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Prikl. Diskr. Mat., 2020, Number 49, Pages 5–17 (Mi pdm710)  

Theoretical Backgrounds of Applied Discrete Mathematics

On images and pre-images in a graph of the composition of independent uniform random mappings

V. O. Mironkin

National Research University Higher School of Economics, Moscow, Russia

Abstract: We study the probability characteristics of the random mapping graph $ f_{[k]} $ — the composition of $k$ independent equiprobable random mappings $ f_1, \ldots, f_k $, where $f_i\colon \{1,\ldots,n\}\to \{1,\ldots,n\}$, $n,k\in\mathbb{N}$, $i=1,\ldots,n$. The following results are obtained. For any fixed $x,y\in S=\{1,\ldots,n\}$, $x\ne y$,
\begin{equation*} \mathbf{P}\{f_{[k]}(x)=f_{[k]}(y)\}=\textstyle\sum\limits_{\begin{smallmatrix}s_1,\ldots,s_{k-1}\in\mathbb{N}\colon
2\geqslant s_1\geqslant\ldots\geqslant s_{k-1} \end{smallmatrix}}\dfrac{q(2,s_{1})}{n^{s_{k-1}-1}}\prod\limits_{i=1}^{k-2}q(s_i,s_{i+1}), \end{equation*}
where $q(a,b)=C_{n}^{n-b} (\dfrac{b}{n})^a \sum\limits_{l=0}^{b}C_{b}^l(-1)^l(1-\dfrac{l}{b})^a$. For any fixed $x\in S$,
\begin{gather*} \mathbf{P}\{ x\in f_{[k]}(S)\}=\frac1{n}\textstyle\sum\limits_{l=1}^{n}{(\dfrac{(n)_l}{n^l} )^k}++\textstyle\sum\limits_{l=1}^{n-2}\sum\limits_{t=1}^{n-l-1}\sum\limits_{m=1}^{n-t-l}(-1)^{m-1}C_{n-1}^m\sum\limits_{\begin{smallmatrix}s_1,\ldots,s_{k-1}\in\mathbb{N}\colon
m\geqslant s_1\geqslant\ldots\geqslant s_{k-1} \end{smallmatrix}}\dfrac{q(m,s_{1})}{n^{s_{k-1}}}\prod\limits_{i=1}^{k-2}q(s_i,s_{i+1})V^{\{k,m\}}_{s_1,\ldots,s_{k-1}}, \end{gather*}
where
\begin{gather*} V^{\{k,m\}}_{s_1,\ldots,s_{k-1}}=\mathbf{P}\{x\in H_{f_{[k]}}^{(t,l)}| D^{\{k\}}_{s_1,\ldots,s_{k-1},1}(y_1,\ldots,y_m),f_{[k]}(y_1)=x \}==\frac{1}{n}\textstyle\prod\limits_{i=m+1}^{t+l+m-1}{( 1-\dfrac{i}{n} )}\prod\limits_{i=1}^{k-1}\prod\limits_{j=s_i+1}^{t+l+s_i-2}{( 1-\dfrac{j}{n} )}\sum\limits_{v=0}^{k-1}\prod\limits_{u=1}^{v}{( 1-\dfrac{t+l+s_u-1}{n} )}, \end{gather*}
$H_f^{(t,l)}$ is $t$-th layer of cycles of length $l$ in graph $G_f$, $D^{\{k\}}_{s_1,\ldots,s_{k}}(y_1,\ldots,y_m)=\textstyle\bigcap\limits_{i=1}^{k} \{|\{f_{[i]}(y_1),\ldots,f_{[i]}(y_m)\}|=s_i\}$, and $(n)_z=n(n-1)…(n-z+1)$. For any fixed $x\in S\setminus S'$ and for any $r\in \{1,\ldots,n-1\}$, $S'\subseteq S$, $|S'|=r$, $z\in \{1,\ldots,n\}$,
\begin{gather*} \mathbf{P}\{\tau_{f_{[k]}}(x)=z,\mathcal{R}_{f_{[k]}}(x)\cap S'=\varnothing \}==(1-(1-\frac{z}{n})( 1-\frac{z-1}{n} )^{k-1})(\frac{(n)_{z-1}}{n^{z-1}} )^{k-1}\frac{(n)_{r+z}}{n^{z-1}(n)_{r+1}}, \end{gather*}
where $\mathcal{R}_{f_{[k]}}(x)$ is the aperiodicity segment of vertex $x$ in the graph of mapping $f_{[k]}$, $\tau_{f_{[k]}}(x)=\min\{ t\in \mathbb{N}\colon {f_{[k]}}^t(x)\in \{ x,{f_{[k]}}(x),…,{f_{[k]}}^{t-1}(x) \}\}$. For any fixed $x,y\in S$, $x\ne y$, and for any $r\in\{1,\ldots,n\}$,
\begin{equation*} \mathbf{P}\{y \in (f_{[k]})^{-r}(x)\}=\frac1n(1-\frac1{n-1}\textstyle\sum\limits_{z\in Q_r\setminus\{1\}}(\dfrac{(n)_z}{n^z})^k), \end{equation*}
where $Q_r=\{m\in \mathbb{N}\colon m|r\}$.

Keywords: equiprobable random mapping, composition of mappings, graph of a mapping, image of a multitude, pre-image of a vertex, initial vertex, layer in a graph, aperiodicity segment, collision.

DOI: https://doi.org/10.17223/20710410/49/1

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Bibliographic databases:

UDC: 519.212.2+519.719.2

Citation: V. O. Mironkin, “On images and pre-images in a graph of the composition of independent uniform random mappings”, Prikl. Diskr. Mat., 2020, no. 49, 5–17

Citation in format AMSBIB
\Bibitem{Mir20}
\by V.~O.~Mironkin
\paper On images and pre-images in a graph of the composition of independent uniform random mappings
\jour Prikl. Diskr. Mat.
\yr 2020
\issue 49
\pages 5--17
\mathnet{http://mi.mathnet.ru/pdm710}
\crossref{https://doi.org/10.17223/20710410/49/1}


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