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<!DOCTYPE html>
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<meta name="author" content="Jan Aerts">
<title>Data Management</title>
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</head>
<body id="_graph_databases" class="article toc2 toc-left">
<div id="header">
<h1>Data Management</h1>
<div class="details">
<span id="author" class="author">Jan Aerts</span><br>
<span id="email" class="email"><a href="mailto:[email protected]">[email protected]</a></span><br>
</div>
<div id="toc" class="toc2">
<div id="toctitle">Table of Contents</div>
<p><span class="toc-root"><a href="index.html">Data Management</a></span></p><ul class="sectlevel1">
<li><a href="_relational_database_management_systems.html">1. Relational Database Management Systems</a>
</li>
<li><a href="_nosql_an_introduction.html">2. NoSQL: an introduction</a>
</li>
<li><a href="_keyvalue_stores.html">3. Key/value stores</a>
</li>
<li><a href="_document_oriented_databases.html">4. Document-oriented databases</a>
</li>
<li><a href="_graph_databases.html"><span class="toc-current">5. Graph databases</span></a>
<ul class="sectlevel2">
<li><a href="_graph_databases.html#_introduction_2">5.1. Introduction</a>
</li>
<li><a href="_graph_databases.html#_nomenclature">5.2. Nomenclature</a>
</li>
<li><a href="_graph_databases.html#_centralities">5.3. Centralities</a>
</li>
<li><a href="_graph_databases.html#_graph_mining">5.4. Graph mining</a>
</li>
<li><a href="_graph_databases.html#_data_modeling_2">5.5. Data modeling</a>
</li>
</ul>
</li>
<li><a href="_arangodb.html">6. ArangoDB</a>
</li>
<li><a href="_arangodb_interfaces.html">7. ArangoDB interfaces</a>
</li>
</ul>
</div>
</div>
<div id="content">
<div class="sect1">
<h2 id="_graph_databases">5. Graph databases</h2>
<div class="sectionbody">
<div class="sect2">
<h3 id="_introduction_2">5.1. Introduction</h3>
<div class="paragraph">
<p>Graphs are used in a wide range of applications, from fraud detection (see the Panama Papers) and anti-terrorism and social marketing to drug interaction research and genome sequencing.</p>
</div>
<div class="paragraph">
<p><span class="image"><img src="assets/hairball.png" alt="hairball"></span></p>
</div>
<div class="paragraph">
<p>Graphs or networks are data structures where the most important information is the <em>relationship</em> between entities rather than the entities themselves, such as friendship relationships. Whereas in relational databases you typically aggregate operations on sets, in graph databases you’ll more typically hop around relationships between records. Graphs are very expressive, and any type of data can be modelled as one (although that is no guarantee that a particular graph is fit for purpose).</p>
</div>
<div class="paragraph">
<p>Graphs come in all shapes and forms. Links can be directed or undirected, weighted or unweighted. They can be directed acyclic graphs (where no loops exist), consist of one or more connected components, and actually consist of multiple graphs themselves. The latter (so-called multi-layer networks) can e.g. be a first network representing friendships between people, a second network representing cities and how they are connected through public transportation, and both being linked by which people work in which cities.</p>
</div>
<div class="paragraph">
<p><span class="image"><img src="assets/graph-types.png" alt="graph types"></span></p>
</div>
</div>
<div class="sect2">
<h3 id="_nomenclature">5.2. Nomenclature</h3>
<div class="paragraph">
<p>A graph consists of vertices (aka nodes, aka objects) and edges (aka links, aka relations), where an edge is a connection between two vertices. Both vertices and edges can have properties.</p>
</div>
<div class="paragraph">
<p>\[G = (V,E)\]</p>
</div>
<div class="paragraph">
<p>Any graph can be described using different metrics:</p>
</div>
<div class="ulist">
<ul>
<li>
<p><em>order</em> of a graph = number of nodes</p>
</li>
<li>
<p><em>size</em> of a graph = number of edges</p>
</li>
<li>
<p>graph <em>density</em> = how much its nodes are connected. In a dense graph, the number of edges is close to the maximal number of edges (i.e. a fully-connected graph).</p>
<div class="ulist">
<ul>
<li>
<p>for undirected graphs, this is:
\[\frac{2 |E|}{|V|(|V|-1)}\]</p>
</li>
<li>
<p>for directed graphs, this is:
\[\frac{|E|}{|V|(|V|-1)}\]</p>
</li>
</ul>
</div>
</li>
<li>
<p>the <em>degree</em> of a node = how many edges are connected to the node. This can be separated into <em>in-degree</em> and <em>out-degree</em>, which are - respectively - the number of incoming and outgoing edges.</p>
</li>
<li>
<p>the <em>distance</em> between two nodes = the number of edges in the shortest path between them</p>
</li>
<li>
<p>the <em>diameter</em> of a graph = the maximum distance in a graph</p>
</li>
<li>
<p>a <em>d-regular</em> graph = a graph where the maximum degree is the same as the minimum degree <em>d</em></p>
</li>
<li>
<p>a <em>path</em> = a sequence of edges that connects a sequence of different vertices</p>
</li>
<li>
<p>a <em>connected graph</em> = a graph in which there exists a direct connection between any two vertices</p>
</li>
</ul>
</div>
</div>
<div class="sect2">
<h3 id="_centralities">5.3. Centralities</h3>
<div class="paragraph">
<p>Another important way of describing nodes is based on their <em>centrality</em>, i.e. how <em>central</em> they are in the network. There exist different versions of this centrality:</p>
</div>
<div class="ulist">
<ul>
<li>
<p><em>degree centrality</em>: how many other vertices a given vertex is connected to. This is the same as node degree.</p>
</li>
<li>
<p><em>betweenness centrality</em>: how many critical paths go through this node? In other words: without these nodes, there would be no way for to neighbours to communicate.
