Would you be explaining the joke if you were to note that no non-trivial solutions exist for n?

For any fixed odd number of friends k you could keep the network consistent by only allowing people to join in groups (call them "cells") of m, where m is even and greater than k.

(Explanation: https://en.wikipedia.org/wiki/Regular_graph)

"Sorry, Gary. Come talk to me when you have 558 friends. Gare? I'm sorry, Gare, those are the rules. Everyone has to be equal. It doesn't matter if you'll never have 558 friends and I only had to find 32. Everyone's equal! Don't you see, Gary! Gary? Gare? What are you doing. Sic semper what? Aughhhh."

that's brilliant. i reckon that could do well :)