From 2d70d1abca4d68668be61a4a435e8aa1d3061e6b Mon Sep 17 00:00:00 2001
From: Jonas Schreiner <jonas.schreiner@posteo.net>
Date: Thu, 26 Jul 2018 00:20:46 +0200
Subject: [PATCH] kleinigkeiten

---
 kapitel/ameisen.tex             |   8 ++--
 kapitel/ea.tex                  |  41 +++++++++++--------
 kapitel/peer2peer.tex           |  67 ++++++++++++++++++++++++++++++++
 kapitel/websuche_datamining.tex |  67 ++++++++++++++++----------------
 lecture_notes.pdf               | Bin 407301 -> 414229 bytes
 lecture_notes.tex               |   1 +
 6 files changed, 130 insertions(+), 54 deletions(-)
 create mode 100644 kapitel/peer2peer.tex

diff --git a/kapitel/ameisen.tex b/kapitel/ameisen.tex
index 2221990..260bf78 100644
--- a/kapitel/ameisen.tex
+++ b/kapitel/ameisen.tex
@@ -27,7 +27,7 @@ Den k"urzesten Weg zu finden ist die emergente Eigenschaft.
 		\frac{\tau_{ij}(t)^\alpha \cdot \eta^\beta_{ij}}{\sum\limits_{l \in \text{allowed}_k(t)} \tau_{il}(t)^\alpha \cdot \eta^\beta_{il}} & \text{ falls }j \in \text{allowed}_k(t) \\ % Bild IZ
 		0 & \text{ sonst}
 	\end{cases} \\
-    \eta_{ij} &= \text{ Sichtbarkeit, oft } \eta_{ij} = \frac 1{d_{ij}}. \text{ Heuristischer \emph{problemspezifischer} Wert.}
+    \eta_{ij} &= \text{ Sichtbarkeit, oft } \eta_{ij} = 1/{d_{ij}}. \text{ Heuristischer \emph{problemspezifischer} Wert.}
 \end{align*}
 \footnotetext{verwendet man heuzutage nicht mehr}
 Die k"unstlichen Ameisen haben ein Ged"achtnis, sie k"onnen sich ihren aktuellen Weg und die beste bislang gefundene Rundreise. Au{\ss}erdem sind sie 
@@ -47,13 +47,13 @@ $Q_1$ ist eine konstante, die ausdr"uckt, wie viel Pheromon von einer Ameise pro
 \emph{Ant-quantity:} k"urzere Kanten werden bevorzugt
 \[
 	\Delta\tau^{(k)}_{ij} (t, t+1) = \begin{cases}
-		\frac{Q_2}{d_{ij}} & \text{Ameise $k$ geht von $i$ nach $j$ zwischen $t$ und $t+1$} \\
+		{Q_1}/{d_{ij}} & \text{Ameise $k$ geht von $i$ nach $j$ zwischen $t$ und $t+1$} \\
 		0 & \text{sonst}
 	\end{cases}
 \]
 
