Hi folks. So let's talk a little bit about the

influence vector now. So in terms of understanding what this

limit of the DeGroot Process looks like, what are people's beliefs convergent to.

So how can we use this DeGroot Model to understand the limit of the learning

process. And let me just sort of reiterate what,

what we looked at before when we looked at you know, t to the t raised times b.

when we're looking at the limit of this, we're looking for some vector s 1 through

s n, which when multiplied by b 1 through b n, gave us back the beliefs, in terms

of what the limiting beliefs are. And so this is essentially giving us a

measure of the influence of each individual how influential they are.

And let's talk a little bit about that. So we're trying to find out who has

influence in this model. and, you know this gave us some preview

of this by telling us what this, this s 1 through s n was a, a unit lefthand side

eigenvector of the matrix. And so that tells us what we get to when

we get convergence and we get consensus. It looks like a, a normalized left-hand

side eigenvector and that gives us the weighted sum of the original beliefs to

figure out what the limit is. So when we look at this you know, let's

just take a peek at one of the matrices we looked at before where person one

weights 2 and 3, three two weights 1 and three weights 2.

And as you begin raising it to different powers, eventually we see after five

periods that it's all non-negative. that's with the aperiodicity, that makes

sure that this thing turns out to be primitive.

We get all non-negative entries eventually, and indeed, as you go to the

limit, we end up with these calculations of 2 5ths, 2 5ths, 1 5th.

As the entries in every row, and so that tells us if we're trying to hit this with

some beliefs b 1, b 2, b 3, how much are we weighting person one belief?

Well 2 5ths they're going to get weighted.

How much are we weighting two, person two beliefs, 2 5ths.

1 5th on person three's. So that tells us those relative weights.

And you can double-check that this is the unit eigen vector of this thing.

So if you multiply 2 5ths, 2 5ths 1 5th times this, what do you get back?

Well, 1 5th, 1 times 2 5ths, 2 5ths, right?

So hit this times this. So if we multiply this thing times 2

5ths, 2 5ths, 1 5th, we get 2 5ths in the first entry, a half of 2 5th and 1 times

a 1 5th is a 2 5ths and a half of a 2 5ths is a 1 5th.

So, indeed we get back the same vector we started with.

Okay. So, when we look at, you know, in

general, the nice thing about this is it tells us what these entries are going to

be in the limit. Which wouldn't be very easy to figure out

just by looking directly at the matrix. right?

So if we look at this matrix and ask what the limit's going to be or we look at

this figure, once we've got some fairly complicated things going on, especially

for a large matrix, it's not going to be easy to figure out what these eventual

entries are going to be. And so the fact that it's a left-hand

unit eigenvector means you could just plug this into your favorite, program,

Matlab, Mathematica, Maple, whatever you like to, to use to do analysis of, of

matrices, and that would give you back, a left-hand side, unit eigenvector.

And then that calculation will allow you to figure out, what the eventual

influence would be. And you know, you could also just raise

this to multiple powers and, and see where, where it's going in terms of the

limit. so in terms of, of what's going on, in

terms of these limiting beliefs, the influence it, you know, it's coming from

the fact that these rows have to converge to the same thing for each row...