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.