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Majority dynamics on random graphs: the multiple states case

Jordan Chellig, Nikolaos Fountoulakis

TL;DR

This work studies synchronous majority dynamics with k≥3 states on the binomial random graph G(n,p). It shows that for any initial distribution S_0 with S_0(v) taking state i with probability λ_i (i.e., S_0∼λ on the simplex Δ_1), unanimity is reached in at most 3 rounds with high probability provided np≫n^{2/3} and limsup p<1, by a first-round elimination of all but the maximal mass classes, a local limit theorem and anti-concentration analysis to quantify first-step changes, and a second-moment/union-bound argument to propagate dominance through rounds 2 and 3. The key technical tool is a detailed LLT-based approximation of the neighbor-count distributions conditional on the initial partition, together with tight concentration bounds for the class sizes and a careful handling of ties. The results extend prior two-state findings to the multi-state setting, linking to related label-propagation analyses and highlighting a robust path to unanimity in dense enough random graphs. The work suggests directions for tightening the p→1 boundary and exploring regimes with near-tied initial configurations or sparser graphs.

Abstract

We study the evolution of majority dynamics with more than two states on the binomial random graph $G(n,p)$. In this process, each vertex has a state in $\{1,\ldots, k\}$, with $k\geq 3$, and at each round every vertex adopts state $i$ if it has more neighbours in state $i$ that in any other state. Ties are resolved randomly. We show that with high probability the process reaches unanimity in at most three rounds, if $np\gg n^{2/3}$.

Majority dynamics on random graphs: the multiple states case

TL;DR

This work studies synchronous majority dynamics with k≥3 states on the binomial random graph G(n,p). It shows that for any initial distribution S_0 with S_0(v) taking state i with probability λ_i (i.e., S_0∼λ on the simplex Δ_1), unanimity is reached in at most 3 rounds with high probability provided np≫n^{2/3} and limsup p<1, by a first-round elimination of all but the maximal mass classes, a local limit theorem and anti-concentration analysis to quantify first-step changes, and a second-moment/union-bound argument to propagate dominance through rounds 2 and 3. The key technical tool is a detailed LLT-based approximation of the neighbor-count distributions conditional on the initial partition, together with tight concentration bounds for the class sizes and a careful handling of ties. The results extend prior two-state findings to the multi-state setting, linking to related label-propagation analyses and highlighting a robust path to unanimity in dense enough random graphs. The work suggests directions for tightening the p→1 boundary and exploring regimes with near-tied initial configurations or sparser graphs.

Abstract

We study the evolution of majority dynamics with more than two states on the binomial random graph . In this process, each vertex has a state in , with , and at each round every vertex adopts state if it has more neighbours in state that in any other state. Ties are resolved randomly. We show that with high probability the process reaches unanimity in at most three rounds, if .
Paper Structure (14 sections, 10 theorems, 181 equations)

This paper contains 14 sections, 10 theorems, 181 equations.

Table of Contents

  1. Majority dynamics on a graph
  2. Results
  3. Outline of the proof and the structure of this work
  4. Some notation and concentration bounds
  5. Some observations about the initial configuration
  6. On the concentration of state classes and the vertex degrees therein
  7. A local limit theorem
  8. Some basic (anti-)concentration properties of the initial configuration
  9. The configuration after one round
  10. Estimating the probability of a certain state after one round
  11. Concentration after the first step
  12. After the first round
  13. Conclusions and some further directions
  14. The second moment calculation

Key Result

Theorem 1.1

Let $\varepsilon >0$ and $\boldsymbol{\lambda} \in \triangle_1$. Consider majority dynamics on $G(n,p)$ with $np\gg n^{2/3}$ but $\limsup_{n\to \infty} p < 1$. For any $n$ sufficiently large, with probability at least $1-\varepsilon$ the following holds: if $S_0 \sim \boldsymbol{\lambda}$, then $S_3

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Claim 2.1
  • proof
  • Claim 2.2
  • proof
  • Claim 2.3
  • proof
  • Claim 2.4
  • proof
  • ...and 38 more