A customer of mine recently had some questions about how the trending tags webpart works in SharePoint 2013. There's isn't a ton of info out there, so I figured I'd share my lessons learned. As always, a thanks goes out to Joe Rodgers - http://blogs.msdn.com/josrod/ .
How the webpart works
When you browse to your mysite, the trending tags webpart appears.
The webpart checks the Distributed Cache to see if data is there about trending tags, and renders it, if there's data. If there isn't, it queries search for the most recent tags AND puts that data into the DC.
The search query will only find tags that have been crawled/indexed, so make sure your crawls are running. Any type of crawl should be sufficient.
To see what search sees for trending tags, see the next section.
The Trending Tags data lives in the DC for 15 minutes. After 15 minutes, the data expires and is evicted from the cache. New data will not enter the cache until the PeopleManager.GetTrendingTags method is called. From what I can tell, only the trending tags web part calls this method. It's a client side call/CSOM, so you could call it manually if you have a need.