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Broadcasting in random recursive dags

Simon Briend, Luc Devroye, Gabor Lugosi

TL;DR

This work analyzes noisy broadcasting on uniform random $k$-dags, where $k$ root bits propagate via a noisy channel and a majority rule is used to color new vertices. By recasting the process as a reinforced urn with $k$ draws, the authors derive a function $g(t)$ and a key parameter $\alpha_k=4\mathbb{E}[(\text{Bin}(k,1/2)-k/2)_+]$ to characterize three asymptotic regimes for the mutation probability $p$: a low-rate regime where $R_n$ converges to one of two asymmetric fixed points, a high-rate regime where $R_n$ converges to $1/2$, and an intermediate regime with $R_n\to 1/2$ but a majority rule that can still outperform random guessing. They establish convergence using Pemantle's results on reinforced processes and provide a general lower bound on the error probability for the majority rule, applicable across $p$. The paper also includes a self-contained appendix proving a tree-case result (for $k=1$) that majority is better than random guessing when $p<1/4$, fixing gaps in earlier work. Overall, the results identify precise threshold phenomena in noisy broadcasting on random dags and quantify when simple majority voting is informative for root-bit reconstruction.

Abstract

A uniform $k$-{\sc dag} generalizes the uniform random recursive tree by picking $k$ parents uniformly at random from the existing nodes. It starts with $k$ ''roots''. Each of the $k$ roots is assigned a bit. These bits are propagated by a noisy channel. The parents' bits are flipped with probability $p$, and a majority vote is taken. When all nodes have received their bits, the $k$-{\sc dag} is shown without identifying the roots. The goal is to estimate the majority bit among the roots. We identify the threshold for $p$ as a function of $k$ below which the majority rule among all nodes yields an error $c+o(1)$ with $c<1/2$. Above the threshold the majority rule errs with probability $1/2+o(1)$.

Broadcasting in random recursive dags

TL;DR

This work analyzes noisy broadcasting on uniform random -dags, where root bits propagate via a noisy channel and a majority rule is used to color new vertices. By recasting the process as a reinforced urn with draws, the authors derive a function and a key parameter to characterize three asymptotic regimes for the mutation probability : a low-rate regime where converges to one of two asymmetric fixed points, a high-rate regime where converges to , and an intermediate regime with but a majority rule that can still outperform random guessing. They establish convergence using Pemantle's results on reinforced processes and provide a general lower bound on the error probability for the majority rule, applicable across . The paper also includes a self-contained appendix proving a tree-case result (for ) that majority is better than random guessing when , fixing gaps in earlier work. Overall, the results identify precise threshold phenomena in noisy broadcasting on random dags and quantify when simple majority voting is informative for root-bit reconstruction.

Abstract

A uniform -{\sc dag} generalizes the uniform random recursive tree by picking parents uniformly at random from the existing nodes. It starts with ''roots''. Each of the roots is assigned a bit. These bits are propagated by a noisy channel. The parents' bits are flipped with probability , and a majority vote is taken. When all nodes have received their bits, the -{\sc dag} is shown without identifying the roots. The goal is to estimate the majority bit among the roots. We identify the threshold for as a function of below which the majority rule among all nodes yields an error with . Above the threshold the majority rule errs with probability .
Paper Structure (16 sections, 12 theorems, 91 equations, 4 figures)

This paper contains 16 sections, 12 theorems, 91 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Related results and our contribution
  4. Different regimes
  5. Convergence of the proportion of red balls
  6. The high-rate-of-mutation regime $\left( \frac{1}{2}-\frac{1}{2\alpha_k}\leq p\leq\frac{1}{2}\right)$
  7. The low-rate-of-mutation regime $\left( 0<p<\frac{1}{2}-\frac{1}{2\alpha_k}\right)$
  8. Is majority voting better than random guessing?
  9. The low-rate-of-mutation regime $\left( 0<p<\frac{1}{2}-\frac{1}{2\alpha_k}\right)$
  10. The high-rate-of-mutation regime $\left( \frac{1}{2}-\frac{1}{2\alpha_k}\leq p\leq\frac{1}{2}\right)$
  11. Extreme rate
  12. Intermediate rate
  13. A general lower bound
  14. Concluding remarks
  15. Appendix
  16. ...and 1 more sections

Key Result

Theorem 1

Let $k$ be an odd positive integer and consider the broadcasting process on a random $k$- dag described above. Assume that initially $R_k > 1/2$.

Figures (4)

  • Figure 1: A realisation of the process up to time 6, for $k=3$, starting with $R_3=1/3$.
  • Figure 2: $g$ as a function of $t\in[0,1]$, for $k=3$, with the choices $p=0.18>1/6$ and $p=0.12<1/6$.
  • Figure 3: A linear lower bound for $|g|$, $k=3$ and $p=0.18$.
  • Figure 4: Comparison of $h$ and $g$, for $k=3$ and $p=0.34$ (rescaled for clarity).

Theorems & Definitions (20)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Pemantle_2007
  • Corollary 1: Pemantle_2007
  • Theorem 2: Pemantle_2007
  • Corollary 2
  • proof
  • ...and 10 more