Add report

Signed-off-by: Yohann D'ANELLO <ynerant@crans.org>

.gitignore

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+*.aux
+*.log
+*.nt
+*.pdf

Report.tex

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+\documentclass{article}
+
+\usepackage{cours}
+
+\title{Keys in Graphs}
+\date{January, $13^{\text{th}}$ 2022}
+
+\begin{document}
+
+\maketitle
+
+\section{Introduction}
+
+This project aims to find Graph keys, as defined in
+\footnote{\url{https://www.researchgate.net/publication/283189709_Keys_for_graphs}}.
+A Graph Key describes the relations that an object can have with their keys, and
+what relations these involved objects can have.
+
+\medskip
+
+For example, a Graph Key for a book can be:
+
+\begin{center}
+\begin{tikzpicture}[y=3cm]
+\node[draw] (0) at (0, 0) {Book};
+\node[] (00) at (-3, -1) {x};
+\node[draw] (01) at (-1, -1) {Person};
+\node[] (02) at (1, -1) {y};
+\node[draw] (03) at (3, -1) {Company};
+\node[draw] (010) at (-2, -2) {Country};
+\node[] (011) at (0, -2) {z};
+\node[] (030) at (3, -2) {t};
+\draw[->] (0) -- (00) node[midway,above,sloped] {title};
+\draw[->] (0) -- (01) node[midway,above,sloped] {author};
+\draw[->] (0) -- (02) node[midway,above,sloped] {subtitle};
+\draw[->] (0) -- (03) node[midway,above,sloped] {publisher};
+\draw[->] (01) -- (010) node[midway,above,sloped] {nationality};
+\draw[->] (01) -- (011) node[midway,above,sloped] {last name};
+\draw[->] (03) -- (030) node[midway,above,sloped] {identifier};
+\end{tikzpicture}
+\end{center}
+
+That's to say, a book can be described with its title and its subtitle, the last
+name and the nationality of its author, and the public identifier of the publisher.
+
+\medskip
+
+To generate these keys, one suggests to find $n$-almost keys using SAKey,
+then to explore involved relations that define a domain and a range, and to
+explore recursively the related fields.
+
+\section{Proposed solution}
+
+The proposed tool is available here:
+\url{https://gitlab.crans.org/ynerant/graph-keys}
+
+\subsection{Requirements}
+
+The project is made with Python 3.9 and Python 3.10, and uses \texttt{BeautifulSoup4}
+and \texttt{SPARQLWrapper} as libraries.
+
+\subsection{Principle}
+
+\medskip
+
+The program takes as an input the class name that we want to explore, and a
+threshold $n$ that is the number of allowed exceptions for SAKey. The class name
+has to be existing in DBPedia since we make only DBPedia queries.
+
+\medskip
+
+First, we load the ontology of DBPedia and load a lot of relations. The goal is
+to define the range and the domain of most relations. For example, we learn that
+the relation \texttt{inCemetery} has the domain \texttt{GraveMonument} and the
+range \texttt{Cemetery}.
+
+\medskip
+
+The next step is to query DBPedia to get all elements of the type of the input
+class, then to get all triples \texttt{?x ?r ?y} that are involving these elements.
+Since datasets can be very big, we limit the output by default to 1000 triples,
+but this value can be changed using the option \texttt{-{}-limit}.
+
+\medskip
+
+We now store all these triples, and give them to the \emph{SAKey} tool, in order to
+extract the $n$-almost keys of the dataset, where $n$ is given in input. These keys
+are relations between our input, and we can now explore further. To get relevant
+data, we made the choice to consider only the relations that has a defined range,
+which is not a primitive (like integers, strings, dates, $\ldots$). In a general
+case, we get only a few but relevant results.
+
+\medskip
+
+In the following, we consider the exploration of Graph Keys for the class
+\texttt{Library}. We run the following command:
+
+\begin{center}
+  \texttt{./main.py Library 5 -{}-limit 3000 -{}-recursion 3}
+\end{center}
+
+The last option will be explained further.
+
+\medskip
+
+For this example, we get the relation \texttt{location}, which has for range
+\texttt{Place}.
+
+\begin{figure}[H]
+\centering
+\begin{tikzpicture}[y=3cm]
+\node[draw] (0) at (0.00, -0) {Library};
+\node[draw] (0-0) at (0.00, -1) {Place};
+\draw[->] (0) -- (0-0) node[midway,above,sloped] {location};
+\end{tikzpicture}
+\caption{Single discovered key}
+\end{figure}
+
+Since we discovered a relation between \texttt{Library} and \texttt{Place}, we
+can now explore the keys of the class \texttt{Place} and extend the key for
+\texttt{Library}.
+
+\medskip
+
+The program takes as optional input the parameter \texttt{-{}recursion}, which
+limits the height of the output graphs. A value of $1$ gives normal keys from
+\emph{SAKey}.
+
+\medskip
+
+When we take the example of \texttt{Library}, we get multiple outputs, like the
+tree bellow:
+
+\begin{figure}[H]
+\centering
+\begin{tikzpicture}[y=3cm]
+\node[draw] (0) at (0.00, -0) {Library};
+\node[draw] (0-0) at (0.00, -1) {Place};
+\node[draw] (0-0-0) at (0.00, -2) {City};
+\node[draw] (0-0-0-0) at (0.00, -3) {Image};
+\draw[->] (0-0-0) -- (0-0-0-0) node[midway,above,sloped] {thumbnail};
+\draw[->] (0-0) -- (0-0-0) node[midway,above,sloped] {capital};
+\draw[->] (0) -- (0-0) node[midway,above,sloped] {location};
+\end{tikzpicture}
+\caption{Sample output of the program}
+\end{figure}
+
+\subsection{Limitations and further works}
+
+While SAKey gives a huge amount of keys, only a few of them are well-typed, in the
+sense that they define a comprehensive range. That gives us a serious limitation
+to our algorithm.
+
+\medskip
+
+Moreover, the current algorithm does not take into account the rules that don't
+have any defined range, which are the most. This isn't really a problem and can
+easily be patched, since this generates more graphs, but does not provide any
+additional input to extend data, as said before.
+
+\medskip
+
+We can notice that graph keys are for the most only paths (degrees of the nodes are
+2 except for the extremal nodes). This is related to the fact that generated keys
+are minimal, and most of them contain only one parameter. Our algorithm should
+be able to generate any type of tree graph.
+
+\medskip
+
+However, our algorithm may not guess all existing graph keys. Current output graphs
+have the property that if we truncate each graph to a given depth, then it stays a
+valid graph key (at least a $n$-almost graph key), which has no reason to be true
+in a general context. To cover this issue, we may extend the SAKey algorithm to
+extend directly the properties with their ranges.
+
+\medskip
+
+To go further, we should take into account the missing properties, and find a way
+to generate more complex graphs, and to find minimal graphs that are not flat.
+
+\end{document}

cours.sty

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+\usepackage{amssymb}
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+\usepackage{amsthm}
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+\usepackage{dsfont}
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+\usepackage{xurl}
+
+\pgfplotsset{compat=1.17}
+\usetikzlibrary{arrows}
+\usetikzlibrary{shapes}
+
+\author{Yohann D'ANELLO}
+
+\pagestyle{fancy}
+\lfoot{Yohann D'ANELLO}
+\rfoot{M2 DS, Université Paris-Saclay}
+
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