<p>You could use the inter- and intra-cluster densities, as defined for undirected graphs in <a href="http://arxiv.org/abs/0906.0612" rel="nofollow">Fortunato'10</a>. Both are based on the notion of graph density, but processed by considering only certain subgraphs. </p>

<p>The intra-cluster density is the proportion of existing links <em>inside</em> a community, relatively to the possible number of such links, if all nodes of the community were connected. If there are m_C links and n_C nodes in community C, then the intra-cluster density is: d_intra(C)=m_C/(n_C(n_C-1)/2).</p>

<p>The inter-cluster density is the proportion of existing links between a community and the rest of the graph, relatively to the possible number of such links if each community node was connected to the rest of the graph. If there are n nodes in the whole graph, and if m_C' is the number of links connecting a node of community C to a node located in another community, then the inter-cluster density of community C is: d_inter(C)=m_C'/(n_C(n-n_C)).</p>

<p>If you need to, the inter-cluster density can be easily modified to characterize the connection between two specific communities (by opposition to: a community and the rest of the graph). If we note m_C1C2 the number of links between communities C1 and C2, and n_C1 and n_C2 their respective numbers of nodes, then we get: d_inter(C1,C2)=m_C1C2/(n_C1*n_C2)</p>

<p>I don't think these measures are directly implemented in igraph (or any other software, for that matter).</p>
dependency between communities(clusters) in igraph
<p>Is there any measure that computes the dependency between communities of a graph in igraph? 
I am looking for a measure of dependency between communities.</p>