A word on convergence

Convergence is an important mathematical aspect of PageRank, which allows
Google to provided unprecedented search quality at comparably low costs. This is
a semi-complex topic but it is important to your understanding of how and why
PageRank works. We've tried to make it as simple as possible but, unless you're
Sergey Brin or Larry Page, you'll still need to concentrate!



Whilst it will take some concentration, it isn't that hard to understand.



We've shown that the outlet values (final values) of one stage of the
calculation become the inlet values (starting values) of the next stage, and
that we keep on doing this (it's known as a recursive procedure). But the really
big question is how and when does the recursive procedure stop?



The answer is "convergence". Provided the dampening factor (d in our equation) is less than one, then convergence will occur. Nominally we set it to 0.85
(because that's the value mentioned in the Stanford papers).



This convergence basically means that whatever values we start at, after running
the calculation a number of times we will end up with the same final values and
that these values will no longer change if we do further iterations of the

calculation. These final values are known as limiting values.



Once the limiting values have been reached, Google no longer needs to expend
processing power on calculating the PageRank. They can finish there!


This is easier to understand with an example. Let's take a look at the following
structure:

PageRank for pages A, B, C, D at various stages of iteration