 Um den \glqq Explorationseffekt\grqq\ zu unterst"utzen kann man zus"atzlich einen Schwellwert $Q$ einf"uhren, 
-anhanddessen mithilfe eines von Ameise $k$ auf Punkt $i$ gew"urfelten Wertes \emph{rnd} bestimmt wird,
+anhanddessen, mithilfe eines von Ameise $k$, auf Punkt $i$, gew"urfelten Wertes \emph{rnd}, bestimmt wird,
 ob die Ameise $p_{ij}^{(k)}(t)$ zum Punktwechsel verwendet, oder einen anderen Punkt besucht. \\\\
 %
 \begin{ttfamily}
@@ -69,7 +69,7 @@ Problem: nach $n$ Iterationen hat jede Ameise eine Tour, die \glqq Beste\grqq{}
 \begin{align*}
 	L^{(k)} &= \text{ L"ange der bislang besten Tour der Ameise }k \\
 	\Delta\tau^{(k)}_{ij} (t, t+n) &= \begin{cases}
-			\frac{Q_2}{L^{(k)}} & \text{falls Ameise $k$ auf ihrer Tour $\{i,j\}$ benutzt} \\
+			\frac{Q_1}{L^{(k)}} & \text{falls Ameise $k$ auf ihrer Tour $\{i,j\}$ benutzt} \\
 			0 & \text{sonst}
 		\end{cases}
 \end{align*}
diff --git a/kapitel/ea.tex b/kapitel/ea.tex
index 866c67a..2e43e23 100644
--- a/kapitel/ea.tex
+++ b/kapitel/ea.tex
@@ -1,6 +1,6 @@
 \section{Evolution"are Algorithmen}
 \begin{itemize}
-    \setlength\itemsep{0.5em}
+    \setlength\itemsep{0.1em}
     \item \emph{Population} von \emph{Individuen}
     \item Reproduktion durch \emph{Vererbung} und \emph{Mutation} (neue Individuen aus den alten erzeugen)
 	\item Selektionsprinzip
@@ -24,7 +24,7 @@ Populationsgr"o"se $\mu$, Zahl der Nachkommen $\lambda$ und Zielfunktion $F$\\
             \STATE $P''$ := erschaffe durch \emph{Rekombination} $\lambda$ Nachkommen aus $P'$
             \STATE $P'''$ := mutiere die Nachkommen in $P''$ mittels \emph{Mutationsoperator}\footnotemark
             \STATE bewerte die Nachkommen mit Eval$_F(P''')$
-            \STATE $P(t+1)$ := selektiere $\mu$ Individuen aus $P(t) \cup P'''$ mit  Selektionsoperator\footnotemark
+            \STATE $P$(t+1) := selektiere $\mu$ Individuen aus $P$(t) $\cup$ $P'''$ mit  Selektionsoperator\footnotemark
             \STATE t := t+1
         \ENDWHILE
         \STATE gib bestes Idividuum aus
@@ -101,43 +101,50 @@ Hillclimber, d.h. die Folge $\pi_0, \pi_1,..$ wird nie schlechter. \\\\
 %
 \emph{Satz}: Die erwartete Anzahl an Auswertungen $f(\pi)$ des (1 + 1) EA f"ur $f \in \{\text{INV},\text{HAM}, \text{LAS} \}$ ist mindestens $\Omega(n^2)$. \\\\
 %
-\emph{Beweis}: Die Wahrscheinlichkeit, dass die erste gew"ahlte Permutation sortiert ist, ist $\frac 1{n!}$. \\
+\emph{Beweis}: Die Wahrscheinlichkeit, dass die erste gew"ahlte Permutation sortiert ist, ist $1/{n!}$.
 Die Permutation ist sortiert, wenn im letzten Schritt der Mutation die sortierte Folge erzeugt wird.
 Es gibt h"ochstens \emph{zwei} Paare, die das tun, und das je m"oglicher Operation.
 Erfolgswahrscheinlichkeit: $\frac 12 \cdot \frac{2}{n(n-1)} \cdot 2$ \\
