<p>It sounds like what you're looking for is <a href="http://en.wikipedia.org/wiki/Modularity_%28networks%29" rel="nofollow">Network Modularity</a>. Given a graph, and a partition (breaking the graph into disjoint subgraphs), the modularity is defined as: </p>

<blockquote>
  <p>The fraction of the edges that fall within the given groups minus the
  expected such fraction if edges were distributed at random.</p>
</blockquote>

<p>Modularity was the basis of some of the first <a href="http://en.wikipedia.org/wiki/Community_structure" rel="nofollow">community detection</a> algorithms on networks, which try to find sets of nodes that are densely connected. Recently, modularity has been shown to be a poor metric for community detection though because of resolution limits that fail to detect small groups or break apart well defined groups in certain cases (see <a href="http://arxiv.org/abs/1107.1155" rel="nofollow">this paper</a>).</p>
Metrics for hierarchical graphs connectivity
<p>this is my first question on Stack Overflow. This is not really a programming question but since most of us have to deal with theoretical problems at some point and there might be some graph theory specialists around, I thought I might give it a go.</p>

<p>I am currently doing some research on multilingual websites and I found some interesting patterns in the website structure. The graphs below are the website graphs of two different multilingual websites. Sorry, I don't have enough rep points to post images so I leave them as links. I used the Force Atlas algorithm for the layout. Vertices are colored according to the page language. The shaded areas correspond to the subgraphs of a specific language.</p>

<p>Here is the graph of the website where different language versions of the same content are very closely linked. Hence the planes representing the different language versions are overlapping.</p>

<p><a href="http://www.ai.soc.i.kyoto-u.ac.jp/~julien/phd/images/tight.png" rel="nofollow">http://www.ai.soc.i.kyoto-u.ac.jp/~julien/phd/images/tight.png</a></p>

<p>In this second graph, we have a website where language versions of a website are almost independent, thus we have almost no overlap.</p>

<p><a href="http://www.ai.soc.i.kyoto-u.ac.jp/~julien/phd/images/loose.png" rel="nofollow">http://www.ai.soc.i.kyoto-u.ac.jp/~julien/phd/images/loose.png</a></p>

<p>So here is my question:</p>

<p><strong>Is there a specific metric to quantify this overlap? If so, what is it named?</strong></p>

<p>Since I used a force-based layout, the number of edges between the language subgraphs. So I guess something like taking the ratio of the number of edges within the subgraph to the number edges going outside/coming inside a specific subgraph might do the trick. I am sure I am not the first to get this idea so I was wondering if this metric had a name. I could then Google it from there :)</p>

<p>Thank you in advance!</p>