From be452cf225eabaf4cc4119a8b68c1507b0d958c5 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Sat, 18 Jan 2014 18:20:40 +0100 Subject: [PATCH] =?UTF-8?q?Verbesserungsvorschlaege=20von=20C.=20Oessner?= =?UTF-8?q?=20(Email=20vom=2015.01.2014)=20eingearbeitet;=20Definition=20v?= =?UTF-8?q?on=20'Random=20Walk'=20hinzugef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/DYCOS/DYCOS-Algorithmus.tex | 56 +++++++++++++------ documents/DYCOS/DYCOS.pdf | Bin 489846 -> 467049 bytes documents/DYCOS/Einleitung.tex | 29 ++++++---- documents/DYCOS/README.md | 8 +++ documents/DYCOS/Sprungtypen.tex | 74 +++++++++++++++----------- documents/DYCOS/abstract.tex | 9 ++-- 6 files changed, 114 insertions(+), 62 deletions(-) diff --git a/documents/DYCOS/DYCOS-Algorithmus.tex b/documents/DYCOS/DYCOS-Algorithmus.tex index a2b0d32..806af33 100644 --- a/documents/DYCOS/DYCOS-Algorithmus.tex +++ b/documents/DYCOS/DYCOS-Algorithmus.tex @@ -2,15 +2,36 @@ DYCOS (\underline{DY}namic \underline{C}lassification algorithm with c\underline{O}ntent and \underline{S}tructure) ist ein Knotenklassifizierungsalgorithmus, der Ursprünglich in \cite{aggarwal2011} vorgestellt -wurde. Er klassifiziert einzelne Knoten, indem $r$ Random Walks der Länge $l$, +wurde. + +Ein zentrales Element des DYCOS-Algorithmus ist der sog. +{\it Random Walk}: + +\begin{definition}[Random Walk, Sprung] + Sei $G = (V, E)$ mit $E \subseteq V \times V$ ein Graph und + $v_0 \in V$ ein Knoten des Graphen. + + %Sei außerdem $f: V \rightarrow \mathcal{P}(V)$ eine Abbildung + %mit der Eigenschaft: + %\[ \forall v \in V \forall v' \in f(v): \exists \text{Weg von } v \text{ nach } v'\] + + Ein Random Walk der Länge $l$ auf $G$, startend bei $v_0$ ist + nun der zeitdiskrete stochastische Prozess, der $v_i$ + auf einen zufällig gewählten Nachbarn $v_{i+1}$ abbildet + (für $i \in 0, \dots, l-1$). + Die Abbildung $v_i \mapsto v_{i+1}$ heißt ein Sprung. +\end{definition} + +Der DYCOS-Algorithmus klassifiziert einzelne Knoten, indem $r$ Random Walks der Länge $l$, startend bei dem zu klassifizierenden Knoten $v$ gemacht werden. Dabei werden die Labels der besuchten Knoten gezählt. Das Label, das am häufigsten vorgekommen ist, wird als Label für $v$ gewählt. DYCOS nutzt also die sog. Homophilie, d.~h. die Eigenschaft, dass Knoten, die nur wenige Hops von einander entfernt sind, häufig auch -ähnlich sind \cite{bhagat}. Der DYCOS-Algorithmus nimmt jedoch nicht -einfach den Graphen für dieses Verfahren, sondern erweitert ihn mit -Hilfe der zur Verfügung stehenden Texte.\\ +ähnlich sind \cite{bhagat}. Der DYCOS-Algorithmus arbeitet jedoch nicht +direkt auf dem Graphen, sondern erweitert ihn mit +Hilfe der zur Verfügung stehenden Texte. Wie diese Erweiterung +erstellt wird, wird im Folgenden erklärt.