 $\Rightarrow \text{ E}[t] \ge \frac 12 n(n-1) = \Omega(n^2) \qquad \square$ \\\\
 %
-\emph{Satz}: Die erwartete Anzahl an Auswertungen $f(\pi)$ des (1+1) EA f"ur $f \in \{\text{INV},\text{HAM}\}$ ist h"ochstens $\mathcal O(n^2 \log n)$.
+\emph{Satz}: Die erwartete Anzahl an Auswertungen $f(\pi)$ des (1+1) EA f"ur $f \in \{\text{INV},\text{HAM},\text{LAS}\}$ ist h"ochstens $\mathcal O(n^2 \log n)$.
 \paragraph{Beweis f"ur f = INV} \ \\\\
 Sei $(i,j)$ ein Indexpaar mit $i < j$ und $\pi(i) > \pi(j)$ \\
-Sei $a$ die Anzahl der Schl"ussel an den Positionen $i+1, \dots, j-1$, die kleiner als $\pi(j)$ sind. \\
-Sei $b$ die Anzahl der Schl"ussel an den Positionen $i+1, \dots, j-1$, die zwischen $\pi(j)$ und $\pi(i)$ liegen. \\
-Sei $c$ die Anzahl der Schl"ussel an den Positionen $i+1, \dots, j-1$, die gr"o\ss er als $\pi(i)$ sind. \\
+Sei $a$ die Anzahl der Schl"ussel an den Positionen $i+1, \ldots, j-1$, die kleiner als $\pi(j)$ sind. \\
+Sei $b$ die Anzahl der Schl"ussel an den Positionen $i+1, \ldots, j-1$, die zwischen $\pi(j)$ und $\pi(i)$ liegen. \\
+Sei $c$ die Anzahl der Schl"ussel an den Positionen $i+1, \ldots, j-1$, die gr"o\ss er als $\pi(i)$ sind.
 %
+\paragraph{Beispiel}
+$(\pi(i) := 7~\underbrace{9~8~10}_{c=3}~\underbrace{6}_{b=1}~\underbrace{3~4}_{a=2}~5 =: \pi(j))$\\
 \begin{addmargin}{0.5cm}
-	\texttt{exch}(i,j) vergr"o\ss ert die Zielfunktion um $2b+1$ \\
-	\texttt{jump}(i,j) "andert die Zielfunktion um $a+b-c+1$ \\
-	\texttt{jump}(j,i) "andert die Zielfunktion um $-a+b+c+1$
+	\texttt{exch}(i,j) vergr"o\ss ert den Wert der Zielfunktion um $2b+1$ \\
+	\texttt{jump}(i,j) "andert den Wert um $a+b-c+1$ \\
+	\texttt{jump}(j,i) "andert den Wert um $-a+b+c+1$
 	\begin{addmargin}{0.5cm}
 		$\to$ mindestens einer der Werte \texttt{jump}(i,j), \texttt{jump}(j,i) ist positiv.
 	\end{addmargin}
 \end{addmargin} \ \\
 %
-Mindestens um 1 wird die Zielfunktion verbessert, falls genau \emph{eine} lokale Operation erfolgreich ausgef"uhrt wird, [genau eine lokale Operation hei\ss t S=0; $Pr[S=0] = \frac 1e$] und diese Operation ist \texttt{exch}(i,j) oder \texttt{exch}(j,i) [$\frac{2}{n(n-1)} \cdot \frac 12$ Wahrscheinlichkeit] oder die Operation ist die \grqq gute\grqq{} von \texttt{jump}(i,j) und \texttt{jump}(j,i) [Wahrscheinlichkeit $\frac 2{2n(n-1)} \cdot \frac 12$]
+Mindestens um 1 wird der Wert verbessert, falls genau \emph{eine} lokale Operation ausgef"uhrt wird,
+[bei $S=0 \text{ mit Wahrscheinlichkeit }  \frac 1e$] \emph{und} diese Operation ist \texttt{exch}$(i,j)$ oder \texttt{exch}$(j,i)$ 
+$\left[  \frac 12 \cdot \frac{1}{\binom n2} = \frac{2}{n(n-1)} \cdot \frac 12 = \frac{1}{n(n-1)}\right]$ oder
+die Operation ist die \glqq gute\grqq{} von \texttt{jump}$(i,j)$ und
+\texttt{jump}$(j,i)$ $\left[ \frac{1}{2} \cdot \frac 12 \cdot \frac {1}{ \binom n2} =\frac{}{2 \cdot n(n-1)} \right]$
 
