0:40

The difference is really going to be in the application.

So this has to be a setting where I can really connect to somebody without them

having to want to, to allow me to connect to somebody.

So it does work in settings like citing somebody's article.

I can do that without their permission, or linking to a webpage.

I can follow somebody on Twitter and so forth without them having to follow me.

So there are setting where you can have this kind of diretic, directed setting.

And so what we'll do is we'll model that as a very simple situation where people

just announce their preferred set of neighbors.

And then a network forms if and he based on which ever links people want to form.

And here we're going to keep track of ordered pairs.

Now, so the fact that ij is in this network, ij is now different from ji, so

this means i is following j. this means j is following i, or, or i

formed a direct link to j, and, and so forth.

And then just look at the Nash set of networks, where each person is forming

some links to some the links that they want.

Given the links that everyone else is forming.

[BLANK_AUDIO] Okay now when we think about these kinds of settings, we have to

think about the flow of payoffs too. So one situation is sort of a one way

flow. If I follow somebody on Twitter and they

don't follow me, then I get to hear what they say but they don't hear what I say.

so that's a setting where the person who pays the cost then hears information from

the, the node that they're accessing. Now two way flow could be that one player

forms a link and bears the cost but both benefit from it.

So it could be that for instance if I add a link to another page on the Internet,

then that's good for me because people can get from my page from the other.

It also helps the other page get new traffic that I direct there.

phone calls, you know, they, they, the, there's you could think of this, as

somebody bears the cost. two people end up but phone calls

actually are across to both people involved if you think about time and so

forth. So there's some situations where we could

think about two way flow. Now the two way flow, there's a paper by

Bala and Goyal in 2000 basically did a directed version of the connections

model. So the model we can think of as being the

same as what we saw from the Jackson Lewinsky version.

But now what's going to be different is instead of having consent to form a

relationship. People can form a relationship

unilateraly. And I can just direct my link to somebody

else. We'll both end up benefiting, so the

benefits will look just like the the connections model we had originally, but

now people can form a link unilaterally. So it's a different formation process but

the same payoff structure. so the person, the benefits structure,

the cost structure is going to be different.

It's only the person forming the link that bears the cost.

So if you just want to go through in terms of efficiency.

The efficiency's going to be exactly the same as before.

Except now that we have half as much costs being born, so instead of having 2c

per link, we're going to have 1c per link.

And then in terms of what that does for the efficient calculations, is it just

factors everything down by a, a factor of 2 in terms of the costs.

So the efficiency in this model is exactly as it was before except for we've

just divided through by 2. and so you get complete networks if costs

are low enough, star networks in the middle range and the empty network when

costs are high enough. One thing here is I put complete and star

in quotes, because complete doesn't mean that every link in both directions is

present. It just means that every two nodes are

connected, and in this case, they're connected by only one link.

It doesn't make sense to have links in both directions because you bear twice

the cost and there's no added information flow in this situation.

So, if you, if you go through this, it's, it's similar architectures.

slightly different cost structure. Somebody has to bear the cost.

But you only want one person bearing the cost between any two individuals, because

things flow in both directions once that's formed.

So efficiency's exactly the same as it was before.

when we begin to look at the Nash stable networks, I'll sort of list through what

you, what you can find. You can check this.

so if, if cost is very low, then the sort of complete network will be Nash stable.

So basically it makes sense to cut an indirect relationship to a direct

relationship. And somebody, if, if the other person's

not doing anything, the other, you know. If one person's not doing it, the second

person will have an incentive to do it. Medium cost range, medium low, all star

networks are Nash stable, plus some other networks.

Whereas a star now could be formed in, in different directions, so it could be that

the centre is forming some of the links. And peripheral agents are forming some of

the others, so you can have a star with multiple directions on the links.

again, you know, it doesn't make sense to form links in both directions because it

doesn't add any benefits, and adds to cost.

so you'll see stars that are going to be Nash stable, and depending on, the

configuration. it, it could be that center's bearing

more cost or the periphery is bearing more cost, there can be combinations of

different types of stars. So any star in that network is going to

be Nash stable. when you get to a higher cost so that c

is bigger than delta, now let's think a little bit about comparing two stars.

So if we look at a situation where we've got a, let's look at two extremes.

One extreme is where the periphery formed a link.

So they all link to the center. Okay.

7:12

Well what this says is this one is the only one of the stars that's going to be

Nash stable. And why is that?

because if you think about it, here the center is bearing a cost by connecting to

somebody, and they're only seeing one delta benefit.

So that's less than the cost, they would rather sever that link.

So they're in a situation where they don't want to be maintaining

relationships to other peripheral agents. If the peripheral people don't bring them

indirect benefits. This one is stable.

Why? Because by accessing the center these

people are getting all kinds of indirect benefits of paths of length too right.

So they've got a bunch of indirect they get a delta plus n minus 2 delta squared

by forming the one link minus the cost. So that peripheral agents are willing to

keep those things because they're they're getting this extra cost.

when we look at a situation where the, when the cost becomes high enough, , then

we end up with a situation where if things can be, have complete networks, be

efficient, but not equalibria. So,Nash stable networks are going to

depend on the exact cost structure. What, what's a little bit different here

then wha, was before is that the, the relative costs are going to select out

who might be forming the links. And so you can have some prediction about

links going in one direction or the other, even though flow might go in both

directions. Okay?

