Hi. Welcome back. We're in our last lecture on networks. Remember we've talked

about the structure of networks. Things like their degree, their path length,

their clustering coefficient, and we talked about the logic on network's form.

In this lecture we're gonna talk about the functionality of that structure. So, when

a network has some structure to it, it has some degree distribution, it has

connectedness, it has a clustering coefficient, and what we can ask is, how

do those properties of that network allow it to carry out different functions? Now

remember when we think about those functions, they're typically emerging.

When people form an. But they're not thinking about the entire network. They're

just thinking about their own connections. So those properties the network structure

itself just emerges from the logical process through which it forms and we

wanna talk about how that structure. Has functionality so it can do particular

things. We're gonna start out by talking about something known as the six degrees

phenomenon. And let me explain where this comes from. It comes from two famous

experiments in social science. So Stanley Milgram, in the'60s, asked 296 people from

Nebraska to get a letter to a stockbroker in Boston. Now, the rule is, they could

only send the letter to someone they knew on a first name basis. And what he found

is that, on average, of the letters that got there, it took about six steps. Now,

Duncan Watts, you know, almost 40 years later, redid this experiment with 48,000

people on the internet. And they had to send an email to someone they knew on a

first name basis. And try to get it eventually to these. You know, target

people all around the globe. And what he found, again, that the average number of

steps was six. So it took, typically, six steps to get from one person to another.

So there's this six degrees phenomena that we want to understand how that can be. So

we're gonna do this. By looking at a variant of the Small World's network. So

we know that social networking have that small world structure to that. We're gonna

use that to explain how you can get six degrees of separation. So when people form

friendship networks, they don't do it with the intent of creating a six degrees of

separation world network. We're gonna show that it just emerges from the structure.

So we're gonna start off by simplifying the small world network as follows. We're

gonna assume that each person has a group friends, see if them belong to a clique.

[inaudible] gonna be friends with each other and then you've got a few random

friends off. To this side. So you get C click friends and R random friends. Let me

show what this looks like. So, here's a clique, and everybody within the clique

we're gonna assume is friends with everybody else. So, it's got a very high

clustering coefficient, and then each person in the clique also has one random

friend, and that random friend belongs to some other clique. Now I need to introduce

a [inaudible] idea. This is called a K neighbor. So, a one neighbor is someone

that you're connected to. So that's be a one neighbor. A two neighbor is someone

who's connected to someone you're connected to. And a three neighbor is

someone who's connected to someone who's connected to someone who's connected to

you. So what you get is you're three steps away. Now, if there's also a connection

between these two people, so this person is both one step away, and three steps

away, we would classify them as a one neighbor. So that the shortest distance

between one person and another. So your three neighbors are the people who are

three steps away, but they're not two steps away, or one step away. So six

degrees of separation is going to mean that someone is six steps away, but not

five, four, three, two, or one. So let me show this graphically, I'm looking at this

person here, the one neighbors are going to be the two people he's directly

connected to. The two neighbors aren't gonna include these two people he's

directly connected to but it will include these two people who are connected to the

people. He's connected to. So one neighbors are who you're connected to. Two

neighbors that were connected to people we're connected to. That's the idea. Now

we're gonna use this to show how you can get six degrees of separation. Here's how

it works. If you look at a person in this random clique network, what they've got is

they've got, who are their one neighbors? It's their clique friends, which we'll

represent by this C. And then their random friends, which we'll represent as being

red. Those are the one neighbors. Now, who are the two neighbors? Well, their two

neighbors are their click friends. Random friends, that's these people. Their random

friends, random friends, which are these two people. And then, finally, their

random friends, click friends, which are these people. So all I've done is they've

got quick friends and random friends, I've just sort of written all of this stuff

out. What about [cough]? What about the click friends, click friends? Well, my

click friends are just equal to my click friends so if I think about how I get my

two neighbors I just click friend. Random, random, random click but I don't add in