\[
C_{B}(i)=\frac{\sum\limits_{j \neq k} g_{jk} (i)}{g_{jk}} \xrightarrow[]{normalize} C'_B = \frac{C_B(i)}{(n-1)(n-2)/2}\] , where the denominator is the number of vertex pairs excluding the vertex itself. \(g_jk(i)\) is number of shortest paths between \(j\) and \(k\), going through i; \(g_jk\) is the total number of shortest paths between \(j\) and \(k\).</p>
</li>
<li>
<p><em>closeness centrality</em>: how much is the node in the "middle" of things, not too far from the center. This is the inverse total distance to all other nodes.
\[C_C(i) = \frac{1}{\sum\limits_{j=1}^N d(i,j)} \xrightarrow[]{normalize} C'_C(i) = \frac{C_C(i)}{N-1}\]</p>
</li>
</ul>
</div>
<div class="paragraph">
<p>In the image below, nodes in A are coloured based on betweenness centrality, in B based on closeness centrality, and in D on degree centrality.</p>
</div>
<div class="paragraph">
<p><span class="image"><img src="assets/centralities.png" alt="centralities"></span></p>
</div>
</div>
<div class="sect2">
<h3 id="_graph_mining">5.4. Graph mining</h3>
<div class="paragraph">
<p>Graphs are very generic data structures, but are amenable to very complex analyses. These include the following.</p>
</div>
<div class="sect3">
<h4 id="_community_detection">5.4.1. Community detection</h4>
<div class="paragraph">
<p>A community in a graph is a group of nodes that are densely connected internally. You can imagine that e.g. in social networks we can identify groups of friends this way.</p>
</div>
<div class="paragraph">
<p><span class="image"><img src="assets/graph-communities.png" alt="graph communities"></span></p>
</div>
<div class="paragraph">
<p>Several approaches exist to finding communities:</p>
</div>
<div class="ulist">
<ul>
<li>
<p><em>null models</em>: a community is a set of nodes for which the connectivity deviates the most from the null model</p>
</li>
<li>
<p><em>block models</em>: identify blocks of nodes with common properties</p>
</li>
<li>
<p><em>flow models</em>: a community is a set of nodes among which a flow persists for a long time once entered</p>
</li>
</ul>
</div>
<div class="paragraph">
<p>The <em>infomap</em> algorithm is an example of a flow model (for a demo, see <a href="http://www.mapequation.org/apps/MapDemo.html" class="bare">http://www.mapequation.org/apps/MapDemo.html</a>).</p>
</div>
</div>
<div class="sect3">
<h4 id="_link_prediction">5.4.2. Link prediction</h4>
<div class="paragraph">
<p>When data is acquired from a real-world source, this data might be incomplete and links that should actually be there are not in the dataset. For example, you gather historical data on births, marriages and deaths from church and city records. There is therefore a high chance that you don’t have all data. Another domain where this is important is in protein-protein interactions.</p>
</div>
<div class="paragraph">
<p>Link prediction can be done in different ways, and can happen in a dynamic or static setting. In the <em>dynamic setting</em>, we try to predict the likelihood of a future association between two nodes; in the <em>static setting</em>, we try to infer missing links. These algorithms are based on a similarity matrix between the network nodes, which can take different forms:</p>
</div>
<div class="ulist">
<ul>
<li>
<p><em>graph distance</em>: the length of the shortest path between 2 nodes</p>
</li>
<li>
<p><em>common neighbours</em>: two strangers who have a common friend may be introduced by that friend</p>
</li>
<li>
<p><em>Jaccard’s coefficient</em>: the probability that 2 nodes have the same neighbours</p>
</li>
<li>
<p><em>frequency-weighted common neighbours (Adamic/Adar predictor</em>): counts common features (e.g. links), but weighs rare features more heavily</p>
</li>
<li>
<p><em>preferential attachment</em>: new link between nodes with high number of links is more likely than between nodes with low number of links</p>
</li>
<li>
<p><em>exponential damped path counts (Katz measure)</em>: the more paths there are between two nodes and the shorter these paths are, the more similar the nodes are</p>
</li>
<li>
<p><em>hitting time</em>: random walk starts at node A ⇒ expected number of steps required to reach node B</p>
</li>
<li>
<p><em>rooted pagerank</em>: idem, but periodical reset to prevent that 2 nodes that are actually close are connected through long deviation</p>
</li>
</ul>
</div>
</div>
<div class="sect3">
<h4 id="_subgraph_mapping">5.4.3. Subgraph mapping</h4>
<div class="paragraph">
<p>Subgraph mining is another type of query that is very important in e.g. bioinformatics. Some example patterns:</p>
</div>
<div class="ulist">
<ul>
<li>
<p>[A] three-node feedback loop</p>
</li>
<li>
<p>[B] tree chain</p>
</li>
<li>
<p>[C] four-node feedback loop</p>
</li>
<li>
<p>[D] feedforward loop</p>
</li>
<li>
<p>[E] bi-parallel pattern</p>
</li>
<li>
<p>[F] bi-fan</p>
</li>
</ul>
</div>
<div class="paragraph">
<p><span class="image"><img src="assets/network-motifs.png" alt="network motifs"></span></p>
</div>
<div class="paragraph">
<p>It is for example important when developing a drug for a certain disease by knocking out the effect of a gene that that gene is not in a bi-parallel pattern (<code>V2</code> in image <code>E</code> above) because activation of node <code>V4</code> is saved by <code>V3</code>.</p>
