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... And in, in terms of what it eventually means, given that we know that this is going to converge to 3 11ths, 4 11ths, 4 11ths, it tells us that the limit, you know, in terms of the weight it puts on whatever person one believes, is 3 11ths. So if person one had a weight of you know, belief of 1, and everybody else believed 0, we'd go to 3 11ths. If it was person two that had that belief it would go to 4 11ths, and and person three would give 4 11ths. So it's telling us, basically, how much eventual weight in the final, belief of society that each person's initial belief have, all right? So it's a very compact and simple measure. Now, when we're looking what this influence measure is, the nice thing in the, the way that we get the fact that it has to be a unit eigenvector comes from the fact that we know whatever we want, in terms of figuring this out, it has to be the same thing since it's a limit of, of doing all this updating. It would have to be the same thing as if we did it after one more, you know, if we did one more updating, it shouldn't change the limit. And so, it has to be that s is equal to s t, and that tells us that, effectively, whatever this influence vector is that we're trying to get to tell us what the eventual beliefs are is going to have to be a left-hand-side unit eigenvector. Now the nice thing about that is it ties us back to these influence measures, the centrality measure. Eigenvector centrality. It's saying that the influence that person i has is a weighted sum of the people who listen to i, t j i times s j. Right, so, so the fact that s is equal to s times t tells us that, effectively, the way that you get influence is being listened to by influential people. That means your belief is going to get into their beliefs, which is going to get into other people's beliefs. And the more influence they have, the more structure they have on the final. So you get high influence by being connected to by high influence high influence individuals. And so again, that relates back to things like power measures Google Page Rank eigenvector centrality and this is now giving us a foundation for why we would want to be looking at an eigenvector as a measure of power or influence. It comes out directly in this model. So it gives us a nice foundation for that. So, the next thing we'll do is, is take a look at putting these to practice. so we'll take some of the, the DeGroot model. Look at the left hand side of the unit eigen vector now of a, of a stochastictized matrix and see what that tells us, can it help us understand what's going on in a particular network setting.