Socket Linking Mechanics (data & simulation)

Many of you have probably seen this gif, http://i.imgur.com/QjUbr.gif, of someone burning through Fusing Orbs trying to generate a 5L or a 6L, with no success. I got curious as to how the links were determined. It wouldn't make sense for it to be strictly random for each link... if each link had a 50% chance of occurring, you would expect to get a 6L every 32 Fusings. But if the chance was much lower, say 20%, it would be difficult to even get a 4L.

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:

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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:

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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!

MOAR.

It actually looks like the chance decreases progressively with each successful link roll.

Two successive links:

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When Link0=True and Link1=True, Link2 frequency is: 0.268 (n=67)

When Link0=False and Link1=True and Link2=True, Link3 frequency is: 0.210 (n=19)

When Link1=False and Link2=True and Link3=True, Link4 frequency is: 0.222 (n=45)

Three successive links:

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When Link0=True and Link1=True and Link2=True, Link3 frequency is: 0.0 (n=18)

When Link1=True and Link2=True and Link3=True, Link4 frequency is: 0.0 (n=4)

So after the first roll at ~66%, the second is about 33% (my previous comment). But on the third roll (to make a 4L), it's down to ~25%. The next, to make a 5L, is very low. Zero out of 22 tries from the data.

I think I'm done... for now.

Interesting. I think that actually works out to be equivalent, mathematically, to what I concocted. Thanks for the reply!

Also the numbers I got are going to off regardless, since whoever rolled that 6S chest got incredibly unlucky. Didn't realize this sort of work had been done already, but I'm not surprised. Oh well, back to grinding, instead!