# Measuring Polarization in Online Debates
## 2 User-Based Model for Reddit
A two-sided debate tree \((\mathsf{SDebT})\) for \(\mathcal{T}\) is a tuple \(\mathcal{T}_{S}=\langle C,r,E,W,S\rangle\) such that \(S\) is a labeling function of nodes \(S:C\rightarrow\{-1,1\}\) defined as follows:_  
* \(S(r)=1\)_;_
* _For all node_ \(c_{1}\
eq r\) _in_ \(C\)_,_ \(S(c_{1})=1\) _if for some node_ \(c_{2}\in C\)_,_ \((c_{1},c_{2})\in E\) _and either_ \(S(c_{2})=1\) _and_ \(W(c_{1},c_{2})>0\)_, or_ \(S(c_{2})=-1\) _and_ \(W(c_{1},c_{2})\leq 0\)_; otherwise,_ \(S(c_{1})=-1\)_._  
To evaluate the side of a comment \(c_{1}\) expressing a neutral position with respect the opinion expressed in \(c_{2}\), i.e., with \(W(c_{1},c_{2})=0\), we consider a similar approach to the proposal of [PERSON] and [PERSON] [12]. In their proposal, they consider neutral opinions to be (slightly) negative. The motivation for this is the realization of [PERSON] et al. [15] that, in newsgroups, people respond more frequently when they disagree. In Reddit, for example, when users agree with a comment, they can simply upvote without leaving any response. When users disagree, on the other hand, they can downvote, but it is more likely that they expose their reasons. Thus, from this perspective, a neutral response would be expressing more disagreement than agreement. Figure 2 shows the \(\mathsf{SDebT}\) structure we obtain for the Reddit debate of Figure 1. The nodes (comments) that support the root comment are colored in cyan, while the rest are colored in navy blue. At this point, we have classified the comments of a Reddit debate according to whether or not they offer support to the root comment and we have labeled the interactions between them according to the sentiment they express. So, to develop a quantitative model for measuring polarization in a Reddit debate, the next step is to introduce and investigate a suitable user-based model that allows us to represent the different interactions between the users to discover groups that either support or reject the root comment of the debate and how far these groups can be considered to be concerning each other. To this end, once the comments are classified into two sides, in favor or against the root comment, we group comments by the user and we consider that the relationship between users' opinions of two users is defined from the agreement and disagreement relationships between the individual comments of these two users. When we represent a debate grouping comments by users, interactions between different users can give rise to  
Figure 2: Two-sided debate tree for the Reddit debate of Figure 1.  
circular agreement and disagreement relationships, and the agreement or disagreement of a user concerning the opinion of another user in the debate should be defined by aggregating the set of single interactions that have occurred between them during the debate. Given a debate \(\Gamma\) on a root comment \(r\) with users' identifiers \(U=\{u_{1},\ldots,u_{m}\}\), we define the _opinion_ of the user \(u_{i}\in U\), denoted \(C_{i}\), as the set of comments of \(u_{i}\) in \(\Gamma\), except for the root comment \(r\). We consider debates in which users are not self-referenced. That is, for all user \(u\in U\) and each pair of comments \(c_{1}=(m_{1},u,sc_{1})\) and \(c_{2}=(m_{2},u,sc_{2})\), we assume \(c_{1}\) does not respond to \(c_{2}\), nor \(c_{2}\) to \(c_{1}\). Considering the particular case of root users (the users who post the root comment), we notice that when they do not participate in the debate, their opinion is empty, denoting that they have only posted the (root) news while staying passive throughout the debate. This is intentional since the root comment plays a special role in the debate, setting its topic. Thus, to be considered a \"true participant\" in the debate, the root user should contribute during the discussion.