So let's talk briefly about an, another version of a directed model, and this is

a one way flow model. So now we have to keep track of directed

flow. And this is when I form a link that I can

access this person, I can listen to them and I can listen, I hear things

indirectly that they listen to or here. So if I connect to somebody and they're

connected to somebody else and so forth. Then the benefits flow back to me from

these connections. But the fact that I connect it to them

doesn't give them this person any benefits unless they've also connected

back to me or have some indirect connection back to me.

Okay, so thing flow in the direction and so the person who bares the cost also

gets the benefits. The other person does not get the

benefits. Okay?

So one example of this is a directed connections model with no decay, where

the delta is basically 1. And in that situation then, the utility

becomes just the number of people that you can access via directed paths, minus

however many links I formed out, outward. So what's my outward degree in the

network times the cost, right. So I'm bearing a number.

So if I have three different links then I'm going to have an out degree of 3 and

how many people I reach is going to depend on how many people they're

connected to. Right?

So in this case, it might be that I can reach, 1, 2, 3 directly, plus 4, 5, 6, 7,

8 in total. And, so my reach would be eight.

Okay, so in that situation, then you've got some number of people that you access

for your, your links. Minus your degree times the cost for

degree is now measured in outward degree. Okay, so, very simple payoff structure.

And efficient networks in this world, as long as c is less than n minus 1, are

going to be wheels. In particular, wheels that have a

particular direction to them. You should be careful because those

wheels also mean something else in, in graph theory.

But Bala and Goyal used to work wheels so, basically this is a directed graph

where each node points to a subsequent node which points back.

And so, each one here can access all of the others.

So if you want to think what's the best way to do this in so that each node can

access every other node. So we got maximum reach, everybody's

reaching n minus 1. We're doing this with only n links, so

everybody's sponsoring one link and we're getting the maximum reach possible in

this world. Right?

So, and each, each link is responsible then for connecting a given individual to

n minus 1 other individuals. Right?

So we, we've got a situation where we have the most efficient architecture.

possible being a wheel. Or if the cost goes above this, then it

doesn't make sense to have any links. And, and then you're better off with the

empty network.

[BLANK_AUDIO]

no stable networks when cost is, is low enough.

then the wheels turn out to be the only Nash stable networks that are strict Nash

equilibria. So that everybody has a strict incentive

to keep the the nodes that they links that they have.

12:35

if, if the cost gets to a higher range then, wheels and empty networks are the

only strict, Nash stable networks. So then you get stuck at the empty

network, because you, nobody wants to form a ring to somebody else unless, they

bring in direct benefits. so, so basically what we've got here is a

situation where again we have some conflict between stability and

efficiency. You can go through and convince yourself

of, of why these things are true. the strictness is important here in

getting these things. There's lots of, of networks which are

Nash equilibria that are not strict Nash equilibria.

So for instance here we've got a situation where this is Nash stable.

You can check that nobody wants to delete any link that they have, so for instance

2 accesses 1 and also accesses 3 and 4 via this link, so they're happy.

3 can reach everybody via the links, but they have to have 2 links because they

have to reach 4 as well as, as 2 and 1, and, and so forth.

You can go through and check this. But, 1 is actually indifferent between

where they place this last link, right? They're indifferent between having it

here. They could also put it to 4 instead.

And they could access everybody through either connecting to 3 or connecting to 4

since there's direct links in both directions between 3 and 4.

Okay? So this is Nash stable, but not strictly

Nash stable. So the if you check on the wheel, if we

go back to the wheel then that's a a situation where nobody.

So if we did this one in terms of a wheel, now anybody who changes the the

links of severed this one and put it somewhere else.

You would actually lose access to somebody.

So they would no longer be able to be given the direction of the network now,

the one way flow part they would no longer be able to access that node.

So strictness is important in, in making those results come out.

Okay, so that's just a glimpse at directed network formation.

Theres going to be different applications, and I think that's one

thing that's important to emphasize here is that which.

Model that we should be using has nothing to do with whether we like unilateral or,

or mutual consent formation as a, as a model.

What's important is, what does the application actually demand?

So, if we're dealing with situations with alliances, or friendships, or social

relationships. A whole series of things you really need

to have mutual consent and, and two way formation processes.

If you're doing things like citing an article, forming a link on a web page.

then you can deal with unilateral network formation, and so it's not a question of

which is a better model. there are different models and they fit

different applications. And some network settings are ones that

are naturally directed and unilateral. others are ones where mutual consent is

really needed, and a lot, a lot of social settings are going to fit into that

category. And so which model you use really depends

on the application. It's not an issue of which one's you

know, sort of a, a nicer model; they're, they're a different models.

Okay. so that sort of takes us through a lot of

modeling. Of, of, of strategic network formation

and some basic looks at different issues. And we'll have a couple of looks at

models for fitting these things in in dealing with data.

And then we're going to turn to diffusion as the next major topic.

So we'll start working. This has been, so far, we've been looking

at network formation, and the next major subject is now, given the network, what's

actually happening on that network? How do we understand behavior on that

network? And what the consequences of different

network formations for the actual behaviors that result.