  Condensation phenomenon is 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 that 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 nonsuperstar degree
asymptotics and height of these random trees in the large network limit.
