## 3 Methodology
Note that for our implementation, we have used the same weight for each comment irrespective of the nature of the post or comment. We took this route in order to avoid analyzing the content of the posts and their comments. However, one may assign different weights to each following relationship arising out of different posts/comments based on the importance and nature of the posts/comments. Figure 1 shows the SIG and the adjacency matrix of two hypothetical subreddits based on user interactions in them. Note that for the first subreddit (labeled \(\mathbf{s}\)), there is a circular relationship between users \(u_{i}\) and \(u_{j}\) in that both \"\(u_{i}\) follows \(u_{i}\)\" and \"\(u_{j}\) follows \(u_{i}\)\" holds. Similarly for the second subreddit labeled \(\mathbf{t}\), the relationship \"\(u_{j}\) follows \(u_{k}\)\" appears twice and hence we have a weight of 2 on the directed edge between \(u_{j}\) and \(u_{k}\), as shown in the adjacency matrix. Figure 2 shows the SIG for a small subreddit called \"Adventure Builders\". Similarly, Figure 3 shows the SIG for a bigger subreddit called \"Castaneda\". It is immediately clear that these graphs do not convey much information due to the large number of edges and nodes in the graph. Looking deeper into this, we found that this is a problem with almost all the subreddits. There is too much redundant information in each subreddit and hence the SIGs are messy and hard to interpret. In order to get around this problem, we introduced three different types of pruning: weight pruning controlled by parameter \(a\), interaction pruning controlled by parameter \(b\), and neighbor pruning controlled by parameter \(c\). Given a SIG \(G_{S}\), we denote the pruned graph by \(G_{S}(a,b,c)\). Pruning reduces redundant information and removes outliers from the SIGs and brings out the consistent communication patterns between the users of a subreddit. A pruned version of the Castaneda subreddit is also shown in Figure 3. This is considerably cleaner and more usable than the original SIG. Next, we introduce and formalize the ideas of implicit leaders, followers, interconnected subreddits, and pruning. Figure 1: Graph representation and adjacency matrix for two subreddits. Figure 2: SIG for Adventure Builders subreddit without pruning. Numbers on nodes refer to unique users and numbers on edges are edge weights. Usernames have been replaced by numbers to anonymize the users. **Implicit Leader**: Given a subreddit \(S\in\mathcal{S}\), \(\mathcal{S}\) is the set of all subreddits. We define an \"(implicit) leader\" in the subreddit as a user who made one or more posts and comments in the subreddit and received a total of at least \(\theta\) responses in return for all their posts and comments combined. Note that \(\theta\) is a parameter that controls who will be called a leader in a subreddit. **Implicit Follow**: An \"(implicit) follower\" is any user who makes a comment on a post/comment in the subreddit. Thus, if user \(u_{j}\) starts a discussion and user \(u_{l}\) comments on the same, we say that \(u_{l}\) follows \(u_{j}\). Note that in this work, we only consider following relations where the user being followed is a leader as defined above. Hence, if user \(u_{l}\) comments/replies to the post/comment made by user \(u_{j}\) and \(u_{j}\) is a leader, then we say that \(u_{l}\) follows \(u_{j}\) for our analysis. Note that a user who is a leader for one subreddit can also be the follower in the same subreddit and hence there can be circular relationships, as noted before. Furthermore, a user who is a leader (follower) in one subreddit, can be a follower (leader) in another subreddit. The fact that a user is allowed to interact with multiple subgroups (subreddits) leads us to the idea of a interconnected subreddit. **Interconnected Subreddits**: A subreddit is interconnected if the users in the subreddit interact with users from other subreddits. The more this interaction occurs, the more \"interconnectedness\" in the subreddit.