Quadratic Scaling in AoE - Why Unholy Death Knights are Kings of Massive Pulls
20.01.2019 um 18:17
Some of you might have seen the Twitch clips circulating showing Unholy DKs reaching above 1 million dps on
. You might also remember the Unholy DK that was brought to the MDI finals,
on a MOTHERLODE!! pull. Behind these mad numbers is a phenomenon called quadratic scaling. This article will outline exactly how it works, and why it allows for such insane DPS.
I've edited the article to accurately represent how our damage scales past 20 targets.
All but very few AoE spells have linear scaling. An example of this would be
. It deals damage to your target, and all surrounding enemies over 2 seconds. It deals the same amount of damage per target, independent of how many targets there are. If you deal 100 damage on 1 target, you would deal 200 damage on 2 targets, 500 damage on 5 targets, and so on. It is easy to see exactly how the damage scales, and we do not get any ridiculous numbers at 20 targets. Sure, the ability will deal a ton of damage, but only 10 times as much as on 2 targets. An Unholy DK has an ability that deals 190 times as much on 20 targets, compared to 2... Furthermore, linearly scaling spells has a damage cap at 20 targets; they simply do not deal more damage even if you add more targets. The Unholy DK's ability continues to scale though, climbing higher and higher for each enemy that join the fight.
Essentially, the principle of quadratic scaling is that it scales much better when we increase the number of targets. When we go from 1 to 2, it doesn’t increase by a factor of 2, but by a factor of 4. When we go from 1 to 10, it increases by a factor of 100. To understand exactly how the Unholy DK can do this, we must delve deeper into the abilities of the Unholy DK.
There are 6 abilities and talents that work together to cause this quadratic scaling, and insane bursts:
Tod und Verfall
. Don’t worry if you’ve never encountered these abilities before, I will make sure to explain everything properly!
is the main star of this story. Unholy DKs apply this debuff to their targets in the form of stacks, going all the way up to 6. The wounds can be burst with other abilities to deal damage to the target.
is a talent that causes your
s to deal AoE damage when burst, to all nearby targets. This damage is capped at 20 targets just like many other similar abilities.
Applying and bursting wounds on a single target wouldn’t do that much for your dps, and it surely wouldn’t scale quadratically. It’s when we are able to apply and burst wounds on all targets, at the same time, that we really get somewhere.
Tod und Verfall
) serve this exact purpose: they apply
s in AoE. For each individual target you are facing, these abilities have a chance of applying
s to that target. If you face twice as many targets, twice as many
s would be applied. When we reach a high number of targets, we see an absolutely insane number of wounds being applied.
Tod und Verfall
are responsible for bursting the wounds.
itself bursts a wound on a single target when used. However, if you are standing in your
Tod und Verfall
, the ability cleaves onto everything, bursting wounds on every single target in the pack! This allows you to quickly convert all those applied wounds to massive amounts of AoE in the space of a few globals!
The Azerite trait
also plays an important role in the damage output. The trait itself can stack all the way up to 30, giving an incredible 7200 strength at 385 Azerite ilvl and 3 stacked traits (which equates to a percentage increase of roughly 115%). This is easily reached in big pulls since we will be bursting a lot more than 30 wounds in the space of just 1 or 2 globals. This helps our AoE scale past 20 targets, where the AoE damage of
Now that we have all the abilities covered, let’s bring up a concrete example and see exactly how this scales. First of all, we shall assume that we can apply one
on each target, each second, and that we pop all these wounds every second as well. This is not exactly what would happen in a real scenario, but the following arguments are still valid. The only thing that changes is the absolute number, the scaling will be exactly the same. We also assume that when a wound is burst, it deals 1 unit of damage to all enemies around it.
Let’s start out simple: we have 2 targets. Each of these targets would cause one wound burst every second, each burst would deal 1 unit of damage, resulting in 2 units of damage every second. For 3 targets, we would have 3 wound bursts, each dealing 2 units of damage resulting in a total of 6 units of damage. For 4 targets, we would get 12 units of damage, and so on. The formula for units of damage as a function of number of targets would be:
units of damage = number of targets * (number of targets - 1)
This multiplying “number of targets” with itself is what we call a “quadratic behaviour”. This behaviour continues up to 20 targets, and thereafter becomes linear. This comes from the 20 target cap that I previously mentioned, which causes each wound to have a maximum amount of total damage it can deal. However, we still get more wounds when we increase the number of targets, and the damage therefore continues to scale. To make the scaling easier to see, here is a table that shows exactly how much damage we get for each target count:
Number of Targets
Units of Damage
This is where you can see the true power of the scaling. Increasing the number of targets from 2 to 10, our damage is increased by a factor of 45! If we go from 2 to 20 targets, it is increased by a factor of 190! This contrasts with most classes that would only deal 5 and 10 times as much damage with their abilities in these scenarios.
To further visualise the scaling, let’s have a look at a graph. Here we are comparing the dps of
, an ability that scales linearly in AoE, with our
deals roughly twice as much damage as a wound being burst. Each second, every target has on average 0.25 wounds applied, and these wounds are then instantly burst. I have chosen not to calculate the added damage from the
strength buff. This is because both these abilities would gain an equal amount from it, and we can therefore compare the two without accounting for it. When we calculate the damage of each ability, we get the following graph:
As you can see,
wins by a large margin when we have few targets. At 2 targets, it deals 8 times as much dps. However, as we start getting more and more enemies,
start to catch up and we reach an intersect at 9 targets. This is not particularly remarkable when it comes to damage. It is the next graph that really shows the difference!
At 20 targets, the quadratic scaling switches to linear scaling for
. However, the 20 target cap on spells also has an effect on
: it simply doens't scale further! Even though
started out with 8 times less DPS compared to
, at 50 targets, it deals 6 times as much damage. What we have seen about
is applicable to most other DPS spec dealing AoE damage. At low target counts, they would beat the Unholy Death Knight easily. However, when we get to the really juicy pulls, they will be far, far behind.
I hope this article has given you some insight into quadratic AoE scaling and how Unholy Death Knights make it work. Remember that the build required to do this is rather niche, as it deals low Single Target damage and mediocre damage on smaller pulls. However, we might see its popularity grow as the
affix is implemented next week (assuming Unholy does not get nerfed…)!
About the Author
I'm Bicepspump, a semi-hardcore player currently playing in Divine Ancient on Kazzak, as well as author of Wowhead's
guides. You can often find me on the
Death Knight discord
discussing rotations and builds. Big shoutout to everyone there that have given me valuable insight through numerous discussions. I take part in writing and optimizing the action priority list for SimulationCraft and continuously test different hypotheses using the SimulationCraft program. Furthermore, I work with the WoWAnalyzer website where I develop modules for the Death Knight specs. Please feel free to contact me on
as 💪Bicepspump💪#6318 if you have any questions or if you found something strange my guides. I'm always happy for a healthy discussion! Outside of playing World of Warcraft, I study for a Masters of Physics at the University of Oxford. I also enjoy rowing and playing Dota 2.
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