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!
Okay, definitely confirmed... don't know why I didn't test this originally anyway. This analysis is from the fusing data again, but looking at each link (after the first one) and its probability of being created, based on the previous link. It's pretty obvious...

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When Link0=True, Link1 frequency is: 0.378 (n=177)

When Link1=True, Link2 frequency is: 0.300 (n=123)

When Link2=True, Link3 frequency is: 0.392 (n=125)

When Link3=True, Link4 frequency is: 0.342 (n=149)


When Link0=False, Link1 frequency is: 0.674 (n=83)

When Link1=False, Link2 frequency is: 0.642 (n=137)

When Link2=False, Link3 frequency is: 0.740 (n=135)

When Link3=False, Link4 frequency is: 0.540 (n=111)
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.
Links aren't done individually. Starting at the first socket, there's a certain chance to link 6 sockects, a chance to link 5, a chance to link 4, etc. It rolls that chance and links that many sockets, then if there are still sockets left, it moves to the next one it hasn't linked yet and does this again.
So if there are 5 sockets, and the first roll gets to link 2, then it'll roll again starting on the 3rd socket (and can't roll above 3, since there's only 3 sockets left).
"I'm a strong, independent blue gem and I don't need no other gems to support me. -sassy finger wag-" - Stormquake
Interesting. I think that actually works out to be equivalent, mathematically, to what I concocted. Thanks for the reply!
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nepharis wrote:
Interesting. I think that actually works out to be equivalent, mathematically, to what I concocted. Thanks for the reply!


I had this thread (pretty old) http://www.pathofexile.com/forum/view-thread/21503 that basically outline the mechanics behind rolling link for items. Only the hard numbers that are not provided by GGG we estimated upon. But it's quite close. So mathematically, its not going to be equivalent to your guess. The numebers can be changed, and the odds change accordingly, and that's how they adjust the chance a 6L item can exists in the economy.
Unviable build tester.
Fuse mechanics:
http://www.pathofexile.com/forum/view-thread/21503
95% Crit Build Without Charges [0.10.1c]:
http://www.pathofexile.com/forum/view-thread/172438
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!
So.... In english were is the % chance to roll a 6L ?
(June 28th, 2013)
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TroySpiral wrote:
So.... In english were is the % chance to roll a 6L ?


No answer to this?
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TroySpiral wrote:
So.... In english were is the % chance to roll a 6L ?


The *estimation* is one third of one percent (so 0.33%) ... but that doesn't mean it actually is that amount, as that estimation is based off a different orb from an older version of the game.


-VG-
Invited to Beta 2012-03-18 / Supporter since 2012-04-08

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