Okay, so want to take a lot at, at a structural model and fitting structural

models if, of network formation, I, and that combine aspects of both strategic

formation and chance meetings. And, the idea here is that, you know we

can build these models to explore the fact that in a lot of settings there's

going to be some choice involved but also some chance involved and we might want to

estimate some things like relative roles...

And you know, the the random models can be too extreme, the strategic models can

be too extreme. We seen the beginnings in terms of the

exponential random graph models of ways to combine some of these things but we

can also in particular instances fit models that are more precise to the

setting involved and more directed at asking a very specific question.

And so for instance, let's ask a question of, when we see homophily how much of

that was due to the choices of the individuals and how much of that was due

to the fact that you're more likely just to be meeting individuals of your own

type rather than choosing to interact with individuals of your own type...

So if we want to ask a question like that.

Can we build a simple model to address that?

And so here, what I want to emphasize is really the techniques for doing this

rather than a specific model. So this is going to be a very specific

and stylus model. But what the what I want to do is just

illustrate that you can use, you can do similar things to where you build what

you think is the right model for a particular application.

And then use that to generate networks. Look at the networks that come out.

Try and match them up with the data and that will allow you to fit parameters to

the model that best match the data and then do statistical tests to see whether,

you know, certain things are really going on.

how much choice is really going on. how much chance is really there.

How much noise is in the data and so on, so forth.

So that's the idea here. And so I want to emphasis basically an

approach rather then taking so seriously the specifics of this particular model.

It's more as an illustatration or an example then as as to be taken seriously

as the model. So in terms of application of homophily

Let's suppose that we've got two types of, two groups, Group A and Group B and

they form fewer cross say race relationships than would be expected

given their population mix. So if we go back to our add health data

and look at one of those high schools. And we see that, then we see a

segregation by race. We could ask is, is this due to

structure? So maybe they just don't meet each other

very often. They don't meet each other very often

because In the school, there's certain kinds of structural patterns in terms of

the way the courses are organized or the way that people will take extra circular

that don't allow for many meetings between different races, or is it due

maybe to the preferences of group A or the preferences of group B or both of

their preferences and so forth... So can we begin to sort these things out.

So, I'm going to just take a look at, at a couple of papers the techniques from a

couple of papers that are with Sergio Quarini and Paulo Pin, from nine 2009 and

10. And what we'll do is just, we'll specify

how much utility a given individual gets as a function of the friendships they

have. And then we'll allow a meeting process

that has randomness in terms of who you're going to meet.

And we'll allow this, both the utilities and the meeting process to depend on your

type, so in this case, say your race or your gender, or your age, or your

profession. Whatever, whatever it might be.

and then begin to see what comes out of that and, and try and match up the

parameters to the to the data. Okay, so let me say a little bit about

the idea here and, and, you know, when we're, when we're thinking about trying

to estimate strategic formation models Generally, what we end up seeing is, is

the result of some choices that were made.

And there's something that's known as revealed preference theory in economics.

Which refers to the fact that you know, we might see say a consumer buying

certain products. And then, based on the fact that they

bought one product at a given price and not another product at a given price.

We begin to try and infer what their preferences over different product

attributes are. So what do they really want if they ended

up buying something and not buying something else?

Okay, and so here what we'll do is we'll be basically inferring preferences by

saying, okay, this person formed these friendships and not the other, another

set of friendships. That gives us some insight into what

their preferences might be. Why did they form these friendships and

not those? Well, it tells us something about the,

what they preferred to form in terms of friendships now again that could be due

to what they have available. And just as in consumer theory you might

have a budget which says okay look these were the things I could've afforded, and

I bought this and not that. here what we're going to have to do is

just sort of infer what are the, what is the rate at which you had opportunities

to form different types of friendships? And so, the chance part is going to be

fitting what were the opportunities that were coming along and then what choices

were made as a function of those, and that'll give us information about what's

actually the, the preferences and, and what were, were the relative

opportunities that they had. So that's the idea.

One thing to emphasize here is this gives us, say, a different kind of look at

things than just direct surveys. So you might, for instance, ask people,

what's your attitude on race or would you like to form friendships across races and

so forth. And the difficulty with asking people

directly is that people often answer in ways that aren't necessarily congruent

with the choices that they make. So this takes seriously what did you

actually do, not what you would say on a survey.

And sometimes there can be differences about this and so this is a different way

of sort of measuring attitudes towards, you know things like race, or gender, or

age, or whatever it might be in that particular context.

Okay, so a simple model. what we're going to have is some set of

types 1 through k, so this might be ethnicity, it might be the, the age of

the individual, it might be a combination of their age, their religion, their

gender, etc. and what we'll have is a very simple

model in terms of the preferences that people have.

So, this is going to be a simple independent link formation model.

So, it's going to be simple in that dimension.

