So in fact what we can say is the action that any individual taken one of these

linear quadratic games of complementarities is something which is

proportional to their Bonacich centrality.

So, higher Bonacich centrality, higher actions, okay.

So we've got everybody takes an action, A over B to begin with, which is just sort

of what they would in isolation with no network.

And then the extra network effect adds in these complementarities.

And how much extra action they get here depends on their Bonacich centrality in

the network. So we get a natural feedback from

complementarities. The actions relate to the total feedback.

Centrality tells us relative number of weighted influences from one node to

another, then captures the complementarities.

And why is that working? So again, you know, these things we're

measuring sort of. How much do I get influence from other

people? From other, from their friends and so

forth? That's exactly what's happening here.

How much is their action influence their, my friend's actions, which then

influences my action, and what do the feedback's look like?

Okay, so we've got this nice solution. So the beauty of their model is that you

end up with the very simple expression for X.

This scales with A over B so this is just multiplying everywhere so we can just

rescale and eliminate that. so, if we think about the Gij is equal to

Wij over B. we, let, let's think of a simple world

where, you either connected to an individual or not.

And then, effectively, the main thing is, you know, who you're connected to, and,

and what's the size of B. And then that will give us a calculation.

And you can directly estimate these things.

so for instance, if we, if we do that calculation, you can do that calculation.

Here's on one network that, for which they did these, these calculations.

You can do it in different settings. so you know, depending on whether, what B

is, if B is 10, that's sort of relatively high cost to taking actions.

Then, what do you get? You get that, a person in the center

position takes an action of 1.75. This person takes an action of 1.88.

This takes 1.72. These people are all, right, this is

going to be a 1.72, 1.88, 1.88. So, depending on how many neighbors you

have and how central you are in this case, the highest action ends up being

for these individuals in, in this position.

if you rescale the B and, and change the B to a different level you get slightly

different numbers. basically, you know, here, you can redo

that for B equals five. So if you lower the cost, people's

actions go up and it more than doubles. And it's more than doubling because

you're getting a feedback. So everybody wants to put in a higher

action but that means that their, their neighbors want to take higher action so

is that an even, increase is there more. So if you hadn't been doing this with the

with the neighbor feedback, you know, just taking the cost in half would have

doubled the action. Now with the feedback we, we get an extra

effect. And indeed for this particular example,

you're still going to get you know, similar structure, but much higher

actions. So what's nice about this model is it

gives us predictions of exactly who's going to take which actions as a function

of their position in the network. And now we've got something which begins

to give us some feeling for why Bonacich's centrality might be an

interesting centrality measure. Its coming out, and its givings us some

idea of what the feedback is and complementarities, and it gains a

strategic complementarities that can be important.

Okay so that takes us through another game with this kind of feedback.

Now you know, the, the nice part about this is that it's, it allows one to do

calculations in terms of a simple network measure.

it's going to be more difficult if we wanted to add in a lot of [UNKNOWN] on

nodes and have different nodes have different preferences.

but we can enrich these models in, in ways that, that allow us then to take

them to data. And, indeed, people have been starting to

work with these models and started to do analyses of, of what predictive behavior

is, looks like, as a function of the network.

And then actually seeing whether that gives us some insight into what's

happening In, in different settings.