Let me consider a new graph with 5 vertices,
let me call them A B C D and E and then let
me put some random edges here and there. So,
that every edge has some incoming links and
some outgoing links. So, this is my graph
fine and now let me create a table for this
A B C D values and then start off with some
values for A B C D E. Idivide my resources
equally to A B C D E resources or gold coins
as in the previous example that we have seen
in our lectures.
So, what will that do? It will be 1 by 5 1
by 5 1 by 5 and 1 by 5 correct. So, if I continue
like this we observed that it converged for
3 nodes. Now, I have taken 5 nodes not so,
easy looking graph will it converge here as
well is my big question, will it converge
here or not. Let us go to the excel sheet
and then theGoogle spreadsheet and then trydoing
the same thing that we did before and we will
see whether it converges or not. So, let usbe
prepared to go and edit there, let us observe
what is happening here so, that the punching
in of the equation will be easier for it.
Now, look at this A is actually equal to C
by 3 correct, why is that from C you see that
there are 3 edges going out. So, the so, the
value of C will be divided between these 3
people. So, I will say A equals C divided
by 3, good nextB is A plus E by 2. You can
check why, maybe you should pause and then
check for each and every value. I have written
it down, I will justgo through itit is for
you to check whether its right or not. B gets
all the values from A, it also gets half the
value from E because it have 2 edges going
from E as you can observe ok.
So, C will be B because there is only one
thing that is coming inside C and Bsends only
1 edge to see; I mean from emanating from
B there are only 1 there is only 1 edge. And,
then D is C by 3 plus E by 2 you can check
why and finally, E is equal to D. So, we will
keep these things and then try to go to ourspreadsheet
and then try to code it. So, what I will do
is I will directly go there to the spreadsheet,
finish this assignments of A B C D E, start
with 1 by 5 1 by 5 1 by 5 and bring you people
there. And, then show you as I pull down the
values on excel, on Google spreadsheet you
will observe that I am populating everything
and we will see whether it converges or not.
So, that I forgot something here. So, let
me just write that down E as you saw has um
incoming link from C as well not justfrom
D. So, you have an incoming link from C. So,
let me make a note of this E is not just equal
to D, it is plus C by 3; a good way to see
if all of them add up to1 is to see if these
things add up to, if these 5 things add up
to, if you add these things you must get;
what should you get guess. You must get A
plus B plus C plus D plus E, see if you are
getting this andyou just think of how and
why exactly do I say they should add to A
plus B plus C plus D plus E; it is a simple
quiz question.
I have now opened the Google spreadsheet and
the values of A B C D and E as you would have
guessed starts from 1 by 5. So, I am going
to populate it with1 by 5 throughout and as
we computed in our previous lecture the value
of A is going to be C divided by 3.I think
you all know howGoogle spreadsheet works,
as I told you earlier you just need to take
the value of this which is C column and the
third row. You should in fact, punch inis
equal to C 3 divided by 3 ok. And then what
was B? B was simplysame as A plus what was
in E divided by 2, isn't it ok. What was my
C?
My C was nothing else but, B you may want
to take a look at my previouslecture if if
you are not following what I am doing. It
is a pretty straightforward processmy D is
going to be D was slightly complicated. It
was C by 3 C by 3 plus E by 2 E by 2 that
is it. And what was E? E was equal to my D
plus C by 3, a good way to check whether it
is right or not is is just to see if it is
equal to the entire row, is this equal to
A 4 plus B 4 plus; what is that A 4 plus B
4plus C 4 plus D 4 plus E 4. It should be
1 perfect, which meansthe values are properly
put; let me delete this now ok.
So, now what I do is I copy this and paste
this and this creates a new iteration, as
you see the value of this changes, but this
value remains the same because it is simplyagain
0.2 by 3 C by 3 and so on and so forth. Let
me just pull this and see whether it converges
or not, it is no wayconverging maybe it does
not converge I do not know.
Let me go ahead and seewell I see somethingpromising,
it looks like it is converging. You see 0.07
and 0.07 here 0.21 and 0.21 although the third
decimal digit is not change, it issomething
elseI sort of see a ray of hope here maybe
it will converge. Let me go on, let me go
on the point is to keep going until you get
the same values.It looks like it is converging,
you see the last 3 the 3 decimal digits after
the point looks like the same .
But, I am not really happy I want to see the
exact converged values. Let me go on like
go on and on I am still not happy, let me
pull this too let us say as you can see I
am in 76th row. I have repeated this 76 times,
if I go on to let us say 200 or let us say
little over 100 I see that the values are
exactly the same right. It looks like it is
exactly the same. When it is exactly the same
no matter where, until what point I go, I
am going to 250 it will still be the same.
You see these 2 cells it was so much and it
still remains the same.
Let us make a note of this, we are observing
that the graphs here that the graph given
graph here with 5 notes A B C D E; when you
start with 1 by 5 1 by 5 1 by 5 it is converging
right, it is converging to these values ok
. So, let us go to the next lecture and see
what can one say about this, why exactly this
is happening .
