# Toward a Standard Approach for Echo Chamber Detection: Reddit Case Study
## 4 Reddit: Politics, Gun Control, and Minorities Discrimination
### EC Analysis: Stability and Persistence over Time  
Up to this point, we have explored users' leanings on the controversy (see Figure 2) and echo chambers (see Figure 3) in static temporal snapshots. Now, we are interested in analyzing their stability and consistency through time. In other words, we aim to answer the following questions:  
1. _How do users' ideology evolve over time? Are users stable and consistent with the same ideology, or do they tend to change opinion?_
2. _How do echo chambers evolve over time? Do members tend to fall again in a polarized community, or are they open to debate with opposite-leaning users?_  
To answer both questions, we model such issues in terms of transition probabilities. In other words, for each user, we compute their probability \(p_{ij}\) to move from _state_\(i\) to \(j\) over contiguous semesters. In the first question, _state_ stands for user ideology (i.e., Pro-Trump, Anti-Trump, and Neutral). In the second one, with _state_ we refer to the leaning of community the user belongs to (e.g., Pro-Trump EC, Anti-Trump EC, Pro-Trump community,...). **i. Ideology Stability over Time. ** In the heat maps in Figure 4, we show the stability of users' ideology over contiguous semesters. As regards Gun Control, transition probabilities could explain why we have not found ECs or communities composed by a majority of Anti-Trump users. Indeed, users have an equal probability of remaining in their position and changing their leaning toward a Pro-Trump one, thus proving to be not strongly polarized. On the contrary, most Pro-Trump users seem to be rooted in their position, with a probability to remain in their state greater than 0.64 in all semesters. For minorities discrimination, the scenario is quite similar. Indeed, even if Anti-Trump users are more polarized across semesters (i.e., \(p_{AA}\geq 0.5\)), some of them change their opinion in favor of the opposite side of the controversy. On the contrary, Political Sphere users aligned with [PERSON] have a strong tendency to move to a Neutral position. However, except for the last couple of semesters, it is pretty unlikely that they change their leaning to Anti-Trump. Instead, with the exception of the last couple of semesters, Anti-Trump users are the ones with both the highest probability to remain in their state and lower to change in favor of Pro-Trump ideas (i.e., respectively \(p_{AA}\geq 0.86\) and \(p_{AP}\leq 0.04\)). Lastly, for what concerns Neutral users, both Gun Control and minorities discrimination insights confirm our hypothesis that they are somewhat undecided. Indeed, in both cases, they have a higher probability of changing state instead of remaining Neutral. On the other hand, Political Sphere Neutral users are definitely more rooted in their position with a tendency toward Anti-Trump beliefs. **ii. Echo Chamber Stability over Time. ** To analyze ECs' consistency over time, we only take into account those two topics (i.e., minorities discrimination, Political Sphere) in which we have detected an EC across different semesters. Indeed, Gun  
Figure 4: For each topic, transition probabilities of users’ ideology over contiguous semesters. \(A\) stands for Anti-Trump users, \(P\) for Pro-Trump ones, and \(N\) for Neutral. Control users fall in a Pro-Trump EC only in the first semester, thus proving to be not stable over time. Additionally, in this analysis, we do not consider the Neutral group of users since they never form an EC or aggregate in a community with a Neutral majority. We decide to compute transition probabilities both for echo chambers (\(EC\)) and communities that do not satisfy ideological or topological constraints (\(C\)). In the heat maps, we show obtained results for minorities discrimination (Figure 4(a)) and Political Sphere (Figure 4(b)). Differently from ideologies, in both topics, echo chambers prove to be strongly consistent over semesters. Indeed, in most samples, ECs have a high probability to not change state, thus remaining in the same-leaning polarized systems.