# From Discourse Relations to Network Edges: A Network Theory Approach to Discourse Analysis
## 4 Networks and Connectivity
### Key Network Indices and Their Relevance to Discourse
This distribution can be derived from the degree distribution \(p_{k}\) as \(q_{k}=\frac{(k+1)p_{k+1}}{\sum_{j\geq 1}jp_{j}}\) ([63]). The quantity \(e_{jk}\) represents the joint probability distribution of the remaining degrees of the two vertices. This quantity is symmetric on a undirected graph and follows the sum rule \(\sum_{jk}e_{jk}=1\) and \(\sum_{j}e_{jk}=q_{k}\). In Table 1 of the Appendix A.1, we present a table with basic statistics for a number of real-world networks. It provides, for several real-world networks, numerical estimates for the important properties of clustering and assortativity. As one can see from Table 1 the clustering coefficient for social networks tends to be of higher value. For example, the network of film actor collaborations has a \(C=0.78\), while a network of collaborations between physicists has been found to have a \(C=0.56\). These are typical values for social networks. Technological and biological networks by contrast tend to have somewhat lower values. The power grid network, for instance, has a clustering coefficient of only about \(0.08\). A high clustering coefficient for the study of discourse networks may serve as an important proxy for pinning down discourse coherence and the way that it is maximized, since it appeals to discourse structures with a higher quality of discourse connectivity. However, to establish such a conclusion would require cross-checking whether discourse networks with a high clustering coefficient are also ranked as highly coherent by speakers. Concerning the assortativity property, one can observe that social networks are in general assortative networks (\(r>0\)). This is quite well known to sociologists, as people have, it appears, a strong tendency to associate with others whom they perceive as being similar to themselves in some way. In sociology, this tendency is called homophily, or assortative mixing. More rarely, one also encounters disassortative mixing, the tendency for people to associate with others who are unlike them. Assortative (or disassortative) mixing is also seen in some nonsocial networks. Papers in a citation network, for instance, tend to cite other papers in the same field more than they do papers in different fields. Web pages written in a particular language tend to link to others in the same language. The assortativity coefficient \(r\) that we study here is a measure of a particular type of homophily, i.e., it measures the _degree_ of assortativity. One may consider the nodes of a graph as belonging to two groups. A group of highly connected nodes and a group of poorly connected ones. The assortativity coefficient \(r\) measures the tendency of nodes to connect (or not) within their respective group. In terms of discourse representation, a highly assortative discourse network may signal the connectedness between highly important nodes. In the case of a highly assortative discourse, one may also claim that a few or even a single common topic or line of argumentation is promoted that is supported by several influential utterances in the discourse. On the other hand, a disassortative discourse network may signal a lack of connectedness between highly important nodes that can be further interpreted as a type of mutual avoidance between nodes with a high degree centrality.