## 3 Methodology  
In order to formalize the structure of Reddit, we modeled it as a graph that is a union of sub-graphs representing subreddits. Mathematically, a graph \(G\) is defined as a triplet \(G=(U,E,W)\), where \(U=\{u_{i};i=1,\ldots,n\}\) is the set of vertices or nodes, \(E=\{e_{ij}:e_{ij}=(u_{i},u_{j}),i\
eq j=1,\ldots,n\}\) is the set of edges, and \(W=\{w(e_{ij});i\
eq j=1,\ldots,n\}\) is the set of weights on the edges. For Reddit, we represent the underlying graph as \(G=\cup G_{S};G_{S}=(U_{S},E_{S},W_{S})\), where \(G_{S}\) represents the graph corresponding to a subreddit \(S\) and (1) \(U_{S}\) is the set of nodes where each node is an individual user in the subreddit, and (2) \(E_{S}\) is the set of edges representing implicit relationships between the users of the subreddit. Thus, \(e_{ij}\in E_{S}\) denotes some type of interaction between the users \(u_{i}\) and \(u_{j}\) in the subreddit \(S\). Depending on the type of interaction, the edges can be either directed or undirected. As pointed out earlier, in this work we considered implicit leader-follower relationships by analyzing interaction patterns between the users in the subreddit. Since a leader-follower relation is clearly not symmetric, the edges denoting this type of relationship are directed edges. Thus, the underlying graph for Reddit is a directed graph using our formalization. The tuple \((u_{i},u_{j})\) denotes the directed edge from the node for user \(u_{i}\) to the node for user \(u_{j}\). (3) \(W_{S}\) is the set of weights. In our case, the weight on an edge \((u_{i},u_{j})\), denoted by \(w(u_{i},u_{j})\), denotes the number of times user \(u_{i}\) followed user \(u_{j}\) implicitly in the subreddit by interacting with user \(u_{j}\). Due to the nature of this graph, if \(S\) is a subreddit, then we call the corresponding graph \(G_{S}\) obtained from the interaction patterns the _subreddit interaction graph (SIG)_ for \(S\). As noted above, the edges between nodes representing users of a subreddit in the SIG represent a follower-following relationship which is inferred from user interactions on the subreddit. The relations that we obtained are specific to subreddits, as we inferred them from communications within a subreddit. In this work, an interaction is defined as the act of commenting on a post. Thus, for a given subreddit, if user \(u_{i}\) makes a post and user \(u_{j}\) comments on it, we create a directed edge between \(u_{i}\) and \(u_{j}\) with direction from \(u_{j}\) to \(u_{i}\). We say that \(u_{j}\) implicitly follows \(u_{i}\) and that \(u_{i}\) is a leader and \(u_{j}\) is a follower. Note that our definition of implicit following opens up the possibility of \"circular\" relationships. Thus, for the above example of \(u_{j}\) commenting on a post by \(u_{i}\), \(u_{i}\) might also make a response to a post/comment by \(u_{j}\), and in that case, we would also have \(u_{i}\) following \(u_{j}\), making the relationship circular. One possibility of representing such a circular follower-following relationship in a SIG is through an undirected edge. However, even though the implicit relationship is circular, it is also asymmetric in the sense that \(u_{j}\) commented on a post by \(u_{i}\), but \(u_{i}\) commented on a _different_ post by \(u_{j}\). Since the posts that characterize the follower relation are different, there is an inherent asymmetry in the relationship. In order to capture this asymmetry, we treat both the relationships as separate and represent them as separate edges in the underlying SIG for the subreddit. We also have the possibility of user \(u_{i}\) (or \(u_{j}\)) commenting on multiple different posts from \(u_{j}\) (or \(u_{i}\)). This creates multiple following relationships between the two users, each being different in lieu of the posts being different. We model this scenario by adding weights (a weight of one in our implementation for each implicit following relationship), thus leading to a weighted directed graph for a subreddit.