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6 changes: 3 additions & 3 deletions manual/analysis/index.html
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Expand Up @@ -328,8 +328,8 @@ <h3 id="comparison-of-centrality-vs-prestige-measures">Comparison of Centrality
<blockquote>
<p>Given that Prestige measures are based on the inbound ties to each node from all others, these indices are meaningfull on directed graphs.</p>
</blockquote>
<div class="expressive-code"><link rel="stylesheet" href="/_astro/ec.j8ofn.css"><script type="module" src="/_astro/ec.8zarh.js"></script><figure class="frame not-content"><figcaption class="header"></figcaption><pre data-language="info"><code><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">Centrality measures are calculated for each node (node Centrality) and for the whole network (group Centrality). Thus, when you click on a centrality option, SocNetV will calculate the corresponding index of every node and the whole network, and it will display the report in a new browser window (using the system's default web browser).</span></div></div><div class="ec-line"><div class="code">
</div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">By default, analysis files are saved in the **socnetv-data/** subfolder.</span></div></div></code></pre><div class="copy"><button title="Copy to clipboard" data-copied="Copied!" data-code="Centrality measures are calculated for each node (node Centrality) and for the whole network (group Centrality). Thus, when you click on a centrality option, SocNetV will calculate the corresponding index of every node and the whole network, and it will display the report in a new browser window (using the system&#x27;s default web browser).By default, analysis files are saved in the **socnetv-data/** subfolder."><div></div></button></div></figure></div>
<aside aria-label="Note" class="starlight-aside starlight-aside--note"><p class="starlight-aside__title" aria-hidden="true"><svg viewBox="0 0 24 24" width="16" height="16" fill="currentColor" class="starlight-aside__icon"><path d="M12 11C11.7348 11 11.4804 11.1054 11.2929 11.2929C11.1054 11.4804 11 11.7348 11 12V16C11 16.2652 11.1054 16.5196 11.2929 16.7071C11.4804 16.8946 11.7348 17 12 17C12.2652 17 12.5196 16.8946 12.7071 16.7071C12.8946 16.5196 13 16.2652 13 16V12C13 11.7348 12.8946 11.4804 12.7071 11.2929C12.5196 11.1054 12.2652 11 12 11ZM12.38 7.08C12.1365 6.97998 11.8635 6.97998 11.62 7.08C11.4973 7.12759 11.3851 7.19896 11.29 7.29C11.2017 7.3872 11.1306 7.49882 11.08 7.62C11.024 7.73868 10.9966 7.86882 11 8C10.9992 8.13161 11.0245 8.26207 11.0742 8.38391C11.124 8.50574 11.1973 8.61656 11.29 8.71C11.3872 8.79833 11.4988 8.86936 11.62 8.92C11.7715 8.98224 11.936 9.00632 12.099 8.99011C12.2619 8.97391 12.4184 8.91792 12.5547 8.82707C12.691 8.73622 12.8029 8.61328 12.8805 8.46907C12.9582 8.32486 12.9992 8.16378 13 8C12.9963 7.73523 12.8927 7.48163 12.71 7.29C12.6149 7.19896 12.5028 7.12759 12.38 7.08ZM12 2C10.0222 2 8.08879 2.58649 6.4443 3.6853C4.79981 4.78412 3.51809 6.3459 2.76121 8.17317C2.00433 10.0004 1.8063 12.0111 2.19215 13.9509C2.578 15.8907 3.53041 17.6725 4.92894 19.0711C6.32746 20.4696 8.10929 21.422 10.0491 21.8079C11.9889 22.1937 13.9996 21.9957 15.8268 21.2388C17.6541 20.4819 19.2159 19.2002 20.3147 17.5557C21.4135 15.9112 22 13.9778 22 12C22 10.6868 21.7413 9.38642 21.2388 8.17317C20.7363 6.95991 19.9997 5.85752 19.0711 4.92893C18.1425 4.00035 17.0401 3.26375 15.8268 2.7612C14.6136 2.25866 13.3132 2 12 2ZM12 20C10.4178 20 8.87104 19.5308 7.55544 18.6518C6.23985 17.7727 5.21447 16.5233 4.60897 15.0615C4.00347 13.5997 3.84504 11.9911 4.15372 10.4393C4.4624 8.88743 5.22433 7.46197 6.34315 6.34315C7.46197 5.22433 8.88743 4.4624 10.4393 4.15372C11.9911 3.84504 13.5997 4.00346 15.0615 4.60896C16.5233 5.21447 17.7727 6.23984 18.6518 7.55544C19.5308 8.87103 20 10.4177 20 12C20 14.1217 19.1572 16.1566 17.6569 17.6569C16.1566 19.1571 14.1217 20 12 20Z"></path></svg>Note</p><div class="starlight-aside__content"><p>Centrality measures are calculated for each node (node Centrality) and for the whole network (group Centrality). Thus, when you click on a centrality option, SocNetV will calculate the corresponding index of every node and the whole network, and it will display the report in a new browser window (using the system’s default web browser).
By default, analysis files are saved in the <strong>socnetv-data/</strong> subfolder.</p></div></aside>
<h3 id="degree-centrality-dc">Degree Centrality (DC)</h3>
<p>The DC measure quantifies how many ties a node has to other nodes in the network. In social network theory, this index is often considered a measure of actor activity. It can be computed in both undirected and directed networks/relations, but is usually best suited for undirected ones.</p>
<p>Mathematically, in undirected graphs, the DC index of each vertex <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span> is the number of edges attached to it.</p>
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</ul>
<p>The following list shows all the <strong>16 triad types</strong>:</p>
<p>Within each row, all the triad types have the same number of arcs present:</p>
<div class="expressive-code"><figure class="frame not-content"><figcaption class="header"></figcaption><pre data-language="code"><code><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">003</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">012</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">102 021D 021U 021C</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">111D 111U 030T 030C</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">201 120D 120U 120C</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">210</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">300</span></div></div></code></pre><div class="copy"><button title="Copy to clipboard" data-copied="Copied!" data-code="003012102 021D 021U 021C111D 111U 030T 030C201 120D 120U 120C210300"><div></div></button></div></figure></div>
<div class="expressive-code"><link rel="stylesheet" href="/_astro/ec.j8ofn.css"><script type="module" src="/_astro/ec.8zarh.js"></script><figure class="frame not-content"><figcaption class="header"></figcaption><pre data-language="code"><code><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">003</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">012</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">102 021D 021U 021C</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">111D 111U 030T 030C</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">201 120D 120U 120C</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">210</span></div></div><div class="ec-line"><div class="code"><span style="--0:#d6deeb;--1:#403f53">300</span></div></div></code></pre><div class="copy"><button title="Copy to clipboard" data-copied="Copied!" data-code="003012102 021D 021U 021C111D 111U 030T 030C201 120D 120U 120C210300"><div></div></button></div></figure></div>
<p>So, when you click on the <strong>Triad Census</strong> menu option, <strong>SocNetV</strong> calculates and displays a vector <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span> of length 16. Each element <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mi>u</mi></msub></mrow><annotation encoding="application/x-tex">T_u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the frequency of a specific triad type in the network, e.g., <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>T</mi><mn>003</mn></msub><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">T_{003} = 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">003</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span>.</p>
<p>Furthermore, the order of the elements of vector T is the same as the aforementioned ordering of the triad types:</p>
<p>T = [ T003, T012, T102, T021D, T021U, T021C, T111D, T111U, T030T, T030C, T201, T120D, T120U, T120C, T210, T300 ]</p>
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