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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.

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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.