired cores with high probability.