https://visone.ethz.ch/wiki/index.php?title=Betweenness&feed=atom&action=historyBetweenness - Revision history2024-03-28T14:20:40ZRevision history for this page on the wikiMediaWiki 1.39.6https://visone.ethz.ch/wiki/index.php?title=Betweenness&diff=681&oldid=prevBrandes at 20:50, 14 May 20112011-05-14T20:50:02Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>\sigma(s,t)</math> is the number of shortest path between <math>s</math> and <math>t</math>, and <math>\sigma(s,t|v)</math> the number of those containing <math>v</math> as an inner node.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where <math>\sigma(s,t)</math> is the number of shortest path between <math>s</math> and <math>t</math>, and <math>\sigma(s,t|v)</math> the number of those containing <math>v</math> as an inner node.</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>That is, under the assumption that information, traffic, resources, trust, etc. spread along shortest paths only, nodes are ranked according to the amount of information, traffic etc. that passes by them. In link-valued graphs, the notion of a shortest path depends on the interpretation of those values. For instance, the attribute might represent a length, and <del style="font-weight: bold; text-decoration: none;">a </del>the length of a path is the sum over the lengths of its links.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>That is, under the assumption that information, traffic, resources, trust, etc. spread along shortest paths only, nodes are ranked according to the amount of information, traffic etc. that passes by them. In link-valued graphs, the notion of a shortest path depends on the interpretation of those values. For instance, the attribute might represent a length, and the length of a path is the sum over the lengths of its links.</div></td></tr>
</table>Brandeshttps://visone.ethz.ch/wiki/index.php?title=Betweenness&diff=680&oldid=prevBrandes: added definition2011-05-14T07:43:30Z<p>added definition</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:43, 14 May 2011</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''Betweenness''' is <del style="font-weight: bold; text-decoration: none;">another means to quantify </del>the structural importance of nodes in a network.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''Betweenness''' is <ins style="font-weight: bold; text-decoration: none;">a medial measure of </ins>the structural importance of nodes in a network<ins style="font-weight: bold; text-decoration: none;">. It assigns higher values to those nodes that are in a position to control the indirect linkages of others</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">It </del>is defined as the <del style="font-weight: bold; text-decoration: none;">percentage of </del>all shortest paths that contain <del style="font-weight: bold; text-decoration: none;">a certain </del>node.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">For each pair of nodes <math>s</math> and <math>t</math>, the ''dependency'' <math>\delta(s,t|v)</math> on a particular node <math>v</math> </ins>is defined as the <ins style="font-weight: bold; text-decoration: none;">fraction </ins>all shortest <ins style="font-weight: bold; text-decoration: none;"><math>st</math>-</ins>paths that contain <ins style="font-weight: bold; text-decoration: none;"><math>v</math> as an inner </ins>node. <ins style="font-weight: bold; text-decoration: none;"> The betweenness score of <math>v</math> is the sum over all pair-dependencies,</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>That is, under the assumption that information, traffic etc. <del style="font-weight: bold; text-decoration: none;">flows only </del>along shortest paths, nodes are ranked according to the amount of information, traffic etc. that <del style="font-weight: bold; text-decoration: none;">flows </del>over <del style="font-weight: bold; text-decoration: none;">them</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math>c_B(v)=\sum_{s\neq v\neq t} \delta(s,t|v) = \sum_{s\neq v\neq t} \frac{\sigma(s,t|v)}{\sigma(s,t)}</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">where <math>\sigma(s,t)</math> is the number of shortest path between <math>s</math> and <math>t</math>, and <math>\sigma(s,t|v)</math> the number of those containing <math>v</math> as an inner node.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>That is, under the assumption that information, traffic<ins style="font-weight: bold; text-decoration: none;">, resources, trust, </ins>etc. <ins style="font-weight: bold; text-decoration: none;">spread </ins>along shortest paths <ins style="font-weight: bold; text-decoration: none;">only</ins>, nodes are ranked according to the amount of information, traffic etc. that <ins style="font-weight: bold; text-decoration: none;">passes by them. In link-valued graphs, the notion of a shortest path depends on the interpretation of those values. For instance, the attribute might represent a length, and a the length of a path is the sum </ins>over <ins style="font-weight: bold; text-decoration: none;">the lengths of its links</ins>.</div></td></tr>
</table>Brandeshttps://visone.ethz.ch/wiki/index.php?title=Betweenness&diff=23&oldid=prevFratz: Created page with ''''Betweenness''' is another means to quantify the structural importance of nodes in a network. It is defined as the percentage of all shortest paths that contain a certain node…'2010-11-29T19:50:04Z<p>Created page with ''''Betweenness''' is another means to quantify the structural importance of nodes in a network. It is defined as the percentage of all shortest paths that contain a certain node…'</p>
<p><b>New page</b></p><div>'''Betweenness''' is another means to quantify the structural importance of nodes in a network.<br />
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It is defined as the percentage of all shortest paths that contain a certain node.<br />
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That is, under the assumption that information, traffic etc. flows only along shortest paths, nodes are ranked according to the amount of information, traffic etc. that flows over them.</div>Fratz