-Ist die Anzahl der falschherum stehenden Paare $m$, so ist die Wahrscheinlichkeit, den Wert der Zielfunktion zu verbessern mindestens 
+Ist die Anzahl der falschherum stehenden Paare $m$ (das hei{\ss}t $m$ \emph{Inversionen}),
+so ist die Wahrscheinlichkeit, den Wert der Zielfunktion zu verbessern mindestens 
 
 \[\frac 1e \cdot m \cdot \left(\frac 1{n(n-1)} + \frac 1{2n(n-1)}\right) = \frac {3m}{2n(n-1)e}\]
 
-$\Rightarrow \frac 23 \cdot \frac 1m \cdot e \cdot n(n-1)$ erwartete Schritte, um den Wert der Zielfunktion zu verbessern. \\
-$\Rightarrow E[t] \le \sum\limits_{m=1}^{\frac 12 n(n-1)} \frac 23 \cdot \frac 1m \cdot e \cdot n(n-1) \le \frac 23 \cdot e \cdot n^2 \cdot \underbrace{\sum\limits_{m=1}^{\frac 12 n(n-1)} \frac 1m}_{\text{Harmonische Reihe}, \le \ln(\frac 12 n(n-1)) + 1} = \mathcal O( n^2 \cdot \log n)$
+$\Rightarrow \frac 23 \cdot \frac 1m \cdot e \cdot n(n-1)$ erwartete Schritte, um den Wert um 1 zu verbessern. \\
+$\Rightarrow \text E[t] \le \sum\limits_{m=1}^{\frac 12 n(n-1)} \frac 23 \cdot \frac 1m \cdot e \cdot n(n-1) \le \frac 23 \cdot e \cdot n^2 \cdot \underbrace{\sum\limits_{m=1}^{\frac 12 n(n-1)} \frac 1m}_{\text{Harmonische Reihe}, \le \ln(\frac 12 n(n-1)) + 1} = \mathcal O( n^2 \cdot \log n)$
 %
 \paragraph{Beweis f"ur f = HAM} \ \\\\
 %
-Ist $HAM(\pi_t) = k$, stehen $n-k$ Schl"ussel an falschen Stellen.\\
+Ist $\texttt{HAM}(\pi_t) = k$, stehen $n-k$ Schl"ussel an falschen Stellen.\\
 Ist der Schl"ussel $i$ an Position $j$ ($\pi_t(j) = i$), $i \neq j$ ist auch der Schl"ussel an Position $i$ an der falschen Stelle. \texttt{exch}(i,j) und \texttt{exch}(j,i) verbessen den Wert der Zielfunktion um mindestens 1. \\
 $\Rightarrow$ es gibt mindestens $n-k$ \glqq gute\grqq{} \texttt{exch}-Operationen. \\
-$\Rightarrow$ Rechnung wie gerade eben: $E[t] \ge 2 en^2 \cdot \sum\limits_{k=1}^n \frac 1k = \mathcal O(n^2 \log n) \qquad \square$
+$\Rightarrow$ Rechnung wie gerade eben: $E[t] \leq 2 en^2 \cdot \sum\limits_{k=1}^n \frac 1k = \mathcal O(n^2 \log n) \qquad \square$
 %
 \paragraph{Beweis f"ur f = LAS} \ \\\\
 %
@@ -161,7 +168,7 @@ $\Rightarrow$ Wahrscheinlichkeit f"ur Verbesserung: $\frac {n-k}{2en(n-1)}$ \\
 $\Rightarrow$ Erwartete Wartezeit f"ur n"achste Verbesserung: $\frac {2en(n-1)}{n-k}$ \\
 $\Rightarrow$ Erwartete Gesamtlaufzeit: $\sum\limits_{k=1}^{n-1} \frac {2en(n-1)}{n-k} = 2en(n-1) \sum\limits_{k=1}^{n-1} \frac 1l = \mathcal O(n^2 \log n)$
 
