Abstract. Motivated by "condensation" phenomena often observed in social networks such as Twitter where one "superstar" vertex gains a positive fraction of the edges, while the remaining empirical degree distribution still exhibits a power law tail, we formulate a mathematically tractable model for this phenomenon which provides a better fit to empirical data than the standard preferential attachment model across an array of networks observed in Twitter. Using embeddings in an equivalent continuous time version of the process, and adapting techniques from the stable age-distribution theory of branching processes, we prove limit results for the proportion of edges that condense around the superstar, the degree distribution of the remaining vertices, maximal non-superstar degree asymptotics, and height of these random trees in the large network limit. Our goal here is to provide a simple model that captures the most salient features of a natural graph that is determined by the Twitter traffic generated by public events. In the Twitter world (or Twitterverse), each user has a set of followers; these are people who have signed-up to receive the tweets of the user. Here our focus is on retweets; these are tweets by a user who forwards a tweet that was received from another user. A retweet is sometimes accompanied with comments by the retweeter. Let us first start with an empirical example which contains all the characterstics observed in a wide array of such retweet networks. Data was collected during the Black Entertainment Television (BET) Awards of 2010. We first considered all tweets in the Twitterverse that were posted between 10 AM and 4 PM (GMT) on the day of the ceremony, and we then restricted attention to all the tweets in the Twitterverse that contained the term "BET Awards." We view the posters of these tweets as the vertices of an undirected simple graph where there is an edge between vertices v and w if w retweets a tweet received from v, or vice-versa. We call this graph the retweet graph. In the retweet graph for the 2010 BET Awards one finds a single giant component (see Figure 1 .1). There are also many small components (with five or fewer vertices) and a large number of isolated vertices. The giant component is also approximately a tree in Date: May 5, 2014,
Bhamidi et al. REF propose and validate a so-called superstar random graph model for a giant component of a retweet graph.
Twitter event networks and the Superstar model
Annals of Applied Probability 2015, Vol. 25, No. 5, 2462-2502
Mathematics