# From Discourse Relations to Network Edges: A Network Theory Approach to Discourse Analysis
## 5 Interrogating the C58 and the STAC Networks  
Figure 10 shows, in the form of histograms, the distribution of the basic properties of 55 discourse networks derived from Greek newspaper articles that were included in C58. The largest of these networks contains 69 nodes, while the smallest contains only 4 nodes (we left out 3 of the original 58 texts included in C58, since they consisted of less than 3 utterances). Similarly, concerning the edges, the largest graph contains 87 edges, while the smallest only 3. Most of the networks are sparse, as the mean value of the fraction of edges is equal to 0.12. There are, however, some exceptions, as the maximum value of the fraction of edges among the 55 networks equals 0.67. Most of the networks have a low clustering coefficient (mean value of C is 0.22). Lastly, the vast majority of the networks are disassortative, since their assortativity coefficients \(r\) are negative. Figure 11 shows the distribution of the z-scores for the 5 basic network patterns depicted in Figure 5. To be specific, for each of the 55 C58 discourse networks, we measure the number of \(S_{i},i=1\ldots 5\) patterns. Then, we create 100 random networks with the same number of nodes and edges as the actual network and measure the occurrence of the same \(S_{i}\) patterns in each of the random networks. For each pattern, and for each actual network, we calculate a z-score as follows. Let \(n_{i}\) denote the number of \(S_{i}\) pattern occurrences in the actual network. Let \(\mu\) and \(\sigma\) denote the mean number and standard deviation of occurrences of pattern \(S_{i}\) in the randomized networks. Then, the z-score is calculated as \(z=\frac{n_{i}-\mu}{\sigma}\). Figure 11 shows that the feed-forward loop \(S2\) is clearly a network motif for a lot of the 55 networks. There are several cases with very high z-scores. (More than 10 in some cases!) The fact that the feed-forward loop, which is the most famous motif in biological networks, is found to appear in linguistic networks as well is certainly intriguing. It is also intriguing that S4, i.e., the dual-cause pattern shown in Figure 5, is clearly an antimotif, as suggested by the large negative z-scores of its distribution. Furthermore, interestingly, the linear chain pattern \(S1\) that represents a sequence of uninterrupted related utterances is neither a motif or an antimotif for the discourse-annotated single-author written texts in C58, since it does not appear more frequently or rarely than it would at random for the vast majority of the discourse graphs. In particular, we compared the observed frequency of the \(S1\) pattern with what would be expected by chance, given the size and composition of the texts in the dataset. Our findings suggest that the usage of this particular linear chain pattern is not a common strategy of the authors of the texts in the C58 dataset. Lastly, the common-cause pattern \(S5\) does not appear to be a network motif or antimotif. Figure 11: C58 networks—histograms of network motif z-scores. Three-node subgraph patterns are denoted as follows: S1—linear chain, S2—the feed-forward loop, S3—a cycle, S4—dual-cause and S5—common-cause. Figure 10: C58 networks—histograms of basic undirected network properties: number of nodes _nsize_, number of edges _nedge_, fraction of edges, i.e., the ratio of _nedge_ to the maximum possible number of edges, mean degree \(k_{av}\), degree standard deviation \(k_{std}\), clustering coefficient \(C\) and assortativity coefficient \(r\). Figure 12 shows, in the form of histograms, the distribution of the basic properties of 299 STAC (chat-only) networks derived from dialogues in the English language. These networks are quite heterogeneous, as the largest of them contains 322 nodes, while the smallest contains only 2 nodes. Similarly, concerning the edges, the largest graph contains 166 edges, while the smallest only 1. The STAC (chat-only) networks are much sparser than the C58 networks, as the mean value of the fraction of edges is equal to 0.02.