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Undecided State Dynamics with Many Opinions

Colin Cooper, Frederik Mallmann-Trenn, Tomasz Radzik, Nobutaka Shimizu, Takeharu Shiraga

TL;DR

The first consensus-time guarantees for USD that hold for \emph{arbitrary} $2\le k\le n$ and for \emph{arbitrary} initial configurations in both the gossip model and the population protocol model are obtained.

Abstract

We study the Undecided-State Dynamics (USD), a fundamental consensus process in which each vertex holds one of $k$ decided opinions or the undecided state. We consider both the gossip model and the population protocol model. Prior work established tight bounds on the consensus time of this process only for the regime $k = O(\sqrt{n}/(\log n)^2)$ (for the population protocol model) and $k = O((n/\log n)^{1/3})$ (for the gossip model), often under restrictive assumptions on the initial configuration. In this paper, we obtain the first consensus-time guarantees for USD that hold for \emph{arbitrary} $2\le k\le n$ and for \emph{arbitrary} initial configurations in both the gossip model and the population protocol model. In the gossip model, USD reaches consensus within $\widetilde O(\min\{k,\sqrt n\})$ synchronous rounds with probability $1-p_{\bot}-n^{-c}$, where $p_{\bot}$ is the gossip-specific probability of collapsing to the all-undecided state in the first round. In the population protocol model, USD reaches consensus within $\widetilde O(\min\{kn,n^{3/2}\})$ asynchronous interactions with high probability. We also present lower bounds that match the upper bounds up to polylogarithmic factors for a specific initial configuration and show that our upper bounds are essentially optimal.

Undecided State Dynamics with Many Opinions

TL;DR

The first consensus-time guarantees for USD that hold for \emph{arbitrary} and for \emph{arbitrary} initial configurations in both the gossip model and the population protocol model are obtained.

Abstract

We study the Undecided-State Dynamics (USD), a fundamental consensus process in which each vertex holds one of decided opinions or the undecided state. We consider both the gossip model and the population protocol model. Prior work established tight bounds on the consensus time of this process only for the regime (for the population protocol model) and (for the gossip model), often under restrictive assumptions on the initial configuration. In this paper, we obtain the first consensus-time guarantees for USD that hold for \emph{arbitrary} and for \emph{arbitrary} initial configurations in both the gossip model and the population protocol model. In the gossip model, USD reaches consensus within synchronous rounds with probability , where is the gossip-specific probability of collapsing to the all-undecided state in the first round. In the population protocol model, USD reaches consensus within asynchronous interactions with high probability. We also present lower bounds that match the upper bounds up to polylogarithmic factors for a specific initial configuration and show that our upper bounds are essentially optimal.
Paper Structure (54 sections, 67 theorems, 340 equations, 2 figures, 1 table)

This paper contains 54 sections, 67 theorems, 340 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Matching lower bounds.
  4. Other Related Works
  5. Proof Overview
  6. Process in a Nutshell
  7. Difficulties in Extending Previous Techniques
  8. (1) Breakdown of concentration-based analysis for large $k$.
  9. (2) Limitations of the $k>\sqrt{n}$ techniques for 3-Majority.
  10. Proof of Lemma 2.1 (I): Behavior of beta and psi (Section 5.2)
  11. Proof of Lemma 2.1 (II): Growth of alphamax (Sections 5.3 and 5.4)
  12. Proof of Lemma 2.1 (III): Gap between two opinions (Sections 5.5 and 5.6)
  13. Concentration Bounds via Freedman's Inequality
  14. Definitions and Main Analysis Tools
  15. Model and Basic Notation
  16. ...and 39 more sections

Key Result

Theorem 1.1

Consider USD on $n$ vertices starting from an arbitrary initial configurationA configuration is an assignment of opinions to vertices. in $\{1,\dots,k,\bot\}^n$ other than the all-$\bot$ configuration, where the number of possible decided opinions $k$ is an arbitrary integer between $2$ and $n$ (inc

Figures (2)

  • Figure 1: Combining the four parts of \ref{['lem:main goal']} to prove the consensus time bound.
  • Figure 2: Behavior of $\beta_t$ over time in the gossip model.

Theorems & Definitions (181)

  • Theorem 1.1: Main Theorem
  • Remark 1.2
  • Theorem 1.3: matching lower bound for the gossip model
  • Lemma 2.1: informal; see \ref{['lem:taubeta', 'lem:taupsi', 'lem:growth of tnpt', 'lem:hitting time for tnpm and npm', 'lem:unique strong opinion', 'lem:towards consensus']} for the formal statements
  • Remark 2.2
  • Definition 3.1: USD Update Rule
  • Definition 3.2: Gossip USD
  • Definition 3.3: Population Protocol USD
  • Remark 3.4
  • Definition 3.5: Bernstein condition; SS25_sync
  • ...and 171 more