The paper is Social Resilience in Online Communities: The Autopsy of Friendster, http://arxiv.org/abs/1302.6109.
There's math in there which I don't understand, but this graph seems to sum it up:
The thesis is that if enough of your friends drop off of a service, you will too, because you won't have anyone to talk to. Then in turn your friends who are left will start to drop off because they miss you and don't have anyone to talk to, etc.
A 2009 review of the new Friendster interface by a Filipino tech enthusiast, "Friendster gets a facelift" seems to match this:
Like most Filipinos online, yes, I have a Friendster account and it was my first attempt at social networking. To be honest though, I have not logged on to my Friendster account in a year. My high school and college friends had all migrated to Facebook, and I found that I could not relate with the Friendster's younger user base.
The question I have is where will the Facebook users migrate to? Techcrunch runs a story that suggests it might be Tumblr next:
But Tumblr does not conform to this calculus, and the reason is that a large percentage of Tumblr users actually don’t WANT an audience. They do not want to be found, except by a few close friends who they explicitly share one of their tumblogs with. Therefore Tumblr’s notoriously weak search functionality is A-OK with most of its user base.
Time alone will tell.