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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â