-\subsubsection{Selektionsverfahren}
+\subsection{Selektionsverfahren}
 
 \paragraph{Turnierselektion}
 \begin{itemize}
diff --git a/kapitel/peer2peer.tex b/kapitel/peer2peer.tex
new file mode 100644
index 0000000..03d03a8
--- /dev/null
+++ b/kapitel/peer2peer.tex
@@ -0,0 +1,67 @@
+\section{Peer-to-Peer-Netzwerke}
+
+\paragraph{Historie}
+\begin{itemize}
+	\item 2004 besteht ein gro"steil des Traffics aus Filesharing. 
+	\item Erstes derartiges Netzwerk: \emph{Napster}
+	\begin{itemize}
+		\item Ein zentralen Server der "uber alle Clients und alle Daten bescheid wei"s
+		\item Client schickt Anfrage an Server, Clients haben die Daten 
+		\item Daten"ubertragung geht direkt "uber Clients 
+		\item Skaliert schlecht, fehleranf"allig 
+		\item Damit eigentlich kein P2P Netz, au"ser dem eigentlichem Download 
+	\end{itemize}
+	\item \emph{Gnutella} soll Probleme von Napster l"osen und funktioniert ohne zentrale Strukturen 
+	\begin{itemize}
+		\item Ist paretoverteilt: von $2$ bis $n$, dann haben $2$ ingesamt $\frac{1}{2}$ (H"alfte) 
+		Anteil, die $n$ dann $\frac{1}{n}$\footnote{Die reichsten 10\% haben 90\% des Geldes}
+		\item Anfragen haben immer ein TTL, damit sie nicht zu lokal bleiben (\glqq 3 ist ein vern"uftiger Wert\grqq)
+		\item Flooding: Netzwerk wird mit Anfragen geflutet, ist zum gro"sen Teil nur noch damit anstelle von 
+		Download besch"aftigt (Asynchronit"at: Download bereits stattgefunden, es kommen trotzdem noch Antworten 
+		von anderen zur"uck)
+		\item Small world hypothesis: Experiment aus den USA bei denen Pakete nicht "uber die Post "uber die ganze 
+		Welt verschickt werden sollten, mit der Anweisung \glqq gib das jemanden den du kennst \glqq. Die meisten Pakete 
+		kamen innerhalb vom $5$ Hops an (daher TTL von $3$)
+		\item Verbesserungen: Random walks (Zeit vs. Anzahl der Anfragen) und Caching (entlang den Pfaden vorherige 
+		Antworten speichern. Ackermann: eine L"osung f"ur passive Replikate)
+		\item Weiteres Problem: keine Struktur in der Datenablage (eine \glqq Nein \grqq Antwort muss nicht 
+		unbedingt stimmen, wurde das ganze Netz durchsucht?)
+	\end{itemize}
+\end{itemize}
+
+\paragraph{DHT}
+\begin{itemize}
+	\item Wenn nur die Daten gehasht werden, gibt es Probleme wenn Peers hinzukommen oder gehen
+	\item Daher auch noch die Peers hashen 
+\end{itemize}
+
+\paragraph{Gradoptimierte Netzwerke}
+\begin{itemize}
+	\item M"oglich: konstanter Grad, log. Durchmesser. Bsp: $n = 15$ Knoten, Durchmesser: $6 =2(\log(n+1)-1)$
+	\item Problem mit B"aumen: es gibt Hotspots die nicht ausfallen dr"urfen
+	\item Besser \emph{Butterfly-Graph} was hotspots angeht, daf"ur schlechter wenn sich Peers spontan anmelden und abmelden wollen, 
+	da immer eine bestimmte Anzahl gebraucht wird um eine neue Schicht erstellen zu k"onnen (Problem der Skalierbarkeit)
+	\item \emph{Viceroy} (eine Schmetterlingsart)
+	\item Besser als $\log$ Durchmesser kann man mit konstantem Grad nicht werden
+	\item Bei CAN war Durchmesser $\mathcal{O}(n^{\frac{1}{2}})$, Viceroy kann das in $\log(n)$
+	\item Wrap-around-Kanten: die untersten Knoten gibt es eigentlich nicht, es sind die selben wie oben (so aber leichter zu 
+	zeichnen). Dadurch hat jeder Knoten Grad $4$ (Beispiel siehe Folien)
+	\item Konstruktion Butterfly-Graphen: siehe Bild
+	\begin{itemize}
+		\item Jeder Knoten hat einer Adresse, $i$ f"ur das Level und ein bitstring der die Spalte angibt
+		\item Erst mal alle vertical verbinden (mit wrap around)
+		\item Danach im bitstring das $i$te Bit flippen und mit dem Knoten im n"achsten level verbinden
+	\end{itemize}
+	\item Wenn man von \texttt{000} nach \texttt{111} will, muss man nach und nach die Bits korrigieren, d.h.
+	\texttt{000}$\to$\texttt{100}$\to$\texttt{110}$\to$\texttt{111} ("uber Kreuzkanten)
+	\item Aufbau Viceroy: 
+	\begin{itemize}
+		\item R"uckgrad: levels (Knoten die in Kreisen zusammengeschlossen sind)
+		\item Neuer Knoten w"ahlt erstmal ein level
+		\item Ein Ring der alle Knoten verbindet
+		\item Butterfly Netzwerk zwischen den levels
+		\item Falls ein Knoten in einem level nicht gefunden wird, dann kommen die Kreise ins Spiel
+		(da nochmal Datenstruktur dr"uber packen die daf"ur sorgt, dass immer ein weiteres Element gefunden
+		wird)
+	\end{itemize}
+\end{itemize}
diff --git a/kapitel/websuche_datamining.tex b/kapitel/websuche_datamining.tex
index 5c19251..845a294 100644
--- a/kapitel/websuche_datamining.tex
+++ b/kapitel/websuche_datamining.tex
@@ -42,14 +42,14 @@ Gib allen Seiten, die den Suchstring $\sigma$ enthalten aus, und die, auf die am
     \item[$\to$] popul"are Seiten, die h"aufig verlinkt, werden zu Autorit"aten zu allen Themen
     \item[$\to$] \glqq gute \grqq Seiten brauchen $\sigma$ ja gar nicht zu enthalten (audi.de--Problematik)
 \end{itemize}
-Hier f"ur uns: es gibt \underline{Autorit"aten} und die, die sie kennen. (Autorit"aten f"ur Autorit"aten, \underline{Hubs}
+Hier f"ur uns: es gibt \emph{Autorit"aten} und die, die sie kennen. (Autorit"aten f"ur Autorit"aten, \emph{Hubs}
 \footnote{Nabe, wie das Zentrum von einem Rad mit vielen Speichen, z.B. beim Fliegen erst mal zu einem gro"sen Flughafen, 
 vor dort aus weiter}) \\
 
