# From Discourse Relations to Network Edges: A Network Theory Approach to Discourse Analysis
## 5 Interrogating the C58 and the STAC Networks
It should be noted that there is a complete absence of clustering. All STAC (chat-only) networks have a clustering coefficient \(C=0\). Moreover, similarly to the C58 networks, the STAC (chat-only) networks are also disassortative (i.e., assortativity coefficients \(r\) are negative for all the networks.) Next, we scanned the STAC (chat-only) networks for the 5 basic network patterns depicted in Figure 5. As shown in Table 12 in the Appendix 12, the actual counts of all the \(S^{*}\) patterns are zero, except for the S4 pattern, which indeed has nonzero values. This is a rather intriguing fact and is in marked contrast to our observations for the C58 corpus networks. As mentioned above, the presence or absence of a specific pattern, although interesting in itself, does not suffice to term a pattern as motif or antimotif. Figure 12: STAC (chat-only) networks—histograms of basic undirected network properties: number of nodes _nssize_, number of edges _nedge_, fraction of edges, i.e., the ratio of _nedge_ to the maximum possible number of edges, mean degree \(k_{\textit{av}}\), degree standard deviation \(k_{\textit{std}}\), clustering coefficient \(C\) and assortativity coefficient \(r\). Thus, Figure 13 shows, for the STAC (chat-only) networks, the z-scores' distribution for the 5 basic network patterns depicted in Figure 5. The plot shows that, unlike in C58, the feed-forward loop \(S2\) is not a network motif in the case of the STAC (chat-only) networks. The z-scores of the \(S2\) distribution are negative, but since all of them are small in absolute values (less than 0.6), we cannot claim that they consist of an antimotif. It is, however, intriguing that S4, i.e., the dual-cause pattern shown in Figure 5, is still an antimotif, as suggested by the large negative z-scores of its distribution. Moreover, the z-scores' distribution suggests that the linear chain \(S1\) and the common-cause \(S5\) patterns are clearly antimotifs of the STAC (chat-only) networks. Figure 14 shows the distribution of the basic properties of 299 English discourse networks derived from the situated discourse corpus described above. While the text files of the corpus are the same as those presented in Figures 12 and 13, additional information of the nonverbal communication of the participants is included and, thus, the resulting networks are much larger than before. The largest of them contains 1628 nodes, while the smallest contains 29 nodes. Similarly, concerning the edges, the largest graph contains 818 edges, while the smallest only 15. These networks are much more sparse than before, i.e., the mean value of the fraction of edges is equal to 0.005. It should be noted that there is a complete absence of clustering. Again, all networks have a clustering coefficient \(C=0\). Additionally, similar to the previous cases, the networks are still disassortative (assortativity coefficients \(r\) remain negative for all the networks). Figure 13: STAC (chat-only) 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. Finally, focusing on the situated STAC networks, Figure 15 displays the z-scores' distribution for the 5 basic network patterns depicted in Figure 5. Despite the increased nonverbal information included in the situated dialogues, the results remain consistent with those of the nonsituated networks, namely that \(S1,S4\) and \(S5\) are antim motifs, since they appear with considerably less frequency than expected by chance, while \(S2\) is neither a motif nor an antimotif. All numerical simulations and data analyses were performed on a workstation equipped with 2 Intel Xeon Gold 6140 Processors (72 cpu cores in total) provided by the MSc program \"Computational Physics\" of the Physics Department, Aristotle University of Thessaloniki. The estimation of network motifs was performed through custom code written in python.