I decided to analyse the rolls from the gif. There are 262 separate fusing rolls (!!!). A quick script to load the image and compare pixel colors at certain locations gave me the the average frequency of links occurring. From the data:

"6S analysis (n=262)

-- Link frequency by location: 0.675, 0.469, 0.477, 0.568, 0.423

-- Overall link frequency: 0.522

Interesting. My first guess as to the method (and which appears to be somewhat accurate), is that each link is rolled based on whether it is connected to another link. Consider, if the link in question is not connected (either the first link to be rolled, or the previous link was not created), it has a 2/3 chance of being created. However, if it *is* going to connect to an already created link, it only has a 1/3 chance.

For example, with a 6 socket item, the first (top) link has a 2/3 chance of being created. Let's say it does. Now the next link, in the top right, only has a 1/3 chance of being created. The RNG gods are with us, it also gets created. Now it's a 3L item already. The third and center link is up, again with 1/3 chance... no dice, it is not created. The bottom left is up, but back to a 2/3 chance... success! The final link, at the bottom, is rolled with a 1/3 chance.

To test my hypothesis, I ran a quick simulation based on the above probabilities. The results:

"Simulation (n=10,000)

-- Link frequency by location: 0.673, 0.452, 0.523, 0.502, 0.499

-- Overall link frequency: 0.530

As you can see, it matches up pretty well with the data from the 6S fusing images. I believe this is in fact how socket linking is rolled in PoE. It explains why links in general are common, but why multi-linked items are more rare.

Furthermore, if this is correct, the chance of rolling a 3L on a 3S item is (2/3)*(1/3) = 2/9, or about 22%. The chance of getting 4L from a 4S item is (2/3)*(1/3)*(1/3) = 2/2, or 7.4%. If anyone has data on these, which either corroborate or refute these data, please let me know.

Cheers!