# From Discourse Relations to Network Edges: A Network Theory Approach to Discourse Analysis
## 4 Networks and Connectivity
### Network Motifs and Discourse Structure
Before responding to the question of whether discourse graphs are characterized by one or more network motifs or antimotifs in Section 5, it is important to realize the types of discourse subgraphs that are mapped to the different \(S*\) network subgraphs. Notice that Figures 6-9 include only the corresponding discourse graphs that respect the Right Frontier Constraint (RFC) and not all possible three-node subgraphs that may satisfy the definition of the network subgraph patterns. However, for our purposes, the distinction between coordinating and subordinating discourse relations does not affect the results of our study (however, see [72] for certain types of RFC violations related to this distinction). \(S1\) represents the uninterrupted sequence of related utterances, and this can be translated in three different discourse graphs. Figure 6 displays the possible three-node subgraphs included in a discourse graph, with either subordinating, coordinating or both types of discourse relations corresponding to the linear chain pattern \(S1\). All three subgraphs are expected to be found frequently in single-author written texts, due to coherence realization in these texts that we expect to promote uninterrupted sequences of related utterance, while the opposite may be expected to occur for the multiparty dialogue texts of the STAC corpus, namely that the participants frequently interrupt the discourse and introduce new topics and/or lines of argumentation. All of these conjectures remain to be confirmed or rejected in Section 5. Figure 6: \(S1\): The linear chain pattern and its correspondence to discourse graphs. \(S2\) corresponds to fully connected discourse subgraphs of three nodes, whereby all three nodes are related to the other two nodes. In terms of subgraph connectivity, \(S2\) refers to maximally connected discourse graphs, which in turn would mean that in these three-node subgraphs, the maximum quality of discourse coherence is served if one admits that network connectivity is considered a direct and strong indication of discourse coherence. Note that all discourse semantic theories either tacitly agree with the statement that the more edges or connections a discourse representation graph has, the more probable it is that the discourse is coherent, or they integrate it as a principle in their theory, e.g., the Maximize Discourse Coherence principle (MDC) in SDRT ([73; 74; 75]). The dual-cause pattern in \(S4\) maps to discourse subgraphs in which \(\pi_{3}\) plays a central role and could be perceived as labeling an utterance that strengthens the quality of discourse coherence. Note that the central role of \(\pi_{3}\) we mention here is not identified with RST's nucleus-satellite distinction by [33] but is attributed to the fact that \(\pi_{3}\) is related to both of the following two nodes in the discourse subgraph. Although \(\pi_{3}\) of the discourse subgraphs that corresponds to \(S3\) is related to the previous two \(\pi\)s, as in \(S2\), there is no indirect relation to \(\pi_{1}\) through \(\pi_{2}\) in which the distance \(d(\pi_{1},\pi_{3})\) equals 2. The distance between two nodes A and B is the length of the shortest path between A and B. The absence of an edge between \(\pi_{1}\) and \(\pi_{2}\) does not mean that they are not related indi  
Figure 8: \(S4\): The dual-cause pattern and its correspondence to discourse graphs. Figure 7: \(S2\): The feed-forward loop pattern and its correspondence to discourse graphs. rectly through an intermediate, though, \(\pi\) but that the number of intermediate nodes, as well as the type of discourse relations that intervenes between them is not specified. Regarding the permissible discourse graph that corresponds to the common cause pattern \(S5\), \(\pi_{1}\) with two outgoing edges plays a prominent role too, since it is directly related to both \(\pi_{2}\) and \(\pi_{3}\), while \(\pi_{2}\) and \(\pi_{3}\) are only indirectly related to each other in the sense described for \(S4\); namely there is no intermediate node between \(\pi_{2}\) and \(\pi_{3}\).