Article Contents
Article Contents

# Power of the crowd

• *Corresponding author: Athanasios Batakis
• Consider a Galton Watson tree of height $m$: each leaf has one of $k$ opinions or not. In other words, for $i\in \{1,\dots,k\}$, $x$ at generation $m$ thinks $i$ with probability ${\bf{p}}_i$ and nothing with probability ${\bf{p}}_0$. Moreover the opinions are independently distributed for each leaf.

Opinions spread along the tree based on a specific rule: the majority wins! In this paper, we study the asymptotic behavior of the distribution of the opinion of the root when $m\to\infty$.

Mathematics Subject Classification: Primary: 37E25, 05C05, 60J80; Secondary: 60G50.

 Citation:

• Figure 1.  The rules

Figure 2.  An example

Figure 3.  The graphs of $f_n$ for $n = 4$ (blue), $n = 3$ (orange)

Figure 4.  The graphs of $f_n$ for $n = 3$ (blue), $n = 5$ (orange), $n = 7$ (green) and $n = 9$ (red), onk__ge $[0,1]$ and $[0,{}^{1}\!\!\diagup\!\!{}_{5}\;]$

Figure 5.  The graphs of $f_n$ for $n = 2$ (blue), $n = 4$ (orange), $n = 6$ (green) and $n = 10$ (red)

Figure 6.  $\frac{1}{4}-w_n$

Figure 7.  $f^\prime_{2n}(\alpha_{2n})$ for $n$ in $[2,250]$

Figure 8.  The graph of $f_2$

Figure 9.  The graphs of $f$ for $p = {}^{1}\!\!\diagup\!\!{}_{2}\;$ (blue), $p = {}^{1}\!\!\diagup\!\!{}_{4}\;$ (orange), $p = {}^{9}\!\!\diagup\!\!{}_{10}\;$ (green)

Figure 10.  The graphs of $f_n$ for $n = 7$ (blue), $n = 9$ (orange), $n = 11$ (green)

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