 \subsubsection{HITS-Algorithmus}
 
 Wie identifiziert man \emph{Autorit"aten} und \emph{Hubs}? (bzw. wie misst man diese?)
-\paragraph{Achtung:} \underline{Clustering} ist etwas anderes, z.B. die Trennung von Seiten, die z.B. \glqq echte Schl"ussel\grqq{} meinen von denen,
+\paragraph{Achtung:} \emph{Clustering} ist etwas anderes, z.B. die Trennung von Seiten, die z.B. \glqq echte Schl"ussel\grqq{} meinen von denen,
 die Kryptographie meinen, oder Fenster von Windows (vor allem im Englischen) \\
 
 Vorstellung des HITS\footnote{Hypertext-Induced Topic Search}-Algorithmus von Kleinberg \\
@@ -77,17 +77,18 @@ auf sie zeigt.
     \begin{algorithmic}[1]
         \STATE $S_\sigma$ := $R_\sigma$
         \FORALL{$p \in R_\sigma$}
-            \STATE $S_\sigma$ := $S_\sigma \cup \Gamma^{+} (p)$\footnote{Menge der Nachbarseiten von $p$ (Seiten, auf die $p$ zeigt), $\Gamma^-(p) sind 
-    Seiten, die auf $p$ zeigen$}
+            \STATE $S_\sigma$ := $S_\sigma \cup \Gamma^{+} (p)$\footnotemark
             \IF{$ \lvert \Gamma^-(p) \rvert \leq d$}
                 \STATE $S_\sigma := S_\sigma \cup \Gamma^-(p)$
             \ELSE
-                \STATE $S_\sigma := S_\sigma \cup d$ (zufaellig aus $\Gamma^-(p)$ ausgewaehlte Seiten\footnote{nennt sich Sampling})
+                \STATE $S_\sigma := S_\sigma \cup d$ (zufaellig aus $\Gamma^-(p)$ ausgewaehlte Seiten\footnotemark)
             \ENDIF
         \ENDFOR
     \end{algorithmic}
     \end{ttfamily}
 \end{algorithm}
+\footnotetext{Menge der Nachbarseiten von $p$ (Seiten, auf die $p$ zeigt), $\Gamma^-(p) sind Seiten, die auf $p$ zeigen$}
+\footnotetext{nennt sich Sampling}
 
 
 (Kleinbergs Experimente von 1999: $t = 200,\; d = 500$, $S_\sigma \approx 1000$ bis $5000$ Seiten)
@@ -95,28 +96,28 @@ auf sie zeigt.
 \begin{itemize}
     \setlength\itemsep{0.1em}
     \item l"osche interne (bzgl. Domain) Links (Navigationslinks) 
-    \item erlaube \underline{aus} einer Domain nur $m$ (ca. $4$ bis $8$) Links \underline{auf} eine Seite 
+    \item erlaube \emph{aus} einer Domain nur $m$ (ca. $4$ bis $8$) Links \emph{auf} eine Seite 
     (wegen: \glqq diese Seite wurde erstellt von...\grqq) 
 \end{itemize}
 \begin{multicols}{2}
     \input{imgs/2_2_hits_result.tex}
     \vfill
 \columnbreak
-    Nun: Unterscheidung zwischen relevanten und blo"s popul"aren. Jetzt auch (iii)\\
+    Nun: Unterscheidung zwischen relevanten und blo"s popul"aren. Jetzt auch (iii).
     Schon auf $S_\sigma$ ist die Sortierung nach dem Eingangsgrad recht gut, im Gegensatz zum Gesamtgraphen.
 \end{multicols}
-\underline{Autorit"at berechnen}
+\emph{Autorit"at berechnen}
 
