PageRank
========

.. rubric:: (due: Thursday, November 30, 8:00 AM)


PageRank algorithm is a method for ranking web pages. This
algorithm assigns to each page :math:`P` in some network of web pages its
ranking which is a number :math:`0 \leq r(P) \leq 1`. Higher ranking
is supposed to indicate a more significant page in the network.

PageRank ranking of a web page is computed as follows. Assume that́
:math:`l_1, \dots, l_k` are all links pointing to a page :math:`P`,
where the link :math:`l_i` starts on a page :math:`P_i`. Then

.. math:: r(P) = \sum_{i=1}^k \frac{r(P_i)}{n(P_i)}

where:

-  :math:`r(P_i)` is the PageRank of the web page :math:`P_i`


-  :math:`n(P_i)` is the number of all links originating from :math:`P_i`.

In addition, PageRank rankings must satisfy the condition that if
:math:`P_1, \dots P_N` are all pages in the network then
:math:`r(P_1) + \dots + r(P_N) =1`.

Project
-------

-  The website of the UB Math Department consists of about 180 web pages.
   Compute PageRank rankings of all these web pages.

-  Let :math:`P_1, \dots, P_N` be the list of all pages of the Math
   Department website ordered according to their rankings, from the
   biggest to the smallest. Create a plot consisting of points with
   coordinates :math:`(i, j)` if there is a link from the page
   :math:`P_i` to the page :math:`P_j`. Use different colors to indicate
   how many links there are  from :math:`P_i` to :math:`P_j`.

-  Describe your observations about the structure of the Math Department
   website. Comment if in this case PageRank rankings make sense, i.e. if they
   correctly indicate which pages are more important.