click, click because those are just the click friends. All right? What about the

three neighbors. Well, I do the same thing. I've got my random friends, random

friends, random friends, my random friends, random friends, random friends,

click friends right? My random friends, click friends, random friends, so who are

my random friends, click friends, random friends? So I'm going to click. I've got

some random friends. My random friends belong to a click and then I've got their

click friends. Random friends, that's who these people are. So I could just write

down all possible combinations. [inaudible] random clique, clique random,

random, random clique, that sort of thing. However, I can't write down random clique,

clique. 'Cause if I have two cliques in a row, my random friends, [inaudible] I've

got a random friend, and he's in a clique, my random friend's clique friends, which

are these people, well, their clique friends are the same people. So random

clique, clique is the same [inaudible] random, as random clique. And clique,

clique random is the same as clique random. And clique, clique, clique. It's

just my seamed clique. So what I have to do is write out all these combinations and

that gives me the total number of three names. Well let's do this in a real case.

So, let's take I've got 140 clique friends and ten random friends. And this is

actually approximately the number of friends that people might have. People

have about 150 friends, most are sort of close to you. So, let's compute the number

of one neighbors. Well that's just equal to 150. What about the number of two

neighbors? Well, I've got my clique friends, random friends. My random

friends, clique friends. And my random friends, random friends. So that's gonna

be, got 140 clique friends, and each has ten random friends. So that's gonna be

1,400. I've got ten random friends, each one has 140 clique friends. So that's

another 1,400. And then I've got. Ten. Random friends each of whom has ten random

friends. So that give me another 100, which gives me 2900. I add all that up.

So, I've got 151 neighbors, I've got 2,902 neighbors. What about three neighbors?

Well, here I've got random, random, random, random, random, click, random,

click, random. End click random, random. And then click random click. Those are all

the possibilities. So if I do this, I'm gonna get ten times ten times ten, which

is 1000. I'm gonna get ten times ten times 140, which is gonna be 14,000, that's a

lot. I'm gonna get another random click random, so that's another 14,000. And I've

got this, which is another 14,000. And then here, I've got 140. Times ten, which

is 1,400 times. 140, which is gonna give me 1-4-0-0-0. And then I'm gonna get +56,

excuse me, 196,000. So when I add all this together, I'm gonna get 229,000 three

neighbors, so that's a lot. [laugh]. I've got 229, 000 three neighbors. 150 one

neighbors. 2,902 neighbors, 229,000. Three neighbors, that's interesting. It's

interesting, cuz it help us understand a phenomena that's been long known

empirically. So, in 1973, Mark [inaudible] wrote a paper called The Strength of Weak

Ties. And what he found in this paper is, if you think of the important things that

happen in your life, like the job you get, who you marry, where you live. All sorts

of important things. It doesn't depend on your one neighbors, your close friends. It

tends to come from your three and your two neighbors and your three neighbors. These

weak ties, these people who you're remotely connected to, end up having a big

effect on your life. Well, let's think about who these three neighbors are. So a

three neighbor could be your roommate's brother's friend, right. One, two. Three

Could be your mother's co-worker's daughter. One two three or could be your

high school roommate's college roommate's, dad. You know one two three so three

[inaudible] aren't that far away and actually can seem [inaudible]. Points of,

sort of, interesting story. Like, I actually got a job with my roommate,

brother's friend. He hired me for his firm. It doesn't seem that far-fetched. In

fact, it's not far-fetched because, as Granovetter shows, that's how most people

get jobs. Why does that happen? Well, let's look. Remember, we've got. 151

neighbors, 2902 neighbors, and 229,003 neighbors. There's so many more. Of these

three neighbors, that their just that much more likely to get you the job. They're

also that much more likely to introduce you to the person you're going to marry.

