# average number of friends on Facebook

Discussion in 'General Math' started by Martijn Harleman, Dec 6, 2011.

1. ### Martijn HarlemanGuest

On Facebook its easy to ee the number of friends your friends have.

On average, do people on Facebook have as much friends as their friends
have?

Bert

Martijn Harleman, Dec 6, 2011

2. ### Ray KoopmanGuest

No. On average, people have fewer friends than their friends have.

Ray Koopman, Dec 9, 2011

3. ### Ray KoopmanGuest

Let f_ij = 1 or 0 according as persons i & j are or are not friends.

f_ij = f_ji (symmetry), f_ii = 0 (convention).

g_i = sum_j f_ij = the number of friends person i has.

h_i = sum_j f_ij*g_j/g_i = the average number of friends
that person i's friends have

sum_i g_i < sum i h_i ?

sum_i sum_j f_ij < sum_i sum_j f_ij*g_j/g_i ?

Those sums are over all {i,j) pairs.
Because f is symmetric, we need to sum over only i > j.

sum_i sum_j<i 2 f_ij < sum_i sum_j<i f_ij*(g_i/g_j + g_j/g_i) ?

2 < (g_i/g_j + g_j/g_i) ?

2 g_i g_j < g_i^2 + g_j^2 ?

0 < (g_i - g_j)^2 ?

If g_i - g_j is nonzero for at least one (i,j) pair
then the assertion is supported.

Ray Koopman, Dec 9, 2011
4. ### Peter WebbGuest

It is a symmetric relationship, ie if x is a friend of y, then y is a friend
of x ?

If so, they are exactly the same.

Related to an older riddle. Considering only heterosexual relationships,
which is larger. The average number of different women an average man has
slept with, or the average number of different men an average woman has
slept with?

Also related to the very old water and wine puzzle.

Peter Webb, Dec 9, 2011