Interesting response!
Yes, the node, edges, travelling salesman, those are my definitions as well.
The only difference is that I've always thought about this is with an undirected graph. That models conventional friendships pretty well, but now that I'm thinking about it, it's probably not a good way to model modern "relationships". Either way, with two nodes it's no longer a travelling salesman, as it becomes a much easier problem to solve.
I would argue that the tyranny of the majority unlikely to not be solved with the current form of social networks. I'm not sure if I have time to write out my argument, but it stems from the fact that the most popular people in the world have very uni-directional relationships with most people; everyone knows of Mr. Beast, very few know Mr. Beast.
What bike is that in the photo?