Putting "six degrees of separation" to the test with graph theory on a very large scale, Facebook analyses the connections between 10% of the worlds population.
Original paper: http://arxiv.org/abs/1111.4503
As one would expect the network is nearly fully connected, with 99.91% of the nodes belonging to a single connected component. The largest subgraph not connected to the major graph consists of about 2000 users.
The degrees of separation for any two facebook users is closer to 4 than 6. In 92% of all pairs of users on facebook; a friend of your friend knows a friend of my friend.
The graph is inherently sparse, duein part to our social ineptness at connecting with any significant fraction of the global population and also by facebook limiting the maximum number of friends to 5000. The median number of friends (degree) globally was 99.
The graph is also highly clustered - "for a median user, 14% of all their friend pairs are themselves friends", and interestingly this "drops rapidly for users with close to 5000 friends, indicating that these users are likely using Facebook for less coherently social purposes and friending users more indiscriminately."
The clique problem is to find the largest subset of people who all know each other, the study calculates this sparsity of the neighborhood graphs by measuring the degeneracy for each degree. Two examples highlight the findings of just how dense the local neighborhood graphs are:
* A node with degree 100 (a user with 100 friends) has an average degeneracy of 15, meaning on average 16 of the 100 friends each know at least 15 other mutual friends.
* Users with 500 friends have an average degeneracy of 53, they have at least 54 friends who all know 53 of their other friends.
The fact that neighborhood graphs center around sizable dense cores is clearly how facebook can offer such good friend suggestions and smart lists - I'm sure could be used to improve sharing between users major cliques (or ahem *circles*).
Pictured is the cumulative degree distribution showing the number of nodes (users) who have degree (number of friends) greater than k. The number ofunique friends added per additional friend was to approximate a linear line of best fit: 355*k-15057. So in theory I have ~205398 unique friends-of-friends.
Some other very interesting revelations: "until you have nearly 700 friends, your (average) neighbor has more friends than you", "a positive correlation does exist between degree and logins... So since your friends have more friends than you do, they also login to Facebook more than you do."
While there are roughly 30 million fewer active female users on Facebook, a random neighbor of a user is slightly more likely to be female. Why? Because the average female degree (198) is larger than the average male degree (172).
All the path length crunching was done on a 24-core machine with 72 GiB of memory and 1 TiB of disk space, the edge graph comprised ~69 billion friend connections between 721 million users and took up 345 GB at 20 bits per arc and was traversed by a new algorithm called HyperANF.