Hi folks. So we're back and we want to talk about

enriching some of the strategic formation models to be have a little more

heterogeneity in them. So that we can help expalin some observed

fact and data. And so we're still in the part of

wrapping up the strategic network formation.

And in particular what we're going to do is enrich things basically just in terms

of cost structure. so cost of forming relationships can

depend on geography and characteristics of nodes.

So, it's easier to be friends with somebody who lives very close by.

It's easier to relate to people with similar backgrounds.

you could also imagine that the benefits would depend on the characteristics of

knowledge. So that people with similar

characteristics find it easier to work together or share, share risk and so

forth. there could also be complementarities in

benefits from diversity. There's a lot of different ways we can

enrich these models. We're going to do this in a very simple

way just to get some ideas out. and the idea here is, is that we can get

the so called small worlds. observations out from cost benefits, so

we want to get simultaneous effects networks tend to have short average path

length. And at the same time have high

clustering. And so we want to look and see whether we

can explain that with strategic model. And let me just give you the, the, the

basic intuitions before we get into the details.

the ideas here are going to be that effectively, the fact that there might be

very low costs to linking to people. Who are very similar or very close by is

going to give high clustering. So we'll get very dense networks on the

local level, just because those relationships are easy to have.

high value to distant connections means that then we'll have a low diameter.

So if I'm not connected to somebody at, at a great distance.

I'm not accessing part of a network that's far away from me and forming

relationships with people. Who, who are distant can give me access

to a lot of information or people I don't have access to before.

So that tends to give high benefits to those which help string the diameter and

the high cost of distant connections means you're not going to have too many

distant links. So you'll have high density on a low

level, a local level. a few long distance links so that's not

to diminish the clustering too much but you'll still have a lower diameter.

Because people will connect far away if you're not already connected.

So that's the basic idea, and let's just go through the logic in a little more

detail. So there's a whole series of models that

have basically looked at the variations on the connections model, where geography

is added in some way. And what I'll do is just take you through

one version of that model, where we have people living on an island.

And people that live on the same island can connect to each other very easily and

there's different islands. So there's, so there's a, a cost little c

for connecting to somebody who's on your same island, and a cost big C to linking

to somebody on another island. But then the benefits that deltas and so

forth are exactly as they were in the original connections model.

And what this will do is give us high clustering within islands, few links

across islands. But we'll still have enough links across

islands to have small distances at least for some perimeter values.

Okay, now the, the island here are metaphors.

For it could be geography but it also could be characteristics so people with

very similar characteristics find it very easy to link to each other.

People with different characteristics find it more costly so the islands are,

are metaphor but a fairly obvious one. Okay so let's have a, a peak at, at some

versions of this. So imagine that we look at a given node

here, in, in network, like this, where we have a, here the five individuals.

in each group here on islands, so these are different islands.

So the J the number J here is equals five and we also have five islands.

So we have five islands and five individuals per island, and we can go

through and look at the, the value to a given individual from their links.

So for instance if this individual is in this particular network, what's their

payoff? Well, there, they have four little c's

because they're connecting to the people on their own island.

They're also getting four direct delta's. they've got a delta squared which comes

from their connection to somebody to away.

They've got seven delta cubes from people three away and 12 delta to the fourth

from people at a distance of four. Okay so, this is the similar to the

connections model but now what we've done is enriched the cost structure, to have

this geography involved. Okay so what, what depending on whether

you have only distant connections or only close by connection or some combination.

The pay outs are going to differ. So in this case, we can see that this

individual here is maintaining a connection to somebody on another island.

They're paying a large cost for that but their seeing additional benefits.

And lower distances than an individual who's not connected across the islands.

So there can be incentives for somebody to connect and also if that person was

not connected then nobody in that island could access anybody on another island.

So, so, if that person was to sever that link, they would lose connections with

the other islands. Okay, so basically what happens here, low

cost to an island. means that you want to connect within

your island. High cost across islands means that you

only want to have limited number of connections across islands.

So here is a situation where for instance if the little c is below 0.04 the big C

is bigger than 1 and still less then 4.5 so you still want to have some outside

connections. Delta is reasonably large, 0.95, then

this is a pairwise stable network. So you can go through and check that

nobody wants to delete a link. And no two individuals who were not

linked would want to add a link, so you can go through and do all those checks

for these parameter values. hopefully I didn't make an error on that,

but you can go ahead and check that[COUGH] and here what we end up with

high clustering and low diameter. So we end up with high clustering given

that many individuals have all of their friends talking to each other.

and, we end up with low diameters because the greatest distance from somebody to

somebody else in this network is 4, right?

So the diameter here, is, is 4. And, you know, so, so, if you, if you

kept enriching this mar to have more and more islands.

you would end up with, you know, very large number of nodes, relatively low

diameter. so in this case, we get high clustering,

low diameter. you know, obviously, this is, is not, is

still a very stark model. It ends up having very particular stable

networks that are going to have certain kinds of regularities and degree and so

forth. Which won't end up matching reality.

But what it does do, is it gives us a different explanation and reasoning

behind why you might see small worlds. And we can begin to, to enrich this kind

of model with some random formation to begin to try and fit things to data.