# Measuring Polarization in Online Debates
## 3 Debate Polarization  
Given a User Debate Graph \(\mathcal{G}=\langle\mathcal{C},\mathcal{E},\mathcal{S},\mathcal{W}\rangle_{n}\), we propose a model to measure the level of polarization in the debate between its users. We identify two characteristics that a polarization measure should capture. First, a polarized debate should contain a bipartition of \(\mathcal{C}\) into two sets \((L,R)\) such that the set \(L\) contains mainly users in disagreement, the set \(R\) contains mainly users in agreement, and both sets should be similar in size. The second ingredient is the sentiment between users of \(L\) and \(R\). A polarized discussion should contain most of the negative interactions between users of \(L\) and users of \(R\), whereas the positive interactions, if any, should be mainly within the users of \(L\) and within the users of \(R\). We believe that both factors are relevant, as the existence of only the first characteristic can indicate simply a set of two _echo chambers_, where within each group, everybody agrees with the rest of the group but nobody pays attention to the other group. So, negative interactions between users of the two sides are the second ingredient for the right measurement of the polarization, such that the more negative the interactions between users of \(L\) and \(R\), the higher the polarization we interpret between both sides. In our measure, the echo chambers case is considered as a case with middle polarization. To capture these two characteristics with a single value, we define two different measures and their combination in a final one referred to as _the bipartite polarization level_. First, given a bipartition \((L,R)\) of \(\mathcal{C}\), we define its level of consistency and how balanced the sizes of \(L\) and \(R\) are as follows:  
\[SC(L,R,\mathcal{G})=LC(L,\mathcal{G})\cdot RC(R,\mathcal{G}),\]  
Figure 3: User Debate Graph for the two-sided debate tree of Figure 2.  
where \(LC(L,\mathcal{G})\) evaluates the disagreement strength of the comments in \(L\) (regarding the root comment) and \(RC(R,\mathcal{G})\) the agreement strength of the comments in \(R\). They are defined as follows:  
\[LC(L,\mathcal{G})=\frac{\sum\limits_{ \begin{subarray}{c}C_{i}\in L,\\ \mathcal{S}(C_{i})\leq 0\end{subarray} }-\mathcal{S}(C_{i})}{|\mathcal{C}|}\text{ and }\]  
\[RC(R,\mathcal{G})=\frac{\sum\limits_{ \begin{subarray}{c}C_{i}\in R,\\ \mathcal{S}(C_{i})>0\end{subarray} }\mathcal{S}(C_{i})}{|\mathcal{C}|}.\]  
The value of \(SC(L,R,\mathcal{G})\) lies in the interval \([0,0.25]\). The minimum value (0) is achieved when either there are no users in \(L\) with \(\mathcal{S}(C_{i})<0\) or no users in \(R\) with \(\mathcal{S}(C_{i})>0\). The maximum value (0.25) is achieved when half of the users are users in disagreement with \(\mathcal{S}(C_{i})=-1\) and are found in \(L\), and the other half are users in agreement with \(\mathcal{S}(C_{i})=1\) and are found in \(R\). Secondly, the sentiment of the interactions between users of different sides is defined as follows:  
\[SWeight(L,R,\mathcal{G})=\frac{\sum\limits_{ \begin{subarray}{c}e_{i}\in \mathcal{E}\cap\\ ((L\times R)\cup(R\times L))\end{subarray} }-c(p(e_{i}))\cdot w(e_{i})}{| \mathcal{E}|}+2,\]  
with \(c(p(e_{i}))=2(p(e_{i})-0.5)^{2}+1/2\), and where \(p(e_{i})\) and \(w(e_{i})\) denote the values of \(p\) and \(w\), respectively, in \(\mathcal{W}(e_{i})=(p,w)\). It is worth noticing that the value of \(SWeight(L,R,\mathcal{G})\) lies in the interval \([0,4]\):  
* The minimum value (0) is achieved when all the interactions \(e_{i}\in\mathcal{E}\) have \(\mathcal{W}(e_{i})=(1,2)\) (pure positive interactions with maximum sentiment value), and all of them are found between \(L\) and \(R\). * The middle value (2) is achieved when the sum of negative values in the summatory of the numerator equals the sum of the positive ones. So, this represents a situation where positive and negative interactions between both partitions compensate for each other.