The following meme has been kicking around reddit and other sites recently:
Wikipedia trivia: if you take any article, click on the first link in the article text not in parentheses or italics, and then repeat, you will eventually end up at "Philosophy". –xkcd #903 alttext
Clearly this is not true for every single article (for instance, it is currently not true for "Philosophy" itself), but it is true for a surprising number of pages. There are hundreds of forum posts by people marveling over the length of the path from cheese to philosophy and drawing deep connections to the scourge of lactose intolerance. And of course, the popularity of the meme has lead to an extensive edit war between people trying to ‘fix’ pages such that they conform to the rule, and people trying to ‘break’ pages out of spite, and those just trying to revert all the changes done by the previous two groups. In my opinion, the fact that pages like “Philosophy” exist seems very unsurprising to me.
Wikipedia conjecture. Consider a directed graph where every node has outgoing degree exactly one. If nodes are added randomly in a scale-free manner (eg the probability of linking to an existing node is proportionate to that node’s incoming degree), then the expected fraction of nodes in the largest connected component will increase monotonically.
Philosophy corollary. Given a sufficiently large such graph, there will be a node which is reachable from a large fraction of the graph. Here “large” is defined as “large enough to stimulated discussion by bored reddit denizens.”
Perhaps I’ll try to prove this after quals, when I have more time for nonsense.