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

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

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

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

Â wrapping up the strategic network formation.

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

Â of cost structure. so cost of forming relationships can

Â depend on geography and characteristics of nodes.

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

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

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

Â knowledge. So that people with similar

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

Â forth. there could also be complementarities in

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

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

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

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

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

Â length. And at the same time have high

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

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

Â basic intuitions before we get into the details.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â Okay, now the, the island here are metaphors.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â from their connection to somebody to away.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â your island. High cost across islands means that you

Â only want to have limited number of connections across islands.

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

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

Â connections. Delta is reasonably large, 0.95, then

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

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

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

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

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

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

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

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

Â somebody else in this network is 4, right?

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

Â kept enriching this mar to have more and more islands.

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

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

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

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

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

Â forth. Which won't end up matching reality.

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

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

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

Â 7:58

so you can go through and, you know, prove things about this model, so in the

Â paper Jackson Rogers 04. We proved some things about this model so

Â here, this is a, a. First of all, you can truncate the

Â connections model. So, you only get value as long as you're

Â within some distance, maximum distance of people.

Â So you don't get infinite if, if I'm at distance of 50 from somebody, I don't get

Â any value from that. So, you can put in some cap d, say for

Â instance a value friends out of a distance three or a distance four et

Â cetera. Then you can go through this Islands

Â model and basically you can show that if the little c is small enough.

Â And the big C isn't too large, then players on each island form a clique.

Â You get a bound on the diameter. So here you're going to get clustering

Â from the, within island connections. You're going to get a bound on the

Â diameter, and if the big C is large enough.

Â Then you won't have too many inter island connections, and you can get a lower

Â bound on clustering. which, is depends on the number of

Â islands and the size of each island. So basically what you can get is, is you

Â know, some proof that this is a process, uh,[INAUDIBLE] .

Â A set of properties which'll hold four parameters values within this kind of

Â geographics connections model. The important thing to take away from

Â this is, is now we see that, we're getting clustering because it's cheap to

Â connect to people who are close by. And we're getting low diameter because

Â it's, there's a high value to connecting to people who, to whom you have only in

Â very long indirect paths. And so, the diameter's going to be

Â limited just to the fact if, if there was too many missing connections then it

Â would pay for somebody to add them. Okay, so in terms of the summary of the

Â strategic formation we've gone through so far we've got efficient and stable

Â networks need not coincide. even when some transfers are possible and

Â with complete information. The details of this depend on the

Â setting, which kinds of transfers we might make.

Â we didn't talk too much about forward looking, but that's something that you

Â can add to these models. and you can match and explain some of,

Â observables, with these kinds of models. so in terms of the strengths of the, of

Â this kind of approach. a big part is that the payoffs allow us

Â to have a welfare analysis. So, we can say something about which are

Â good networks, which are bad ones. And we can identify trade offs between

Â individual incentives to form relationships and societal goals.

Â And so we, we, we end up with is, is a real understanding of whether or not, the

Â process of, of, networking formation that's out there is leading to good ones

Â or bad ones. And we're also tying the nature of the

Â externalities to network formation. So are there positive extrenalities, are

Â there negative ones, how does this depend on context.

Â and so when we also end up accounting for some observable facts like clustering and

Â the low diameter. there's an answer of, of why this might

Â happen because of, of certain features rather than sort of a, a mechanical model

Â which imitates it. We have an idea of, of, what kinds of

Â fundamental assumptions about human behavior would lead to this.

Â 11:25

Now of course the, the problems with the economic approach that we've talked about

Â so far. is that the, a lot of the models we've,

Â we've looked at in order to solve them analytically, have been very very simple.

Â So they tend to be very stark over the regular lots of symmetry.

Â And we want to add hydrogeneity to those. if, if we want to enrich those models

Â we're going to have to add something, add some heterogeneity.

Â simulations can help then If you want to take these to data.

Â You can enrich the models, simulate them and see what happens.

Â there's also a question of whether or not, you know, things are basically

Â currently at random or whether people are really making determined choices.

Â And I think depending on the application, thing might be swayed very much toward

Â individual stategic choices. And in other applications it might be

Â very much at random. And so depending on the application you,

Â you might want some mixture of those two. in, in terms of, of applying these

Â things, one challenge is figuring out what the payoffs are.

Â So what are the payoffs? how do we relate network structure and

Â outcomes to payoffs? How do we identify that?

Â That's not an easy question, and it depends very much on the context.

Â So you have to really begin to think about what is it, what is it that

Â motivates people to form relationships? Why do they maintain certain kinds of

Â relationships? What are they, what are they getting from

Â those things? What's influencing their behavior.

Â So that's an important element that needs to go into these kinds of models.

Â So, models that start to marry the strategic random network models that

Â we've seen before. And the, you know, put these two types of

Â models together, are, are really needed. the strengths of the random network

Â models, the, in terms of being able to match data or, or fit to data.

Â are, are, are to some extent the weaknesses of the economic approach, and

Â vice versa. So we've got basically two sets of models

Â with very complimentary types of, of properties.

Â And mixing these together would then allow for us to do this kind of welfare

Â and efficiency analysis. Understand why things are happening, take

Â the model to data and do so across a wide range of applications.

Â these kinds of models are being developed.

Â we'll talk about some of those in, in some other videos.

Â