## 3 Topic-Oriented Multi-Document Summarization (TOMDS) Model
### Abstractive Stage
#### 3.3.2 Primary-Secondary Relationship Matrix  
Discourse structure analysis helps to understand the structure and semantics of a discourse. It not only assists in generating concise and informative summary information but also effectively identifies the topic of the discourse. Given a sentence, a paragraph, or a document, we can generate a corresponding discourse parsing tree by discourse structure analysis and determine the nucleus information and rhetorical relationships of its spans to guide generating summaries. We use an existing rhetorical structure parser [39] to generate an RST discourse parse tree for each input paragraph. First, the input paragraph \(R_{i}^{s}\) is split into \(n_{i}\) sentences, such that \(R_{i}^{s}=\{S_{i1},S_{i2},\ldots,S_{in_{i}}\}(i=1,2,\ldots,L^{\prime})\), and then split the sentences \(S\) in the paragraph into \(l\)-th EDUs, expressed as:  
\[S_{ih}=\{E_{ih1},E_{ih2},\ldots,E_{ihl_{ih}}\}(h=1,2,\ldots,n_{i}).\]  
We adopt the scoring method of [PERSON] et al. [40], defined \(s_{label}(k,q)\) as a scoring function used to set nucleus and rhetorical relationship labels for two adjacent spans at the paragraph or sentence sub-level. Here, two adjacent EDUs (\(E_{ihk}\) and \(E_{ihk+1}\)) under the \(j\)-th sentence of the \(i\)-th paragraph are taken as an example. \[ \begin{split} s_{label}(k,q)=W_{q}MLP\Big{(}\Big{[}E_{ihk};E_{ihk +1};\sum_{k=1}^{k}E_{ihk};\sum_{k=k+1}^{k-l}E_{ihk}\Big{]}\Big{)},\\ (0<k<l)\end{split}  \tag{11}\]  
where \(W_{q}\) is the nucleus or rhetorical relation label projection layer, and MLP is a multi-layer perceptron that uses a single feed-forward network and a ReLU function as the activation function. Moreover, \(\sum_{k=1}^{k}E_{ihk}\) and \(\sum_{k=k+1}^{k-l}E_{ihk}\) are the current \(E_{ihk}\) left and right spans. We choose the label for the maximum value of the above equation. The label is defined as follows:  
\[\hat{p}=\underset{q\in Q}{argmax}[s_{label}(k,q)] \tag{12}\]  
where \(Q\) represents a set of valid nucleus label combinations [N-S, S-N, N-N] and rhetorical structure labels {Circumstances, Solutionhood, Elaboration,...} for predicting nucleus and rhetorical relationships. Similarly, if we want to determine the nucleus and rhetorical relationship labels of two adjacent sentences (\(S_{ih}\) and \(S_{ih+1}\)) under the \(i\)-th paragraph, then  
\[s_{label}(h,q)=W_{q}MLP([S_{ih};S_{ih+1};\sum_{h=1}^{j}S_{ih};\sum_{h=h+1}^{h =n}S_{ih}]) \tag{13}\]  
Figure 3 shows the discourse segmentation and RST discourse parse tree construction and conversion. (a) is a paragraph we randomly selected from the WikiSum dataset. (b) is the RST discourse tree constructed based on (a). In order to better represent the primary and secondary relationships between EDUs, we use a simple and effective model to convert the RST discourse tree into a discourse tree based on the primary and secondary relationships, as shown in Figure 3c. The serial number in the circle corresponds to the EDU serial number. If there is a directed edge between two serial numbers, we consider that there is a primary and secondary relationship between the two EDUs and the words they contain. We record the primary and secondary relationships in the matrix as T\({}_{i}^{s}\). Let P represent a given input paragraph R\({}_{i}^{s}\)'s primary and secondary relationship matrix, where P\({}_{xy}\) represents the relationship weight between \(x\)-th and \(y\)-th words. Here, we simplify the definition of weight, as shown in Equation (14). \[\text{P}_{xy}\ =\ \left\{ \begin{array}{l}1\ \text{primary}-\text{secondary} \ \text{relationship}\ \text{exists}\\ 0\ \text{no primary}-\text{secondary}\ \text{relationship}\end{array} \right. \tag{14}\]  
In constructing the primary and secondary relationship matrix, our method ignores the concept of individual words. As long as there is a directed edge between two EDUs, the words they contain are assigned a value of 1. Otherwise, it is set to 0.