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

Â influence vector now. So in terms of understanding what this

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

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

Â process. And let me just sort of reiterate what,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â sure that this thing turns out to be primitive.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â just by looking directly at the matrix. right?

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

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

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

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

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

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

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

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

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

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

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

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

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

Â 3:59

And in, in terms of what it eventually means, given that we know that this is

Â going to converge to 3 11ths, 4 11ths, 4 11ths, it tells us that the limit, you

Â know, in terms of the weight it puts on whatever person one believes, is 3 11ths.

Â So if person one had a weight of you know, belief of 1, and everybody else

Â believed 0, we'd go to 3 11ths. If it was person two that had that belief

Â it would go to 4 11ths, and and person three would give 4 11ths.

Â So it's telling us, basically, how much eventual weight in the final, belief of

Â society that each person's initial belief have, all right?

Â So it's a very compact and simple measure.

Â Now, when we're looking what this influence measure is, the nice thing in

Â the, the way that we get the fact that it has to be a unit eigenvector comes from

Â the fact that we know whatever we want, in terms of figuring this out, it has to

Â be the same thing since it's a limit of, of doing all this updating.

Â It would have to be the same thing as if we did it after one more, you know, if we

Â did one more updating, it shouldn't change the limit.

Â And so, it has to be that s is equal to s t, and that tells us that, effectively,

Â whatever this influence vector is that we're trying to get to tell us what the

Â eventual beliefs are is going to have to be a left-hand-side unit eigenvector.

Â Now the nice thing about that is it ties us back to these influence measures, the

Â centrality measure. Eigenvector centrality.

Â It's saying that the influence that person i has is a weighted sum of the

Â people who listen to i, t j i times s j. Right, so, so the fact that s is equal to

Â s times t tells us that, effectively, the way that you get influence is being

Â listened to by influential people. That means your belief is going to get

Â into their beliefs, which is going to get into other people's beliefs.

Â And the more influence they have, the more structure they have on the final.

Â So you get high influence by being connected to by high influence high

Â influence individuals. And so again, that relates back to things

Â like power measures Google Page Rank eigenvector centrality and this is now

Â giving us a foundation for why we would want to be looking at an eigenvector as a

Â measure of power or influence. It comes out directly in this model.

Â So it gives us a nice foundation for that.

Â So, the next thing we'll do is, is take a look at putting these to practice.

Â so we'll take some of the, the DeGroot model.

Â Look at the left hand side of the unit eigen vector now of a, of a

Â stochastictized matrix and see what that tells us, can it help us understand

Â what's going on in a particular network setting.

Â