It's a pretty complicated group with a binary operation of something called "banter", I am not yet experienced with this however.
On a side note, I think I have refuted the friendship paradox, for I can't have fewer friends than my friends if I have no friends to begin with.
The friendship paradox was first observed by the sociologist Scott L. Field in 1991 and it states that most people have fewer friends than their friends have, on average. Mathematically, it is explained through graph theory, and is mainly related to the AM-GM and the Cauchy-Schwarz inequalities. Summarysing from Wikipedia:
Theorem: The average degree of a friend is strictly greater than the average degree of a random node.
Proof:If you have a social network (which trivially I don't) you can represent it using a graph
G=(V,E) where the set
V of vertices corresponds to people in the social network, and the set
E of edges corresponds to the friendship relation between pairs of people. It is assumed that friendship satisfies symmetry, that is, if
X is a friend of
Y, then
Y is a friend of
X; it is also modelled that the average number of friends of a person in the social network is the average of the degrees of the vertices in the graph. In graph theory, the degree of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice, that is,
v∈V∑deg(v)=2∣E∣. The average number of friends that a typical friend has (i.e not someone like me) can be modeled by choosing, uniformly at random, an edge of the graph (pair of friends) and an endpoint of the edge (one of the friends), and calculating the degree of the selected endpoint gives
2∣E∣∑v∈Vdeg(v)2=μ+μσ2,
where
σ2 is the variance of the degrees in the graph and
μ is the average number of friends of a random person in the graph. For a graph that has vertices of varying degrees (as is typical for a social network), both
μ and
σ2 are positive, which implies that the average degree of a friend is strictly greater than the average degree of a random node.
Oh damn, Mathemagicien going savage on Steve Jobs.