Solutions: Homework 1, Math 152-205, additional problem:

[There are many possible answers to questions involving "intuitive 
explanations." No answer is particularly preferred.]

>>For this problem, you should consult the class blog if you missed class
>>on the first day.
>>
>>A network has two webpages, page A and page B.  The links are as follows:
>>
>>Page A has 0 links to page A and 2 links to page B.
>>Page B has 6 links to page A and 2 links to page B.
>>
>>
>>(Part a) Denote the PageRanks of A and B by  xA  and  xB .  Explain
>>intuitively what the meaning is of each of the following PageRank equations:
>>
>>xA + xB = 1

This (1) makes xA and xB unique, (2) normalizes them so you can tell
what fraction of PageRank each has, (3) expresses them in terms of
"probabilities", (4) etc. etc. etc.

>>xA = (0/2) xA + (6/8) xB

xA gets 0/2 of xA's "vote," and 6/8 of xB's "vote"

>>xB = (2/2) xA + (2/8) xB

similarly

>>
>>
>>(Part b) Guess which webpage will have a higher page rank.  (This is a
>>guess;  there are no right or wrong answers, the only wrong answer is
>>not to make a guess.)
>>

Any guess is fine.  It's better if you think a bit.

>>
>>(Part c) Solve these equations any way you want.  Explain intuitively
>>why one webpage has a higher page rank than the other (this should not
>>be a guess).
>>

xA = (6/8) xB = (6/8) (1-xA) shows that  xA = 3/7  and  xB = 4/7.
xB is larger since  A votes entirely for B, while B doesn't vote
entirely for A.

>>
>>(Part d) If Page B has 6,000,000,000 links to page A but all other link
>>numbers remain the same, what do you expect the PageRanks to be,
>>roughly speaking?  Which page will have the larger PageRank?
>>
A still votes entirely for B, but now B votes almost entirely (but not
quite) for A.  So the PageRanks will be nearly equal (xA,xB each roughly
1/2), but xB will be very slightly bigger.