\\ Für diese Erweiterung wird zuerst wird Vokabular $W_t$ bestimmt, das charakteristisch für eine Knotengruppe ist. Wie das gemacht werden kann und warum nicht einfach jedes Wort in das Vokabular aufgenommen wird, @@ -38,25 +59,25 @@ jedes Wortes berücksichtigt. Entsprechend werden zwei unterschiedliche Sprungtypen unterschieden, die strukturellen Sprünge und inhaltliche Mehrfachsprünge: -\begin{definition} +\begin{definition}[struktureller Sprung] Sei $G_{E,t} = (V_t, E_{S,t} \cup E_{W,t}, V_{L,t}, W_{t})$ der um die Wortknoten $W_{t}$ erweiterte Graph. Dann heißt das zufällige wechseln des aktuell betrachteten Knoten $v \in V_t$ zu einem benachbartem Knoten $w \in V_t$ - ein \textbf{struktureller Sprung}. + ein struktureller Sprung. \end{definition} \goodbreak Im Gegensatz dazu benutzten inhaltliche Mehrfachsprünge tatsächlich die Grapherweiterung: -\begin{definition} +\begin{definition}[inhaltlicher Mehrfachsprung] Sei $G_t = (V_t, E_{S,t} \cup E_{W,t}, V_{L,t}, W_{t})$ der um die Wortknoten $W_{t}$ erweiterte Graph. Dann heißt das zufällige wechseln des aktuell betrachteten Knoten $v \in V_t$ zu einem benachbartem Knoten $w \in W_t$ und weiter zu einem zufälligem Nachbar $v' \in V_t$ von $w$ - ein \textbf{inhaltlicher Mehrfachsprung}. + ein inhaltlicher Mehrfachsprung. \end{definition} Jeder inhaltliche Mehrfachsprung beginnt und endet also in einem Strukturknoten, @@ -65,11 +86,11 @@ springt über einen Wortknoten und ist ein Pfad der Länge~2. Ob in einem Sprung der Random Walks ein struktureller Sprung oder ein inhaltlicher Mehrfachsprung gemacht wird, wird jedes mal zufällig neu entschieden. Dafür wird der Parameter $0 \leq p_S \leq 1$ für den Algorithmus -gewählt. Mit einer Wahrscheinlichkeit von $p_S$ wird eine struktureller +gewählt. Mit einer Wahrscheinlichkeit von $p_S$ wird ein struktureller Sprung durchgeführt und mit einer Wahrscheinlichkeit von $(1-p_S)$ ein modifizierter inhaltlicher Mehrfachsprung, wie er in -\cref{sec:sprungtypen} erklärt wird, gemacht. Dieser -Parameter gibt an, wie wichtig die Struktur des Graphen im Verhältnis +\cref{sec:sprungtypen} erklärt wird, gemacht. Der +Parameter $p_S$ gibt an, wie wichtig die Struktur des Graphen im Verhältnis zu den textuellen Inhalten ist. Bei $p_S = 0$ werden ausschließlich die Texte betrachtet, bei $p_S = 1$ ausschließlich die Struktur des Graphen. @@ -90,11 +111,11 @@ Pseudocode vorgestellt. \Ensure Klassifikation von $V_t \setminus V_{L,t}$\\ \\ - \ForAll{Knoten $v$ in $V_t \setminus V_{L,t}$} + \ForAll{Knoten $v \in V_t \setminus V_{L,t}$} \State $d \gets $ defaultdict - \For{$i$ von $1$ bis $r$} + \For{$i = 1, \dots,r$} \State $w \gets v$ - \For{$j$ von $1$ bis $l$} + \For{$j= 1, \dots, l$} \State $sprungTyp \gets \Call{random}{0, 1}$ \If{$sprungTyp \leq p_S$} \State $w \gets$ \Call{SturkturellerSprung}{$w$} @@ -132,10 +153,13 @@ verwaltet der DYCOS-Algorithmus zwei weitere Datenstrukturen: assoziatives Array geschehen. Wörter, die nicht in Texten von $v$ vorkommen, sind nicht im Array. Für alle vorkommenden Wörter ist der gespeicherte Wert zum - Schlüssel \enquote{Wort} die Anzahl der Vorkommen von - \enquote{Wort} in den Texten von $v$. + Schlüssel $w \in W_t$ die Anzahl der Vorkommen von + $w$ in den Texten von $v$. \item Für jedes Wort des Vokabulars $W_t$ wird eine Liste von Knoten verwaltet, in deren Texten das Wort vorkommt. + Diese Liste wird bei den inhaltlichen Mehrfachsprung, + der in \cref{sec:sprungtypen} erklärt wird, + verwendet. \item An einigen Stellen macht ein assoziatives Array, auch \enquote{dictionary} oder \enquote{map} genannt, Sinn. 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z!>DLHhx-Qy|I^0D`v=7_i~M`0(m%6vGIIEDHwZ=HK}BEvaaT~@@?$q$K&lw~-#vXtq+#;Ex-am{>cU0!lK4tV4;c#{7kK?t z|03H*@h@bEU3vNvs*7zLQCX%YdsAebzBbqpy9+91kxaq~sY_*?pf=Hvxa&Qo$r!WC z*OxLSs}IMQ{}@GWDcIMy>n@y*P?*3TCN>`d_#G9gKs=%grIFQvc2?{R2x|1_IEE&U zQLD-OwkFqx=^Tqt%{GU`$3#_QeZUO+2q|a*wSw3T`EdSuE%UxEvm?Hf1TD!q%E)C(E}+=`aIL?zYn&u2i9)iFU3c?dg;2H^wG_2XEKndxUD+P6!%`NQ zHEXG&CL}DTn%g&pok&hevzc>P7yrpKD_qvqKA-;z?V0WF08dLy3kpCJ1}N#ZKFNNs zThI$y+MCWn@(^&&zEfm{Z)egsT5Kelg`;9{<#h@}Yv8=Vmm`wPUYyuJHFmK@!<|BL_Gu2B1y%#yy&Aw#<%+PlDc=5}LN!;yt6o+nnj{`@px5{!R^D5vgqw=U z^rBYTIBhxxTTBrh+oMe1yz+fn_KfMF%dmWKcf&)r7^mm2T>0d#5;GuG?+xr%j40Pg ztf^AbIOtlVno2DMQeQLnq2!QVOnbb`P#sSy|KzX@>PE(AMNkX@I3s4mTJ`F%qD%+l zvm)K~=fd3@Lmn69!N?io?=$*XSX>O&G=+oA^6E{}g~{NEk$HHyft-lc)DlXPi2n~| CF_TUJ diff --git a/documents/DYCOS/Einleitung.tex b/documents/DYCOS/Einleitung.tex index 5ee868e..704ed3b 100644 --- a/documents/DYCOS/Einleitung.tex +++ b/documents/DYCOS/Einleitung.tex @@ -1,14 +1,20 @@ -\subsection{Motivation} -Teilweise gelabelte Netzwerke sind allgegenwärtig. Publikationsdatenbanken +Im Folgenden werden in \cref{sec:Motivation} einge Beispiele, in denen +der DYCOS-Algorithmus anwendung finden könnte, dargelegt. In +\cref{sec:Problemstellung} wird die Problemstellung formal definiert +und in \cref{sec:Herausforderungen} wird auf besondere Herausforderungen +der Aufgabenstellung hingewiesen. + +\subsection{Motivation}\label{sec:Motivation} +Teilweise beschriftete Graphen sind allgegenwärtig. Publikationsdatenbanken mit Publikationen als Knoten, Literaturverweisen und Zitaten als Kanten -sowie Tags oder Kategorien als Labels; +sowie von Nutzern vergebene Beschriftungen (sog. {\it Tags}) oder Kategorien als Labels; Wikipedia mit Artikeln als Knoten, Links als Kanten und Kategorien als Labels sowie soziale Netzwerke mit Eigenschaften der Benutzer als Labels sind drei Beispiele dafür. Häufig sind Labels nur teilweise vorhanden und es ist wünschenswert die -fehlenden Labels zu ergänzen. +fehlenden Labels automatisiert zu ergänzen. -\subsection{Problemstellung} +\subsection{Problemstellung}\label{sec:Problemstellung} Gegeben ist ein Graph, der teilweise gelabelt ist. Zusätzlich stehen zu einer Teilmenge der Knoten Texte bereit. Gesucht sind nun Labels für alle Knoten, die bisher noch nicht gelabelt sind.