 Seiten mit gro\ss er Autorit"at zur Anfrage $\sigma$ sollten nicht nur gro\ss en Eingangsgrad haben;
-diejenigen, die auf sie zeigen, sollten sich erheblich "uberlappen; \underline{Hubs} \\
+diejenigen, die auf sie zeigen, sollten sich erheblich "uberlappen; \emph{Hubs} \\
 
 Hubs und Autorit"aten verst"arken sich gegenseitig. \\
-\underline{Jede} Seite $p$ hat ein Autorit"atsgewicht $x^{\langle p \rangle}$, je gr"o{\ss}er, umso besser geeignet als Autorit"at auf $\sigma$.
+\emph{Jede} Seite $p$ hat ein Autorit"atsgewicht $x^{\langle p \rangle}$, je gr"o{\ss}er, umso besser geeignet als Autorit"at auf $\sigma$.
 Jede Seite $p$ hat ein Hubgewicht $y^{\langle p \rangle}$, je gr"o{\ss}er, umso wertvoller als Autorit"at f"ur Autorit"aten bez"uglich $\sigma$ ist $p$.
 
 \begin{tabular}{ll}
-Je gr"o\ss er der & $x$-Wert, um so besser zu $\sigma$ passt die Seite. \\
-	& $y$-Wert, ein um so besserer Hub liegt vor.
+Je gr"o\ss er der & $x$-Wert, umso besser passt die Seite zu $\sigma$. \\
+                  & $y$-Wert, ein um so besserer Hub liegt vor.
 \end{tabular}
 
 \allowdisplaybreaks
@@ -148,7 +149,6 @@ Nun gilt: $\sum\limits_{p \in S_\sigma} (x^{\langle p \rangle})^2 = 1$ und $\sum
             \STATE $x^{\langle p \rangle} := \sum\limits_{q:\, (p \to q)} y^{\langle p \rangle}$
             \STATE $y^{\langle p \rangle} := \sum\limits_{q:\, (p \to q)} x^{\langle p \rangle}$
             \STATE normiere x und y auf 1
-            \STATE gib die Seite(n) auf mit h"ochster Autorit"at in $x$
         \ENDFOR
     \end{algorithmic}
     \end{ttfamily}
@@ -170,28 +170,29 @@ Da $AA^T$ und $A^TA$ symmetrisch: alle Eigenwerte reell. Wegen der Normierung ko
 von $A^TA$ und $y \text{ gegen den von } AA^T$, d.h. dem Eigenvektor, der zum gr"o{\ss}ten Eigenwert geh"ort und L"ange 1 hat. \hfill$\square$\\
 
 Experimente zeigen: $k \approx 20$ bis $30$ reicht aus (Expander: relativ gro\ss e Teilmenge von Knoten hat viele Nachbarn au\ss erhalb der Knotenmenge).
-$S_\sigma$ ist Teil des WWW, und der WWW-Graph \underline{ist} ein Expander.
+$S_\sigma$ ist Teil des WWW, und der WWW-Graph \emph{ist} ein Expander.
 
 \noindent
-\underline{Literatur}: Jon M. Kleinberg: Authoritative Sources in a Hyperlinked Environment, J.ACM 1999 % TODO: wohin das?
+\emph{Literatur}: Jon M. Kleinberg: Authoritative Sources in a Hyperlinked Environment, J.ACM 1999 % TODO: wohin das?
 
