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diff --git a/vorlesungen/slides/8/markov/google.tex b/vorlesungen/slides/8/markov/google.tex new file mode 100644 index 0000000..d1ec31d --- /dev/null +++ b/vorlesungen/slides/8/markov/google.tex @@ -0,0 +1,123 @@ +% +% google.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Google-Matrix} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\r{2.4} +\coordinate (A) at (0,0); +\coordinate (B) at (0:\r); +\coordinate (C) at (60:\r); +\coordinate (D) at (120:\r); +\coordinate (E) at (180:\r); + +\foreach \a in {2,...,5}{ + \fill[color=white] ({60*(\a-2)}:\r) circle[radius=0.2]; + \draw ({60*(\a-2)}:\r) circle[radius=0.2]; + \node at ({60*(\a-2)}:\r) {$\a$}; +} +\fill[color=white] (A) circle[radius=0.2]; +\draw (A) circle[radius=0.2]; +\node at (A) {$1$}; + +{\color<6>{red} + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C); +} + +{\color<7>{red} + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) to[out=-150,in=-30] (E); +} + +{\color<8>{red} + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) to[out=-90,in=30] (A); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) to[out=-30,in=90] (B); +} + +{\color<9>{red} + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (C); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (A); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E); +} + +{\color<10>{red} + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A); + \draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) to[out=90,in=-150] (D); +} + +\end{tikzpicture} +\end{center} +\vspace{-10pt} +\renewcommand{\arraystretch}{1.1} +\uncover<5->{ +\begin{align*} +H&=\begin{pmatrix} +\uncover<6->{0 } + &\uncover<7->{0 } + &\uncover<8->{{\color<8>{red}\frac{1}{2}}} + &\uncover<9->{{\color<9>{red}\frac{1}{3}}} + &\uncover<10->{{\color<10>{red}\frac{1}{2}}}\\ +\uncover<6->{{\color<6>{red}\frac{1}{2}}} + &\uncover<7->{0 } + &\uncover<8->{{\color<8>{red}\frac{1}{2}}} + &\uncover<9->{0 } + &\uncover<10->{0 }\\ +\uncover<6->{{\color<6>{red}\frac{1}{2}}} + &\uncover<7->{{\color<7>{red}\frac{1}{2}}} + &\uncover<8->{0 } + &\uncover<9->{{\color<9>{red}\frac{1}{3}}} + &\uncover<10->{0 }\\ +\uncover<6->{0 } + &\uncover<7->{0 } + &\uncover<8->{0 } + &\uncover<9->{0 } + &\uncover<10->{{\color<10>{red}\frac{1}{2}}}\\ +\uncover<6->{0 } + &\uncover<7->{{\color<7>{red}\frac{1}{2}}} + &\uncover<8->{0 } + &\uncover<9->{{\color<9>{red}\frac{1}{3}}} + &\uncover<10->{0 } +\end{pmatrix} +\\ +\uncover<11->{ +h_{ij} +&= +\frac{1}{\text{Anzahl Links ausgehend von $j$}} +} +\end{align*}} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Aufgabe} +Bestimme die Wahrscheinlichkeit $p(i)$, mit der sich ein Surfer +auf der Website $i$ befindet +\end{block} +\uncover<2->{ +\begin{block}{Navigation} +$p(i) = P(i,\text{vor Navigation})$, +\uncover<3->{$p'(i)=P(i,\text{nach Navigation})$} +\uncover<4->{ +\[ +p'(i) = \sum_{j=1}^n h_{ij} p(j) +\]} +\end{block}} +\vspace{-15pt} +\begin{block}{Freier Wille} +\vspace{-12pt} +\[ +G = \alpha H + (1-\alpha)\frac{UU^t}{n} +\] +Google-Matrix +\end{block} +\end{column} +\end{columns} +\end{frame} |