\\ @@ -30,12 +36,13 @@ für alle Knoten, die bisher noch nicht gelabelt sind.\\ $\tilde{f}|_{V_{L,t}} = f$. \end{definition} -\subsection{Herausforderungen} +\subsection{Herausforderungen}\label{sec:Herausforderungen} Die Graphen, für die dieser Algorithmus konzipiert wurde, -sind viele $\num{10000}$~Knoten groß und dynamisch. Das bedeutet, es -kommen neue Knoten und eventuell auch neue Kanten hinzu bzw. Kanten -oder Knoten werden entfernt. Außerdem stehen textuelle Inhalte zu den +sind viele $\num{10000}$~Knoten groß und dynamisch. \enquote{Dynamisch} +bedeutet in diesem Kontext, dass neue Knoten und eventuell auch neue +Kanten hinzu kommen bzw. Kanten oder Knoten werden entfernt werden. +Außerdem stehen textuelle Inhalte zu den Knoten bereit, die bei der Klassifikation genutzt werden können. -Bei kleinen Modifikationen sollte nicht alles nochmals berechnen +Bei kleinen Änderungen sollte nicht alles nochmals berechnen werden müssen, sondern basierend auf zuvor -berechneten Labels sollte die Klassifizierung modifiziert werden. +berechneten Labels sollte die Klassifizierung angepasst werden. diff --git a/documents/DYCOS/README.md b/documents/DYCOS/README.md index 61afed7..1f94f70 100644 --- a/documents/DYCOS/README.md +++ b/documents/DYCOS/README.md @@ -2,3 +2,11 @@ Ausarbeitung zum Proseminar "Netzwerkanalyse" am KIT. Die Ausarbeitung soll 10-12 Seiten haben und die Präsentation 25-30 Minuten dauern + 10-15 Minuten Diskussion. + +TODO +----- + +* label -> Beschriftung +* Abschnitt "Problemstellung" überarbeiten +* Abbildung verlinken +* Algorithmen erklären diff --git a/documents/DYCOS/Sprungtypen.tex b/documents/DYCOS/Sprungtypen.tex index 107016c..7b526f9 100644 --- a/documents/DYCOS/Sprungtypen.tex +++ b/documents/DYCOS/Sprungtypen.tex @@ -18,7 +18,7 @@ gezeigt wird. \label{alg:DYCOS-structural-hop} \end{algorithm} -Bei inhaltlichen Mehrfachsprüngen ist jedoch nicht sinnvoll so direkt +Bei inhaltlichen Mehrfachsprüngen ist jedoch nicht sinnvoll so strikt nach der Definition vorzugehen, also direkt von einem strukturellem Knoten $v \in V_t$ zu einem mit $v$ verbundenen Wortknoten $w \in W_t$ zu springen @@ -30,52 +30,66 @@ Gemeint sein können z.~B. das Bauwerk, das Entwurfsmuster der objektorientierten Programmierung oder ein Teil des Gehirns. Deshalb wird für jeden Knoten $v$, von dem aus man einen inhaltlichen -Mehrfachsprung machen will folgendes Clusteranalyse durchgeführt: -\begin{enumerate}[label=C\arabic*),ref=C\arabic*] - \item[C1] Gehe alle in $v$ startenden Random Walks der Länge 2 durch - und erstelle eine Liste $L$, der erreichbaren Knoten $v'$. Speichere +Mehrfachsprung machen will folgende Textanalyse durchgeführt: +\begin{enumerate}[label=C\arabic*,ref=C\arabic*] + \item \label{step:c1} Gehe alle in $v$ startenden Random Walks der Länge $2$ durch + und erstelle eine Liste $L$ der erreichbaren Knoten $v'$. Speichere außerdem, durch wie viele Pfade diese Knoten $v'$ jeweils erreichbar sind. - \item[C2] Betrachte im folgenden nur die Top-$q$ Knoten, wobei $q \in \mathbb{N}$ - eine zu wählende Konstante des Algorithmus ist.