 \subsubsection{PageRank}
 
-Eine Alternative zum Hits-Algorithmus ist der PageRank-Algorithmus. \\
-%
-\begin{ttfamily}
-Initialisiere die Relevanz der Seiten auf $\frac 1{\text{Anzahl Seiten}}$ \\
-W"ahle $d$, zB $d = \frac 45$ \\
-while kein Terminierungskriterium erf"ullt do
-	\begin{addmargin}{0.5cm}
-	for all Seiten S do
-		\begin{addmargin}{0.5cm}
-		Neue Relevanz von S ist $\frac{1-d}{\text{Anzahl Seiten}} + d \cdot \sum\limits_{\forall R: (R,S) \in E} \frac{\text{Relevanz von R}}{\text{Ausgangsgrad von R}}$
-		\end{addmargin}
-	end for
-	\end{addmargin}
-end while
-\end{ttfamily}
+Eine Alternative zum Hits-Algorithmus ist der PageRank-Algorithmus.
+
+\begin{algorithm}[H]
+    \caption{\texttt{PageRank}}
+    \begin{ttfamily}
+    \begin{algorithmic}[1]
+        \STATE Initialisiere die Relevanz der Seiten auf $\frac 1{\text{Anzahl Seiten}}$
+        \STATE W"ahle $d$, zB $d = 4/5$
+        \WHILE{Terminierungskriterium nicht erf"ullt}
+            \FORALL{Seiten S}
+                \STATE Neue Relevanz von S ist $(1-d) \cdot \frac{1}{\text{Anzahl Seiten}} + d \cdot \sum\limits_{\forall R: (R,S) \in E} \frac{\text{Relevanz von R}}{\text{Ausgangsgrad von R}}$
+            \ENDFOR
+        \ENDWHILE
+    \end{algorithmic}
+    \end{ttfamily}
+\end{algorithm}
 
 \subsubsection{Zentralit"atsma\ss e}
 
@@ -200,18 +201,18 @@ end while
 \[C_D(e) = \frac{\text{deg}(e)}{n-1}\]
 
 Wobei deg$(e)$ den Grad des Knotens beschreibt (Anzahl der Nachbarn). \\
-\underline{Bedeutung}: Je mehr Nachbarn ein Knoten $e$ hat, desto wichtiger ist $e$.
+\emph{Bedeutung}: Je mehr Nachbarn ein Knoten $e$ hat, desto wichtiger ist $e$.
 
 \paragraph{Betweeness Centrality}
 
 \[C_B(e) = \frac{2}{(n-1)(n-2)} \cdot \sum\limits_{s\neq e\neq t, t\neq s} \frac{\sigma_{st}(e)}{\sigma_{st}}\]
 
 $\sigma_{st}$ beschreibt die Anzahl der k"urzesten Wege von $s$ nach $t$, $\sigma_{st}(e)$ beschreibt die Anzahl derer, die dabei "uber $e$ verlaufen. \\
-\underline{Bedeutung}: Je mehr Bekanntschaften 'direkt' "uber einen Knoten $e$ verlaufen, desto wichtiger ist $e$.
+\emph{Bedeutung}: Je mehr Bekanntschaften 'direkt' "uber einen Knoten $e$ verlaufen, desto wichtiger ist $e$.
 
 \paragraph{Closeness Centrality}
 
 \[C_C(e) = \frac{\sum\limits_{t\neq e}\text{dist}(e,t)}{n-1}\]
 
 dist$(e,t)$ bezeichnet hier die L"ange des k"urzesten Pfades zwischen $e$ und $t$. \\
-\underline{Bedeutung}: Je 'direkter' viele Leute einen Knoten $e$ kennen, desto wichtiger ist $e$.
+\emph{Bedeutung}: Je 'direkter' viele Leute einen Knoten $e$ kennen, desto wichtiger ist $e$.
diff --git a/lecture_notes.pdf b/lecture_notes.pdf
index a990d4b4707f173c32f10448a19ea8f793f76550..c162945ee0dd44ff26f9d2bcff1e083098ed36e9 100644
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diff --git a/lecture_notes.tex b/lecture_notes.tex
index d43bb73..3031cab 100644
--- a/lecture_notes.tex
+++ b/lecture_notes.tex
@@ -49,5 +49,6 @@
 \input{kapitel/websuche_datamining}
 \input{kapitel/ameisen}
 \input{kapitel/ea}
+\input{kapitel/peer2peer}
 \ifthenelse{\boolean{englishDocument}}{}{ \renewcommand{\bibname}{Literatur} }
 \end{document}
-- 
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