0:00

Okay, so let's took take a look now at some examples of the DeGroot learning and

Â influence vector in practice. And first, we can start by just some

Â simple situations where we see how people's position in the network

Â translates into the influence. And one thing that, that sort of

Â interesting is that people who put huge weight on themselves high weight on

Â themselves are going to end up maintaining beliefs while other people

Â are changing over time. And so, groups that are highly

Â introspective and, and get listened to by other groups are also going to end up

Â having their weight be very high in the society.

Â So, one thing that's sort of interesting about this model and, and perhaps a, a

Â little strange is that groups that, that don't listen much to the outside but, but

Â end up being listened to a lot by the outside end up having a large influence.

Â So it's, it's not only listening a lot to yourself, but also then having people

Â listen to you. That combination can give somebody a very

Â high influence, because they don't update their beliefs much, and they end up

Â eventually impacting others beliefs. so when we look at another example of

Â influences. Suppose that, that actually what people

Â did was equally weight their, their connections, and let's also suppose that

Â you know, talking the that we think of this as friendships, so they're mutual.

Â So, Tij is, is greater than 0 if and only if Tji is, is equal to 0.

Â So, if I listen to you, you listen to me. So we, we have conversations and I put

Â equal weight on all my connections. so if you think then of a situation where

Â di is i's out degree. then Tij is going to be 1 over di, for

Â each i and j. So, everybody that I am talking to, I put

Â equal weight on, so however many friends I have, I give one if I have ten friends,

Â I give 1 10th weight to each one of them. Okay, so let's think of that as a very

Â simple version of this model. so if we let D be the overall total sum

Â of out degrees in this model, where everybody is weighting their friends

Â equally, then the claim is going to be that a persons influence is just going to

Â be proportional to their degree. So, this gives a a foundation for degree

Â centrality. In a world where you had conversations

Â with people and you weighted everybody that you talked to equally then you're,

Â your influence would be related proportional to your degree.

Â 2:35

And you know, recall that s is the left-hand side unit eigenvector, so you

Â needed to satisfy this equation. And so, what you need to verify is that s

Â i is then going to be equal to this sum over j of Tji times sj, all right?

Â And so, let's check that this actually works.

Â So, we've got this claim that this is the eventual limit and we want to verify that

Â this equation holds. So, if we plug in for si sort of try and

Â check that we're going to get di out over D, well what's this sum?

Â This sum is the sum over all the people that i listens to, and because the

Â listening is reciprocal, right? So, Tji is greater than 0 if and only if

Â Tij is greater than 0. So we're going to sum this over how many

Â people i listens to 1 over dj, and the we want to verify that if we stick in the

Â S's for this we'll get back the right answer.

Â So we've got dj over D. Well, these two things are going to

Â cancel the two djs. And then we've got a sum which is

Â proportional to the number of individuals that i is listening to, that's directly.

Â So we've got 1 over d summed that many times that's going to be di over D.

Â So in D, check, we've got back the solution.

Â So, in a situation where you put equal weight on all your friends and weight,

Â and listening is reciprocal, you get back degree centrality.

Â So, in that situation eigenvector centrality and degree centrality actually

Â coincide. Okay, so let's have a look at an example

Â now, and this is papered by David Krackardt where he looked at an advice

Â network in a company and a paper from 19 87.

Â And this is one where we've got a few enough nodes, and we've got information

Â so we can actually calculate out what the the S's, and in particular this is a a

Â picture of the network. Now, it's a directed network, so certain

Â people could actually listen to others without them listening back.

Â So, this would be a directed network not necessarily back and forth.

Â And there's some individuals who actually aren't connected.

Â So, some individuals are not getting listened to at all and their influence is

Â going to turn out to be zero. In this situation, so it's not a strongly

Â connected network. Nonetheless, the S is still going to work

Â as the the right answer to this. so if you go through and figure out what

Â the S is. So if you solve for the S of this, there

Â are some individuals, for instance 6 who didn't get listened to and 13 and so

Â forth, end up with 16 and 17, so some people end up with no influence, nobody

Â listens to them and comes to them for advice.

Â but basically we, we end up with the advice levels varying from 0 up to let's

Â see, we've got a 0.2 here. so we end up with different levels of

Â influence. And what the other columns in this table

Â represent is sort of, these are levels of the individuals in a hierarchy in this

Â company. so level one, this is the CEO, the head

Â of the company. Level two, we've got people at the second

Â highest management level. Then level three, is the third-highest

Â level. And what we see is actually there's some

Â people that are more influential in terms of this network than the, the individual

Â at the top, actually one of the people at level two has a higher influence vector

Â than the person at the top. And we can begin to, you know, look at

Â different people, different people in, at, at level three have different levels

Â of influence. And you know, you can look at that by the

Â different department, their age, their tenure, and so forth.

Â And this information is going to be complimentary to some of those things

Â that doesn't just necessarily correlate with how old they are or how long they

Â have been in the company or which department they're in.

Â These numbers are telling you something different about the, the relative

Â influence of these individuals have. so what that does is it just shows, you

Â know, one example of where you can begin to take this DeGroot model.

Â It gives us a foundation for looking at this particular left-hand side unit

Â eigenvector as a, a measure of influence. And you know, if you ran the DeGroot

Â process on this particular network. And had people updating over time in

Â their beliefs, that would tell you what their beliefs would eventually converge

Â to. And this was done under the assumption

Â that so we don't know exactly what these weights are.

Â That people put equal weights on each one of their friends that they set.

Â So person 17, for instance has you know five out arrows.

Â So they put 1 5th weight on each one of those.

Â