<p>Not sure this is what you are asking, but...</p>

<p>Communities are a form of clustering for networks. The basic idea is that nodes (vertices) in a given community are "more connected" to other nodes in that community, than they are to nodes in other communities. In your simple example, node 46 is connected to nodes 269 and 1854, but these three nodes are not connected to any other nodes, so they form a community. Similarly, nodes 11 and 911 are connected to each other, but not to any other nodes, so they form a community. The definition of "more connected" depends on the algorithm used to identify the communities (to do the clustering).</p>

<p><strong>EDIT</strong> Response to OP's comment.</p>

<p>From the documentation:</p>

<blockquote>
  <p>This function tries to find densely connected subgraphs, also called
  communities in a graph via random walks. The idea is that short random
  walks tend to stay in the same community.</p>
</blockquote>

<p>Here is an example:</p>

<pre><code>library(igraph)
# create a sample graph
g &lt;- graph.full(5)
for (i in 0:3) {
  g &lt;- g %du% graph.full(5)
  g &lt;- add.edges(g,c(5*i+1,5*(i+1)+1))
}

wc &lt;- walktrap.community(g)
colors &lt;- rainbow(max(membership(wc)))
set.seed(1)   # for reproducible layout
plot(g,vertex.color=colors[membership(wc)], 
     layout=layout.fruchterman.reingold)
</code></pre>

<p><img src="https://i.stack.imgur.com/rCLQM.png" alt=""></p>

<p>In this example, each subgroup (community) is highly interconnected, and while the clusters are connected to each other, they are less so. So a random walk that starts in nodes 1-5 is more likely to circulate among those nodes than it is to get to any of the other nodes.  Hence nodes 1-5 form a community.</p>

<p>The algorithm is described in detail <a href="http://arxiv.org/abs/physics/0512106" rel="nofollow noreferrer">here</a>.</p>
Interpreting community structure in graph theory
<p>Okay, so I have a file like this.</p>

<pre><code> 5 1211 11
 18 25 11
 12 281 11
 522 569 11
 46 269 11
 46 1854 11
 544 2324 11
 544 1955 11
 10 795 11
 246 982 11
 37 1500 11
 2 1154 11
 11 911 11
 200 281 11
 512 663 11
 197 663 11
 181 202 11
 1 124 11
 14 636 11
 14 1616 11
 578 1743 11
</code></pre>

<p>The first two columns represent the nodes (people) and the third column represent a particular pattern that they follow ( same in this case ) while they send messages over a time period. The nodes actually represent people who work in the same office. Now, I plotted a graph for them and<img src="https://i.stack.imgur.com/hu5mx.png" alt="This is the graph of the nodes following a particular pattern."></p>

<p>Now, I used the community command with the algorithm as walktrap.community using R. I got the graph again as, <img src="https://i.stack.imgur.com/KhWQz.png" alt="after applying communities"></p>

<p>I really wish to know what these groupings mean. I know they have been grouped by taking into account the modularity. But what does these groupings actually represent?
I read about this on a lot of research papers but didnt find anything relevant. </p>