If I wanted to write an algorithm to calculate the similarity between two users (who are not friends, or have any common friends) on Facebook, how would I go about doing it?

A few years ago I began working on the development of a "social network" called HowYouKnowMe where the objective is to have user generated autobiographies created and (historically) stored online for everyone.  The premise of this project, as indicated by its tagline: 'The story of your life.', was to have the people who know you, with whom you have spent real time, share that story, in whole or in part, to then be categorized and uploaded to your life story archive.  Then, your story could be told, or "published", by way of a specific event (i.e. a concert, presidential inauguration, Super Bowl), relationship (best friend, co-worker, spouse), affiliation (basketball team, fan of a music group), or just straight up chronological timeline.     The project progressed and my partner and I looked at different functions that we could perform as a deviation of the data being collected.  One such feature was what I can only describe as a "floating bubble relationship matrix" to demonstrate the depth and pictorially represent the nature of any particular relationship between two (2) people.  Giving weight to these input variables were factors such as how many years (since) people have met, the frequency in which they (shared time together) interacted, the type (email, phone call, shared physical location) of time shared, the basis of the relationship as blood relative, business associate, college roommate, casual acquaintance, someone with whom you shared a moment on an airplane or on line at McDonald's, any or all of the above.  These bubbles would constantly update or "float" as a real time HowYouKnowMe galaxy.  The illustration of how close or far apart any two people could be would be represented both in the distance between their respective bubbles and the size of their own bubble which may represent their commonly shared "value" with one another.  (For example, by sheer time on earth, known as your age, the bubble for a 2 year old would be smaller than the bubble of a 20 year old.  Therefore, the ability for a 20 year old to really "know" a 2 year old is inherently limited by the amount of possible time, 2 years, that they could have spent together.)   Anyway, I hope my answer helps to give you an additional perspective when and where measuring human relationships via a mathematical calculation.  I can probably share a demo of this postulate based on the HowYouKnowMe beta and release the algorithm we used.  But my point is that I do not feel that the integrity of the data that would be found on fb could provide the input you may need to discover and ultimately represent any true similarity between two users.  Unless you are basing the exercise purely off of their online likenesses derived from their digital personalities.