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In your opinion, how does facebook calculate mutual friends?
Has it cached all mutual friends for each user?
Or does it let MySQL calculate through a query each time it displays them?
Or does it make a query to the database and then calculate from a list?
Or is there another way?
I believe it is pre-computed and not done when you load the page.
How? There are a couple of options to look at this problem.
A common way is to look at the data as a graph, where each person is a vertex, and an edge defines a friendship. Iterating through each existing pair of persons, and finding 2 edges paths. The vertexes you went over, are the mutual friends. If you really want to get into it, I would suggest starting with learning about Graph Theory
When dealing with such an amount of data, using MySql and queries won’t work. On small amounts of data, it will.