In a research paper published on the 17th of January, 2014, several Princeton researchers found the end of Facebook's popularity to be taking place within the next few years. Using a model in which they track the popularity of MySpace based on the term - the name of the brand - appearing in Google a number of times over the past decade. This Princeton paper saw it fit to extrapolate a direct correlation between Facebook (the word)'s popularity on Google and the possibility that the network itself may be taking a dive.
Facebook's own Data Scientist Mike Develin decided that he'd take essentially he same approach to seeing how long Princeton would last as they did seeing how long Facebook would last. The results were, as Develin suggests, "alarming", to say the least.
Instead of seeing how many times "Facebook" appeared in search results over the past decade, Develin looked to Google Scholar to see how many scholarly articles had been published by Princeton since 1940. You'll see a rather large dip in the past few years.
Develin used Facebook-based pages for several Ivy-league schools to also prove that Princeton wasn't doing especially well. This chart shows the general number of "likes" each of the following schools have had over the past several years: Princeton, Harvard, and Yale.
Jumping in to Princeton's own enrollment records, Develin also found that as the school grows, so too does its Google Trends search indexing fall. Develin suggests - making a rather strong point as he does so - that correlation does not alway equal causation.
Develin also points toward what he calls actual engagement trends for Facebook, showing that, at least from their perspective, they're not going away any time soon. Even IF people don't search for the word "Facebook" in Google as much as they used to.