Measuring Polarization in Online Debates

Social networks can be a very successful tool to engage users to discuss relevant topics for society. However, there are also some dangers that are associated with them, such as the emergence of polarization in online discussions. Recently, there has been a growing interest to try to understand this phenomenon, as some consider that this can be harmful concerning the building of a healthy society in which citizens get used to polite discussions and even listening to opinions that may be different from theirs. In this work, we face the problem of defining a precise measure that can quantify in a meaningful way the level of polarization present in an online discussion. We focus on the Reddit social network, given that its primary focus is to foster discussions, in contrast to other social networks that have some other uses. Our measure is based on two different characteristics of an online discussion: the existence of a balanced bipartition of the users of the discussion, where one partition contains mainly users in agreement (regarding the topic of the discussion) and the other users in disagreement, and the degree of negativity of the sentiment of the interactions between these two groups of users. We discuss how different characteristics of the discussions affect the value of our polarization measure, and we finally perform an empirical evaluation over different sets of Reddit discussions about diverse classes of topics. Our results seem to indicate that our measure can capture differences in the polarization level of different discussions, which can be further understood when analyzing the values of the different factors used to define the measure.

Transversals and Bipancyclicity in Bipartite Graph Families

A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n \geqslant 4$, every balanced bipartite graph on $2n$ vertices in which each vertex in one color class has degree greater than $\frac{n}{2}$ and each vertex in the other color class has degree at least $\frac{n}{2}$ is bipancyclic. We prove a generalization of this theorem in the setting of graph transversals. Namely, we show that given a family $\mathcal{G}$ of $2n$ bipartite graphs on a common set $X$ of $2n$ vertices with a common balanced bipartition, if each graph of $\mathcal G$ has minimum degree greater than $\frac{n}{2}$ in one color class and minimum degree at least $\frac{n}{2}$ in the other color class, then there exists a cycle on $X$ of each even length $4 \leqslant \ell \leqslant 2n$ that uses at most one edge from each graph of $\mathcal G$. We also show that given a family $\mathcal G$ of $n$ bipartite graphs on a common set $X$ of $2n$ vertices meeting the same degree conditions, there exists a perfect matching on $X$ that uses exactly one edge from each graph of $\mathcal G$.