</div>
</div>
</div>
<div class="sect2">
<h3 id="_data_modeling_2">5.5. Data modeling</h3>
<div class="paragraph">
<p>In general, vertices are used to represent <em>things</em> and edges are used to represent <em>connections</em>. Vertex properties can include e.g. metadata such as timestamp, version number etc; edges properties often include the weight of a connection, but can also cover things like the quality of a relationship and other metadata of that relationship.</p>
</div>
<div class="paragraph">
<p>Below is an example of a graph:
<span class="image"><img src="assets/graph.png" alt="graph"></span></p>
</div>
<div class="paragraph">
<p>Basically all types of data can be modelled as a graph. Consider our buildings table from above:</p>
</div>
<table class="tableblock frame-all grid-all stretch">
<colgroup>
<col style="width: 14.2857%;">
<col style="width: 14.2857%;">
<col style="width: 14.2857%;">
<col style="width: 14.2857%;">
<col style="width: 14.2857%;">
<col style="width: 14.2857%;">
<col style="width: 14.2858%;">
</colgroup>
<thead>
<tr>
<th class="tableblock halign-left valign-top">id</th>
<th class="tableblock halign-left valign-top">name</th>
<th class="tableblock halign-left valign-top">address</th>
<th class="tableblock halign-left valign-top">city</th>
<th class="tableblock halign-left valign-top">type</th>
<th class="tableblock halign-left valign-top">nr_rooms</th>
<th class="tableblock halign-left valign-top">primary_or_secondary</th>
</tr>
</thead>
<tbody>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">1</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">building1</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">street1</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">city1</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">hotel</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">15</p></td>
<td class="tableblock halign-left valign-top"></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">2</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">building2</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">street2</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">city2</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">school</p></td>
<td class="tableblock halign-left valign-top"></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">primary</p></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">3</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">building3</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">street3</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">city3</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">hotel</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">52</p></td>
<td class="tableblock halign-left valign-top"></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">4</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">building4</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">street4</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">city4</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">church</p></td>
<td class="tableblock halign-left valign-top"></td>
<td class="tableblock halign-left valign-top"></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">5</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">building5</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">street5</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">city5</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">house</p></td>
<td class="tableblock halign-left valign-top"></td>
<td class="tableblock halign-left valign-top"></td>
</tr>
<tr>
<td class="tableblock halign-left valign-top"><p class="tableblock">…​</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">…​</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">…​</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">…​</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">…​</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">…​</p></td>
<td class="tableblock halign-left valign-top"><p class="tableblock">…​</p></td>
</tr>
</tbody>
</table>
<div class="paragraph">
<p>This can also be represented as a network, where:</p>
</div>
<div class="ulist">
<ul>
<li>
<p>every building is a vertex</p>
</li>
<li>
<p>every value for a property is a vertex as well</p>
</li>
<li>
<p>the column becomes the relation</p>
</li>
</ul>
</div>
<div class="paragraph">
<p>For example, the information for the first building can be represented as such:</p>
</div>
<div class="paragraph">
<p><span class="image"><img src="assets/examplegraph.png" alt="examplegraph"></span></p>
</div>
<div class="paragraph">
<p>There is actually a formal way of describing this called RDF, but we won’t go into that here…​</p>
</div>
</div>
</div>
</div>
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<p>← Previous: <a href="_document_oriented_databases.html">Document-oriented databases</a> | ↑ Up: <a href="index.html">Data Management</a> | Next: <a href="_arangodb.html">ArangoDB</a> →</p>
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Last updated 2023-11-13 10:40:23 +0100
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