\footnote{Sowohl für den DBLP, als auch für den -CORA-Datensatz wurde in \cite[S. 364]{aggarwal2011} $q=10$ gewählt.} \label{list:aggregate.2} - \item[C3] Wähle mit Wahrscheinlichkeit $\frac{\Call{Anzahl}{v'}}{\sum_{w \in L} \Call{Anzahl}{v'}}$ - den Knoten $v'$ als Ziel des Mehrfachsprungs. + \item \label{step:c2} Betrachte im folgenden nur die Top-$q$ Knoten bzgl. der + Anzahl der Pfade von $v$ nach $v'$, wobei $q \in \mathbb{N}$ + eine zu wählende Konstante des DYCOS-Algorithmus ist.\footnote{Sowohl für den DBLP, als auch für den +CORA-Datensatz wurde in \cite[S. 364]{aggarwal2011} $q=10$ gewählt.} + Diese Knotenmenge heiße im Folgenden $T(v)$ und $p(v, v')$ + sei die Anzahl der Pfade von $v$ über einen Wortknoten nach $v'$. + \item \label{step:c3} Wähle mit Wahrscheinlichkeit $\frac{p(v, v')}{\sum_{w \in T(v)} p(v, w)}$ + den Knoten $v' \in T(v)$ als Ziel des Mehrfachsprungs. \end{enumerate} -Konkret könnte also ein Inhaltlicher Mehrfachsprung sowie wie in +Konkret könnte also ein inhaltlicher Mehrfachsprung sowie wie in \cref{alg:DYCOS-content-multihop} beschrieben umgesetzt werden. +Der Algorithmus bekommt einen Startknoten $v \in V_T$ und +einen $q \in \mathbb{N}$ als Parameter. $q$ ist ein Parameter der +für den DYCOS-Algorithmus zu wählen ist. Dieser Parameter beschränkt +die Anzahl der möglichen Zielknoten $v' \in V_T$ auf diejenigen +$q$ Knoten, die $v$ bzgl. der Textanalyse am ähnlichsten sind. + +In \cref{alg:l2} bis \cref{alg:l5} wird \cref{step:c1} durchgeführt. +In \cref{alg:l6} wird \cref{step:c2} durchgeführt. Bei der +Wahl der Datenstruktur $M_H$ ist zu beachten, dass man in +\cref{alg:21} über Indizes auf Elemente aus $M_H$ zugreifen können muss. + +In \cref{alg:l8} bis \cref{alg:l13} wird ein Wörterbuch erstellt, +das von $v' \in T(v)$ auf die relative +Häufigkeit bzgl. aller Pfade von $v$ zu Knoten aus den Top-$q$ abbildet. + +In \cref{alg:15} bis \cref{alg:22} wird ein Knoten $v' \in T(v)$ mit +einer Wahrscheinlichkeit, die seiner relativen Häufigkeit am Anteil +der Pfaden der Länge 2 von $v$ nach $v'$ über einen beliebigen +Wortknoten entspricht ausgewählt und schließlich zurückgegeben. \begin{algorithm} \caption{Inhaltlicher Mehrfachsprung} \label{alg:DYCOS-content-multihop} \begin{algorithmic}[1] - \Procedure{InhaltlicherMehrfachsprung}{Knoten $v$} - \State \textit{//Alle Knoten bestimmen, die von $v$ aus über Pfade der Länge 2 erreichbar sind} - \State \textit{//Zusätzlich wird für diese Knoten die Anzahl der Pfade der Länge 2 bestimmt,} - \State \textit{//durch die sie erreichbar sind} - \State $reachableNodes \gets$ defaultdict + \Procedure{InhaltlicherMehrfachsprung}{Knoten $v \in V_T$, $q \in \mathbb{N}$} + \State $reachableNodes \gets$ defaultdict\label{alg:l2} \ForAll{Wortknoten $w$ in $v.