They're also that much more likely to tell you about a great new place to live or a

place to go on vacation. It's just the sheer numbers. So, this puzzle, this sort

of strength of weak ties puzzle, the study that sort of loose connections get you

things isn't a puzzle once we write down a model and do a little bit math. Let's look

at other network structures. So, here's a network of collaboration among scientists,

collaborations among scientists. I want you to see that. These, that there's some

people who collaborate more with others. They're more central to the production of

knowledge. And if we think back to our. Internet model, or worldwide web model, we

saw that we got that power law distribution. So there was some nodes that

were connected to a lot. And they were, most nodes were connected to few. What are

the functionalities of this sort of network? Look, here's an interesting

functionality. Suppose I think about random node failures. So suppose nodes on

the internet are gonna fail randomly. Well most nodes are connected to very few. Most

nodes are over here. So that means if you have random failure, this node is gonna be

incredibly robust. So no one said, hey, let's. Make connections in such a way that

makes the internet robust, but the fact that it emerges from the structure of the

network. What about targeted failures? What if you want to shut down internet?

What if you want to target failure, then you go after these, lots and lots of

connections. So although the internet is really robust in handling failure but it's

not at all robust to targeted failure. That's a functionality that emerges from

the preferential [inaudible] rule. Nobody built them in. They just happened. So what

have we learned? We learned that it's sort of fun to talk about networks. There's

pictures but we can really unpack it in a formal way by constructing models and

networks. Cause models and networks can focus on the logic. How does the network

form. The structure. What are the statistical properties within networks?

And then finally the functionality. What does the network do? Right. Does the

network robust to random failures or is it robust to strategic failures? Does it give

us six degrees of separation or 400 degrees of separation? Is it connected or

non-connected? So there's all these functionalities that emerge from the

network structure. And the network structure in turn is a result of. The

individual logic for how people make connections, or how firms make

connections, or how. Web pages make connections. [laugh]. One last thing,

before I conclude this set of lectures on networks. Now that we have networks we

understand the functionality of those networks. We can think about interventions

into the network. So here's, again, a social network that suppose you want to

ask that there's some disease that's going to spread. Now remember we talked about

our model of vaccinations and you see that you have to vaccinate as a function of the

R0 so the higher R0 is the more people you have to vaccinate. But that was assuming

that people were randomly connected. And then, everybody's sort of, randomly

meeting other people. But in real social networks, you'll see there's some like

this person here, and these people here, that are much more central to the node.

M-, much more central to the graph. They're connected to lots of people. These

might be schoolteachers, these might be bus drivers. So if you think about

vaccination, rather than saying, okay, blanket. We've got to vaccinate twenty

percent of people or 30 percent of people, based on R zero. Instead, you could look

at the social network and say, oh, you know what? We needed to vaccinate these

people, these people, these people, these people. So by profession, [inaudible] by

profession, who are the most important people to vaccinate to prevent this thing

from spreading? So by combining our network model with our disease [inaudible]

model, we can actually come up with lower vaccination rates to stop the spread of

diseases. So again, this is why you wanna be a many model thinker. Cuz if you've got

lots of models in your head, you can then combine those models in interesting ways.

So we have the vaccination model, which says, the more virulent the disease, the

more people we have to vaccinate. Now we've got this network model says, well,

no, everybody's not connected to everybody with equal probability. The random graph

model isn't true of social networks. So then you realize, that what really matters

isn't vaccinating everybody, but vaccinating the key people to prevent the.

Disease from spreading. And what you'd like to do is make the network, by

snipping off people, disconnected. Because if it's disconnected then it can't spread.

Okay, so we've learned a lot about networks, their logic, their structure,

and their function. And we've seen how we can [inaudible] for the disease model. But

if we take a lot of the models we've done in class, you can also throw networks in.

So a lot of research has been done in the last 10-15 years in social sciences, has

been to add networks onto things, like economic performance, school performance,

things like that, to show how these sort of interactions between individuals have

an effect on what's happening at the macro level. Alright, thanks.