So, there's an Individuals and what we're going to work with in terms of a network

here Is actually going to be weighted and directed.

And, it's going to be a stochastic matrix.

So we'll, we'll work with the notation T as sort of a, a trust matrix.

So here what's, what individuals do is they have some, these are going to be

weights that I put on different individuals when updating my belief.

Okay. And so what I do is person I, everybody

starts at time 0 with some initial belief.

Right? So we've all got whatever information we

had from the past, whatever experiences we have.

So we all start with some prior, big question, will there be a recession next

year. Or, is there global warming or you know

is this politician a good politician? so we're all start with our beliefs and

we're going to put these in zero one. You could have these be vectors, you know

the model actually extends quite naturally to have multiple dimensional

versions of beliefs and Beliefs on many things and, and so forth.

We'll just work with a simple case where you've got a belief and we'll keep it in

0, 1. So this is my belief of what's the

probability that there is global warming. Okay, now the belief at time t that I

have is just going to be a weighted average of the beliefs of my neighbors

and my neighbors, this is captured through the Tij, right, so I put some

weight, person I puts some weight on j's belief, which is captured by Tij.

So stochastic here, is telling us that the sum of Tij's when we sum across J, so

this thing is each one of the Tij is not negative so I can't put negative weight

on equal, I put some positive weight and out of the people I listen to I decided

I'm going assign a total weight of one. So it could be that in this model, it

could be that Tii is positive right, so I put weight on my own past belief.

somebody that never listens to anybody would have Tii equal to 1.

Right. That would be that I just listen to

myself, I never pay attention, my belief just stays what it is and you can't

convince me of anything and I'm extremely stubborn.

but if I listen to anybody else then I'm going to put some weight on their belief,

some weight on my own belief. And what people were doing is forming a

new belief by repeatedly talking to others.

And incorporating that and, and updating their beliefs.

Okay. So, a very simple, natural updating

process. So, let's take a quick look at an

example. So let's suppose that person one listens

to everybody equally, right so a third on everybody.

So, here we see person one putting weight one third on one, one third on two, one

third on three. Person two puts weight a half on one and

a half on two. So weights one and two equally, but

doesn't listen to three at all 3 listens to 1, and herself so we get half on on, 1

half on 3. Okay.

So that would be the t matrix associated with this and sometimes it's going to be

useful to keep track of the diagrams in terms of the cycles and so forth and here

we can that you know, now we're going to have self links, self loops.

So some people are listening to themselves, and we've got waits.

So now we have a weighted directed matrix, and uh,different individuals can

pay different attention. This is one where each of the individuals

happens to put equal weight on each of their friends.

You don't have to have that, we'll look at different examples afterwards, but

this is a, a simple starting point. Okay?

So now what we can do is begin to see how updating works under this.

So suppose that, for instance, the initial beliefs of the three individuals

were person 1 started with a belief of 0, person 3 started with a belief of 0, and

it was only person 2 that started with a positive belief.

[COUGH] Okay, so person one puts an equal weight on each of these.

So they are going to weight a third of zero, a third of zero and third of one.

Their next period of belief, so at time zero, they have this, belief of person

one. At time

One, is now a third. So after one iteration of updating,

they've switched to a belief of a third. Okay.

So you can just do this for each individual.

This person is waiting a half of a one and a zero.

They're going to go to a half. This person just saw two zeros, right?

So they're still stuck at zero. So they didn't update at all.

And we can see that the amount that the people updated depended on which weights

they had and who they were listening to. And, but this person's belief now, in

the, in the second period Now becomes positive because at this point now, you

know, it took one period for this person to start having a positive belief.

Now, once this person has a positive belief, this person's belief starts going

up and what we can see is, through this averaging process people's beliefs are

going to be tend to be pulled towards each other.