It's not going to be trying to recreate richer parts of the network but it's

going to allow to separate out some of the preference aspects from, from some

other aspects. And so what people value, is they care

about how many same-type friendships they have, and how many different-type

friendships they have. So, really simple model.

You just care about how many friendships do I have with t-, people that look like

me, how many friendships that I have of people that are of a different type, and

I get some benefit from just that. Okay, so very, the simplest possible

formulation you can imagine. And in particular, what you get in terms

of utility, is then some number of, of same and different type friendships

weighted by a parameter, gamma i, where gamma i is capturing how much do you

weight a different friendship compared to a same type friendship, okay?

So, it's a preference bias. If this was 1, then all I care about is

the total number of friendships. I don't care what their mix is.

If this is bigger than 1, then I actually care for diversity.

I care more to have friendships with other types than same types.

If it's less than 1, then I get a higher benefit from same-type friendships than

different-type friendships, right? So.

Gama i is going to be the critical perimeter.

In terms of representing preference bias. And then we also have this other

perimeter. Alpha.

And what is Alpha going to keep track of. Alpha, is going to be generally less than

one. Is going to be some diminishing returns

to friendships. So my first friendship might be very

valuable to me. My second one additional value and so

forth. By the time I get to my 10th 12th

etcetera these friendships are becoming less valuable and so the fact that alpha

might be less than one would give a concave function.

So as you look at at the utility as a function of total numbers the utility's

going to tend to be concave if alpha is less than 1.

So we've got a situation where, as alpha's less than 1, then we've got

curvature in that utility function. Okay.

so let's let t i be the total number of, of friendships that we're forming.

And, so basically, people are socializing, they have an opportunity to

form friendships. They meet people of different types, and,

in this model let's let qi be the fraction of own types that you're going

to meet and be able to form friends with. And then 1 minus qi is the relative

number of other types that you're going to form.

And so if you spend, if TI is the total number of friends that you form, then the

relative number, this is going to be your S size, is going to be, the fraction that

were of same type and, the DI is going to be the fraction that were different type,

times your total friendship. And so here in this model is a very

simple model I just have opportunities coming and the cost is just going to be

its going to be costly for me to form some number of friendships I'll cut off

that total number but then the mix I get is just going to depend on the relative

meeting rate. So I, I meet people at some rates, and I

take whatever friendships come, but it's expensive for me to form friendships, and

so after some time period I stop socializing or trying to find new

friends. Okay?

So ti is just going to maximize, it's going to be a maximizer of this overall

utility function. Where you've got, same type firend,

different type friend, and so forth. And the rate at which the come is Qi for

same type, 1 minus Qi for different types.

And that's going to be coming out of the random part of the process.

And right now, then what we could do, is say, we can figure out if we knew what

gamma was, and was alpha was, and what Q is, and C, and so forth We could solve

this function and say, how many total friendships would a given individual like

to have? And how would that depend on those

relative parameters, okay? Okay, so to maximizes this function.

If you solve that, you can get an expression for what ti is in terms of the

over the other parameters. So, maximizing that function.

take the derivative with respect to TI, set it equal to zero.

Alpha's less than one. This is necessary and sufficient for the

solution. and then we'll also add some noise to the

given decision. So it might be that a given individual

for whatever reason has more or fewer opportunities or more or less values.

So they're going to, we're just going to add noise in terms of the, the

friendships that have given individual forms so a person a of type i is going to

have an extra error term, epsilon a and so the total number of friendships of any

given individual forms is just going to be some noisy thing about this solution.

Okay. So very simple model in terms of the the

formation. But now we can see if we write down a

simple model of How much utility you get from some aspect of, of the network.

we maximize that. And we get a solution for what.

How many in this case. What degree.

So we can think of this really as the degree of aging eye.

This is the degree that they would like to have.

In terms of this model. and then what they end up with is some

noisy variation on what they would like to have.

given the parameters of the model. Okay so how do we actually identity the

parameters from the data. so what we can do is in the data we'll

actually observe the ti's the tai's so we'll see, how many we'll use the add

health data. So when we look at the actual networks of

friendships in these high schools we can see how many friendships did each

individual form. And so we observe this directly in the

data, and that's going to vary with the qi's, so as a function of qi, the

function of the alphas, the gamma i and so forth.

That gives us a tai. And so one thing to notice, is that when

we look at this expression for tai, This is increasing in q i if gamma is less

than one. Right?

So, if gamma is less than one, then you've got a plus one q i and then you've

got minus gamma. So, you've got q i Times 1 minus gamma i,

in here. And so if gamma i is less than 1, then

you've got a positive expression for t a, t is a function of q.

So, more of my if the fraction of people I'm meeting is more of my own type, I

should form more friendships And so that's what's going to allow us to begin

to fit what gamma i is, right? So the idea is t i should be a function

of q i, and how quickly it varies with q i is going to be dependent on what gamma

i is. Okay?

