A "circle of friends" is a set of (site) members where each member has exactly two "friendship" relations in the whole set

Hmm, I think so - each member can have many "friendships", but the answer will only have two "friendships" per person in the solution (i.e. superfluous friendships will be ignored).

I think this definition fits your description, but I'm not sure. My mental image is of a circle of persons, each holding hands - is that the right image?

Yes, but with some constraints - the first friend in the pathway must not be known to the last friend in the pathway (i.e. I know both of them, but they don't know each other).

map{$a=1-$_/10;map{$d=$a;$e=$b=$_/20-2;map{($d,$e)=(2*$d*$e+$a,$e**2
-$d**2+$b);$c=$d**2+$e**2>4?$d=8:_}1..50;print$c}0..59;print$/}0..20
[download]

For example, A is befriended with B, which is befriended with C and E, where C is befriended with D, and D befriended with A (the original circle), but E is also befriended with F which in turn is befriended with A.

So there's 2 circles of friends, and A and B are in both: A-B-C-D-A and A-B-E-F-A.

Your example still fits with my (tentative) definition. For both circles, it is true that each member has exactly two friendship relations in it. My definition does not say anything about the number of overall friendship relations on the whole site, but only about friendship relations within a circle.

But I think I need a second thing in the definition, but I'm not sure which one is "better":

• A "circle of friends" must have at least three members
• There must be at least one person for every member of the circle without a direct "friend" relation

The two are not equivalent, because the first rule allows circles of three persons, while the latter allows circles starting at four members.