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The final topic of this module on Eigenproblems, as well as the final topic of this course
as a whole, will focus on an algorithm called PageRank. This algorithm was famously published
by and named after Google founder Larry Page and colleagues in 1998. And was used by Google
to help them decide which order to display their websites when they returned from search.
The central assumption underpinning page rank is that the importance of a website is related
to its links to and from other websites, and somehow Eigen theory comes up. This bubble
diagram represents a model mini Internet, where each bubble is a webpage and each arrow
from A, B, C, and D represents a link on that webpage which takes you to one of the others.
We're trying to build an expression that tells us, based on this network structure, which
of these webpages is most relevant to the person who made the search. As such, we're
going to use the concept of Procrastinating Pat who is an imaginary person who goes on
the Internet and just randomly click links to avoid doing their work. By mapping all
the possible links, we can build a model to estimate the amount of time we would expect
Pat to spend on each webpage. We can describe the links present on page A as a vector, where
each row is either a one or a zero based on whether there is a link to the corresponding
page. And then normalise the vector by the total number of the links, such that they
can be used to describe a probability for that page. For example, the vector of links
from page A will be 0 1 1 1. Because vector A has links to sites B, to C, and to D, but
it doesn't have a link to itself. Also, because there are three links in this page in total,
we would normalize by a factor of a third. So the total click probability sums to one.
So we can write, L_A = (0, a third, a third, a third). Following the same logic, the link
vectors in the next two sites are shown here. And finally, for page D, we can write L_D
is going to equal, so D is connected to B and C, but not A, two sides in total, (0,
a half, a half, 0). We can now build our link matrix L by using each of our linked vectors
as a column, which you can see will form a square matrix. What we're trying to represent
here with our matrix L is the probability of ending up on each of the pages. For example,
the only way to get to A is by being at B. So you then need to know the probability of
being at B, which you could've got to from either A or D. As you can see, this problem
is self-referential, as the ranks on all the pages depend on all the others. Although we
built our matrix from columns of outward links, we can see that the rows describe inward links
normalized with respect to their page of origin. We can now write an expression which summarises
the approach. We're going to use the vector r to store the rank of all webpages. To calculate
the rank of page A, you need to know three things about all other pages on the Internet.
What's your rank? Do you link to page A? And how many outgoing links do you have in total?
The following expression combines these three pieces of information for webpage A only.
So r_A is going to equal the sum from j = 1 to n, where n is all the webpages of the link
matrix relevant to webpage A and location j, multiplied by the rank at location j. So
this is going to scroll through each of our webpages. Which means that the rank of A is
the sum of the ranks of all the pages which link to it, weighted by their specific link
probability taken from matrix L. Now we want to be able to write this expression for all
pages and solve them simultaneously. Thinking back to our linear algebra, we can rewrite
the above expression applied to all webpages as a simple matrix multiplication. So r = Lr.
Clearly, we start off not knowing r. So we simply assume that all the ranks are equally
and normalise them by the total number of webpages in our analysis, which in this case
is 4. So r equals a quarter, a quarter, a quarter, a quarter.
Then, each time you multiply r by our matrix L, this gives us an updated value for r. So
we can say that ri+1 is going to be L times ri.
Applying this expression repeatedly means that we are solving this problem iteratively.
Each time we do this, we update the values in r until, eventually, r stops changing.
So now r really does equal Lr. Thinking back to the previous videos in this module, this
implies that r is now an eigenvector of matrix L, with an eigenvalue of 1. At this point,
you might well be thinking, if we want to multiply r by L many times, perhaps this will
be best tackled by applying the diagonalization method that we saw in the last video. But
don't forget, this would require us to already know all of the Eigen vectors, which is what
we're trying to find in the first place. So now that we have an equation, and hopefully
some idea of where it came from, we can ask our computer to iteratively apply it until
it converges to find our rank vector. You can see that although it takes about ten iterations
for the numbers to settle down, the order is already established after the first iteration.
However, this is just an artifact of our system being so tiny. So now we have our result,
which says that as Procrastinating Pat randomly clicks around our network, we'd expect them
to spend about 40% of their time on page D. But only about 12% of their time on page A,
with 24% on each of pages B and C. We now have our ranking, with D at the top and A
at the bottom, and B and C equal in the middle. As it turns out, although there are many approaches
for efficiently calculating eigenvectors that have been developed over the years, repeatedly
multiplying a randomly selected initial guest vector by a matrix, which is called the power
method, is still very effective for the page rank problem for two reasons. Firstly, although
the power method will clearly only give you one eigenvector, when we know that there will
be n for an n webpage system, it turns out that because of the way we've structured our
link matrix, the vector it gives you will always be the one that you're looking for,
with an eigenvalue of 1. Secondly, although this is not true for the full webpage mini
Internet, when looking at the real Internet you can imagine that almost every entry in
the link matrix will be zero, i.e,, most pages don't connect to most other pages. This is
referred to as a sparse matrix. And algorithms exist such that multiplications can be performed
very efficiently. One key aspect of the page rank algorithm
that we haven't discussed so far is called the damping factor, d.
This adds an additional term to our iterative formula. So r i + 1 is now going to equal
d, times Lri + 1- d over n. d is something between 0 and 1. And you can think of it as
1 minus the probability with which procrastinating Pat suddenly, randomly types in a web address,
rather than clicking on a link on his current page. The effect that this has on the actual
calculation is about finding a compromise between speed and stability of the iterative
convergence process. There are over one billion websites on the Internet today, compared with
just a few million when the page rank algorithm was first published in 1998. And so the methods
for search and ranking have had to evolve to maximize efficiency, although the core
concept has remained unchanged for many years. This brings us to the end of our introduction
to the page rank algorithm. There are, of course, many details which we didn't cover
in this video. But I hope this has allowed you to come away with some insight and understanding
into how the page rank works, and hopefully the confidence to apply this to some larger
networks yourself.
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