Minas Gjoka, Michael Sirivianos, Athina Markopoulou, and Xiaowei Yang, a team of authors at UCI, wrote a paper, Poking Facebook: Characterization of OSN Applications, which looks at data from Facebook application installation and use.
First, they seem to have gotten a pretty successful crawl, which is saying something since Facebook is pretty selfish with data. Here is a PDF of application installation, both according to facebook stats and their crawled dataset, which match up pretty well:
They also modeled the histogram of installed-apps-per-user as preferential, running a simulation with "users as bins" and different apps as "different colored balls", iteratively assigning balls to bins. For instance, saying that 100 users have installed the Zombies application, would translate to "gray balls appear in 100 bins".
For each iteration, one goes through each "ball" (application installation), starting with the "most popular color" (application with the most installations). For each ball one then assigns an additional "bin that doesn't already contain that color" (picks a new user that hasn't already installed the app) according to a probability:
Where balls(i) is the number of applications a user i has installed, and B is the set of users that hasn't already installed the application. init is a parameter to moderate the initial activity, and rho is the preferential exponent, chosen in simulations to be 1.6. In the end you get a sort of heavy-tailed behavior, with most users installing a couple apps and a few who go nuts with application installs. It fits pretty well:
One of the fun parts of these sort of data is the outliers-- the users who go nuts on something. (Netflix users rating thousands of movies, etc.) It looks like in the crawled data there are a few users with 500 apps installed!
In the paper there is also fit to the "coverage of applications"-- that is, how many of the ranked apps we need to go through before we have all the users with one of those apps installed, and it appears the simulation reaches coverage a little too quickly, so perhaps the most popular applications are taking too many users in the simulation.
What's somewhat surprising to me is that this isn't at all based on the behavior of a user's friends, but of the entire Facebook network at large. I suspect that in reality that does govern user behavior, but for large-scale patterns one can overlook it. This might be different for actually modeling how an application catches on. (Using other features like network effects are listed as "future work" for the authors in refining the model.)
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