Okay, so let's took take a look now at some examples of the DeGroot learning and influence vector in practice. And first, we can start by just some simple situations where we see how people's position in the network translates into the influence. And one thing that, that sort of interesting is that people who put huge weight on themselves high weight on themselves are going to end up maintaining beliefs while other people are changing over time. And so, groups that are highly introspective and, and get listened to by other groups are also going to end up having their weight be very high in the society. So, one thing that's sort of interesting about this model and, and perhaps a, a little strange is that groups that, that don't listen much to the outside but, but end up being listened to a lot by the outside end up having a large influence. So it's, it's not only listening a lot to yourself, but also then having people listen to you. That combination can give somebody a very high influence, because they don't update their beliefs much, and they end up eventually impacting others beliefs. so when we look at another example of influences. Suppose that, that actually what people did was equally weight their, their connections, and let's also suppose that you know, talking the that we think of this as friendships, so they're mutual. So, Tij is, is greater than 0 if and only if Tji is, is equal to 0. So, if I listen to you, you listen to me. So we, we have conversations and I put equal weight on all my connections. so if you think then of a situation where di is i's out degree. then Tij is going to be 1 over di, for each i and j. So, everybody that I am talking to, I put equal weight on, so however many friends I have, I give one if I have ten friends, I give 1 10th weight to each one of them. Okay, so let's think of that as a very simple version of this model. so if we let D be the overall total sum of out degrees in this model, where everybody is weighting their friends equally, then the claim is going to be that a persons influence is just going to be proportional to their degree. So, this gives a a foundation for degree centrality. In a world where you had conversations with people and you weighted everybody that you talked to equally then you're, your influence would be related proportional to your degree. And you know, recall that s is the left-hand side unit eigenvector, so you needed to satisfy this equation. And so, what you need to verify is that s i is then going to be equal to this sum over j of Tji times sj, all right? And so, let's check that this actually works. So, we've got this claim that this is the eventual limit and we want to verify that this equation holds. So, if we plug in for si sort of try and check that we're going to get di out over D, well what's this sum? This sum is the sum over all the people that i listens to, and because the listening is reciprocal, right? So, Tji is greater than 0 if and only if Tij is greater than 0. So we're going to sum this over how many people i listens to 1 over dj, and the we want to verify that if we stick in the S's for this we'll get back the right answer. So we've got dj over D. Well, these two things are going to cancel the two djs. And then we've got a sum which is proportional to the number of individuals that i is listening to, that's directly. So we've got 1 over d summed that many times that's going to be di over D. So in D, check, we've got back the solution. So, in a situation where you put equal weight on all your friends and weight, and listening is reciprocal, you get back degree centrality. So, in that situation eigenvector centrality and degree centrality actually coincide. Okay, so let's have a look at an example now, and this is papered by David Krackardt where he looked at an advice network in a company and a paper from 19 87. And this is one where we've got a few enough nodes, and we've got information so we can actually calculate out what the the S's, and in particular this is a a picture of the network. Now, it's a directed network, so certain people could actually listen to others without them listening back. So, this would be a directed network not necessarily back and forth. And there's some individuals who actually aren't connected. So, some individuals are not getting listened to at all and their influence is going to turn out to be zero. In this situation, so it's not a strongly connected network. Nonetheless, the S is still going to work as the the right answer to this. so if you go through and figure out what the S is. So if you solve for the S of this, there are some individuals, for instance 6 who didn't get listened to and 13 and so forth, end up with 16 and 17, so some people end up with no influence, nobody listens to them and comes to them for advice. but basically we, we end up with the advice levels varying from 0 up to let's see, we've got a 0.2 here. so we end up with different levels of influence. And what the other columns in this table represent is sort of, these are levels of the individuals in a hierarchy in this company. so level one, this is the CEO, the head of the company. Level two, we've got people at the second highest management level. Then level three, is the third-highest level. And what we see is actually there's some people that are more influential in terms of this network than the, the individual at the top, actually one of the people at level two has a higher influence vector than the person at the top. And we can begin to, you know, look at different people, different people in, at, at level three have different levels of influence. And you know, you can look at that by the different department, their age, their tenure, and so forth. And this information is going to be complimentary to some of those things that doesn't just necessarily correlate with how old they are or how long they have been in the company or which department they're in. These numbers are telling you something different about the, the relative influence of these individuals have. so what that does is it just shows, you know, one example of where you can begin to take this DeGroot model. It gives us a foundation for looking at this particular left-hand side unit eigenvector as a, a measure of influence. And you know, if you ran the DeGroot process on this particular network. And had people updating over time in their beliefs, that would tell you what their beliefs would eventually converge to. And this was done under the assumption that so we don't know exactly what these weights are. That people put equal weights on each one of their friends that they set. So person 17, for instance has you know five out arrows. So they put 1 5th weight on each one of those. So that gives us one example of the influence of, of, of, of ha, coming out here. And again you know, we see the relation of this to the eigenvector centrality and that's coming back from this relation of, of matrices to powers which was underlined eigenvector centrality. Okay, so that's one of the applications of this. that gives us some perspective on this DeGroot model. you know we'll, we'll wrap up some of our discussion of learning next and then, and then after that we'll start talking more about games on networks.