\Call{getWordNodes}{ }$} \ForAll{Strukturknoten $x$ in $w.\Call{getStructuralNodes}{ }$} \State $reachableNodes[x] \gets reachableNodes[x] + 1$ \EndFor - \EndFor - - \State \textit{//Im folgenden wird davon ausgegangen, dass man über Indizes wahlfrei auf} - \State \textit{//Elemente aus $M_H$ zugreifen kann. Dies muss bei der konkreten Wahl} - \State \textit{//der Datenstruktur berücksichtigt werden.} - \State $M_H \gets \Call{max}{reachableNodes, q}$ \Comment{Also: $|M_H| = q$, falls $|reachableNodes|\geq q$} - \State \textit{//Dictionary mit relativen Häufigkeiten erzeugen} - \State $s \gets 0$ + \EndFor\label{alg:l5} + \State \label{alg:l6} $M_H \gets \Call{max}{reachableNodes, q}$ \Comment{Also: $|M_H| = q$, falls $|reachableNodes|\geq q$} + \\ + \State \label{alg:l8} $s \gets 0$ \ForAll{Knoten $x$ in $M_H$} \State $s \gets s + reachableNodes[x]$ \EndFor \State $relativeFrequency \gets $ Dictionary \ForAll{Knoten $x$ in $M_H$} \State $relativeFrequency \gets \frac{reachableNodes[x]}{s}$ - \EndFor - \State \textit{//Wähle Knoten $i$ mit einer Wahrscheinlichkeit entsprechend seiner relativen} - \State \textit{//Häufigkeit an Pfaden der Länge 2} - \State $random \gets \Call{random}{0, 1}$ + \EndFor\label{alg:l13} + \\ + \State \label{alg:15} $random \gets \Call{random}{0, 1}$ \State $r \gets 0.0$ \State $i \gets 0$ \While{$s < random$} @@ -83,8 +97,8 @@ Konkret könnte also ein Inhaltlicher Mehrfachsprung sowie wie in \State $i \gets i + 1$ \EndWhile - \State $v \gets M_H[i-1]$ - \State \Return $v$ + \State $v \gets M_H[i-1]$ \label{alg:21} + \State \Return $v$ \label{alg:22} \EndProcedure \end{algorithmic} \end{algorithm} diff --git a/documents/DYCOS/abstract.tex b/documents/DYCOS/abstract.tex index 150b84d..282b88b 100644 --- a/documents/DYCOS/abstract.tex +++ b/documents/DYCOS/abstract.tex @@ -7,10 +7,9 @@ Struktur des Graphen sowie textuelle Informationen, die den Knoten zugeordnet sind. Die in \cite{aggarwal2011} beschriebene experimentelle Analyse ergab, dass er auch auf dynamischen Graphen mit $\num{19396}$ bzw. $\num{806635}$ Knoten, von denen nur $\num{14814}$ bzw. $\num{18999}$ -beschriftet waren, innerhalb von weniger als einer Minute auf einer -CPU eines Intel Xeon 2.5GHz mit 32G RAM ausgeführt -werden kann.\\ -Zusätzlich wird auf Schwächen von \cite{aggarwal2011} hingewiesen -und mögliche Verbesserungen vorgeschlagen. +beschriftet waren, innerhalb von weniger als einer Minute auf einem +Kern einer Intel Xeon 2.5GHz CPU mit 32G RAM ausgeführt werden kann.\\ +Zusätzlich wird \cite{aggarwal2011} kritisch Erörtert und +und es werden mögliche Erweiterungen des DYCOS-Algorithmus vorgeschlagen. \textbf{Keywords:} DYCOS, Label Propagation, Knotenklassifizierung