And, in particular, if you actually look, this is a picture of the add health data.

So these high schools here there are 84 schools.

And each dot here represents a certain race.

Group within a particular school. So for instance, this dot here is a group

of white students that formed. So it was a particular school, and in

that school the white students formed about a little over between 60 and 65% of

the population. this school right here, this is a, a

group of black students in a high school where their groups' size is a fraction of

the school was just below, between 20 and 30 percent, a little closer to 30 percent

and this then tells us on average how many friendships did they form?

They formed on average about eight friendships.

This group formed on average you know about three and a little bit of change,

and so forth. What we see here is that indeed if you do

just the slope between here, you see that the slope is 2.3.

So, there's an increase as a function of your group size.

So the more prevalent your group is in the population ,that's going to lead to

higher qis. And indeed, we see that there's a higher

group of friends as, higher friendship, as a function of the the size of the

group. So the easier it is to meet your own type

the more friends different groups are forming, and so we will be able to

actually identify that gamma perimeter from this data, and in particular when

you look at this thing, you know the slope here is 2.3, the T statistic on

that is 7.3, so... You're you're quite a number of standard

deviations away from from zero so so we're actually seeing a highly

significant slope here. So we will be able to identify the fact

that groups that have higher proportions are forming more friendships which would

indicate that they're getting higher utility under this model.

and we can then estimate what the gammas are based on that.

And in particular I'll just sort of you know, show you the best fit lines.

If you look at the best fit lines for different parameters, you'll for

different races you'll end up seeing different slopes, and that'll allow us to

back out gamma-ise the gammas for different Races, because each one of them

has, having a different relationship between how big their size is, their

group size, and then how many friendships they are forming.

Okay. So the last part of the puzzle in terms

of figuring out the randomness in this kind of model is where do the Qis come

from. So we've got the, how many friendships

each... Person would want to form as a function

of the parameters of the utility function and the rate at which they meet different

individuals, but now we want to ask, what's the rate at which they meet

different individuals? Okay, and the important thing here is

that the rate at which they're going to meet different individuals is going to

depend on the decisions of the other agents.

Agents, okay? So if everybody was trying to form the

same number of friendships, and we're just sort of mixing in the population,

then if my group formed 30% of the population and some other group formed

70% of the population, then I would meet my own group at, at a rate 30%.

And I would meet agents of different types at a rate 70%.

But if, the other types, imagine if the other types are actually trying to form,

they form many more friendships. They're spending more time circulating

and mixing, then they're going to be easier to meet and my type is going to be

relatively less likely to meet, and so it's not just...

a function of the relative sizes, it's also a function of how many friendships

different groups are trying to form. And so we need to solve this overall as

an equilibrium given that the, the, that the t's are going to be determined by

these relative rating, the q's, and the q's are going to be determined also by

the actual decisions of the agents. So in particular let's think of the

meeting process and we'll think of this as a giant party.

So we can think of this like a cocktail party.

So let's think of a different given individual say is a green agent.

And this green agent is bouncing around in a party where there are green agents

and red agents. And so what's going to happen.

imagine that the incoming proportion of reds is 80%.

And green's it is 20%. But if the red's spend more time, trying

to form friendships. And are generally forming more

friendships. It's going to be easier to form

friendships with red's then green's. And so even though it say let's say .8.

0.2 coming in. The mixture in here could be, say, 90%

10%, or, or even more skewed than that if the reds are spending, say, twice as much

time in the, in this party than, than the greens are.

So, the rate at which they come in and, and go out, is not necessarily going to

be the same as what the relative stock of people is, if the greens are exiting much

more rapidly than the reds are. So, as we go through this process then,

you know, this group, given green node bounces in to somebody, meets one

friendship, meets two. Three, four, so it's got three red

friends and one green friend, and it decides, okay, that's enough, I'm

satiated. You know, I formed four friendships, and

that's enough for me. And then it decides to exit.

a red might find this to be, if, if gamma i is less than 1, then reds are meeting

reds at a higher rate. They might want to stay longer.

And that's basically the idea of the model.

Ok? So we've got this q i is the rate at

which i meets i. one minus q i the rate at which you meet

the different types and the way in which this is going to be modeled is the q i

the rate at which you meet your own type in terms of this process Is going to be

dependent on the stock, how many of those individuals are actually in a room.

But will also allow this to be biased, so that even when I'm in the room, it might

be that that I'm biased in terms of meeting my own type.

So maybe I'm in this large room, but I actually look for greens and try and find

greens. In which case I'm going to meet greens at

a faster rate than actually they're, they're in the room.

And so, if beta i is exactly equal to 1, then the rate at which I meet people is

just, what's the stock of these people in this party.