The concept that, like the 6 degrees of separation idea, that every node on Everything is, at most, six links away from every other node. Now, given the amount of links per node, and the number of nodes, the probability this is true might be able to be calculated (as sockpuppet tries in another writeup in this node). However, this only tells us if it is theoretically possible for this to occur or not.

In reality, this is most likely not the current situation. Clicking on random node still seems to have a good chance of putting you in a Webster node that is as yet completely unlinked to the rest of the nodescape. Because this is not a rare occurence, this suggests there are still a significant number of unconnected nodes remaining from the big import of Webster 1913. And because most of the travelling between nodes that people do is via hard link and soft link, these unconnected nodes are not going to be added at a rate that makes complete connection of the nodescape something that will occur soon.

This does not even take into account the nodeshells that are sitting about, also unlinked.

I would suspect that a sizable chunk of the nodescape is connected together, though. The addition of new nodes, assuming that time is taken to properly softlink the new nodes into the nodescape, could help to increase the interconnectedness.

Here's a little back-of-the-envelope calculation. From a recent Everything Daily Report (as I go to check my email) there were 3,719,603 links, and 287,745 nodes. (I am using commas for readability because the numbers are large.) This is, on average, 12.9 links per node (which may include multiple writeups). This seems kind of low when you think about it, but it points out how rough this estimate will really be: the distribution of links/node is probably closer to bimodal than normal: there are lots of pithy writeups with only a handful of links, and also lots of meaty writeups with tons of links. This is especially true because a large percentage of our writeups are Webster 1913's (98,233 of these, or roughly 1/4 of all writeups!) which tend to have very few hardlinks.

We also have to allow for the fact that a certain percentage of links, as we all know, lead nowhere. Assume for the sake of argument that this is maybe 20% of all hardlinks. Then the expected number of nodes you can possibly reach from a single node is (12.9 * 0.8)^6, or 1,219,312. That's pretty encouraging, because it's way more than we have.

But that number is too high, because it assumes that there are no cycles in our directed graph. (Here I am using "graph" as in graph theory.) If you think of this as a tree rooted at a particular node, lots of nodes at level n of the tree will have links back to nodes on the same level or lower, which ultimately makes the tree less bushy, and reduces the number of nodes the tree finally reaches. If we go with 10.3 as our working link number (12.9 * 0.8 from before), the probability that any node on level n of the tree will not refer back to the others, to leading order, is (1 - (10.3^n)/287,745). So the percentage of the 1,219,312 unique nodes we expect to end up with is roughly the product of these, for n = 1 to 5 (we get the first link for free assuming the node isn't self-referential). That product is about 0.57, so the actual number of nodes we expect to get to taking into account dead links and recombination in the tree is 0.57 * 1,219,312 = 695,007.

This is well beyond the 287,845 nodes we actually have, suggesting that we don't have enough statistical information (i.e. a complete distribution) about the inter-linkedness of E2 to be sure. But because we have overshot it is suggestive that 6 is enough almost all of the time, which is all we can really hope for with a B.O.E. calculation anyway.

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