Learn how to model social and economic networks and their impact on human behavior. How do networks form, why do they exhibit certain patterns, and how does their structure impact diffusion, learning, and other behaviors? We will bring together models and techniques from economics, sociology, math, physics, statistics and computer science to answer these questions. The course begins with some empirical background on social and economic networks, and an overview of concepts used to describe and measure networks. Next, we will cover a set of models of how networks form, including random network models as well as strategic formation models, and some hybrids. We will then discuss a series of models of how networks impact behavior, including contagion, diffusion, learning, and peer influences. You can find a more detailed syllabus here: http://web.stanford.edu/~jacksonm/Networks-Online-Syllabus.pdf You can find a short introductory videao here: http://web.stanford.edu/~jacksonm/Intro_Networks.mp4
5.1: Diffusion
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Social and Economic Networks: Models and Analysis
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À partir de la leçon
Diffusion on Networks
Empirical Background, The Bass Model, Random Network Models of Contagion, The SIS model, Fitting a Simulated Model to Data.
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Matthew O. JacksonProfessor
Economics
Okay, folks, so now we're ready to start looking at the last major part of the
course, which is understanding how networks influence behavior, so taking
the networks as a given and then understanding their implications for
behaviors. And in terms of the starting point, we'll
start with diffusion and then talk about other aspects.
And so we're moving into this part of the course and, you know, we can think of
sort of breaking this in. First, we'll start with diffusion.
Then, we'll think about learning and then we'll talk about situations where what I
do actually depends on what my neighbors are doing.
So, there's some strategic interaction going on.
So, we'll take those each in turn. And, one thing to sort of emphasize is
here we looked at formation, and here we're looking at taking the networks as
given, and then studying the behaviors. obviously these things influence each
other, so what I expect to happen, given a network.
Will influence which links I, I form and which links I form influence the
behavior. So, these things are, are really
co-determined or, or co-evolve. And yet we're going to take them one at a
time. And ultimately to understand the
formation process, you would have to make predictions about what's going to happen
once the network formed, so before we treated the utility function or.
The benefits coming from a network as a given.
Now this will put some meat on it so we know what will happen once the network's
formed. That then gives payoffs to people.
Then we can go back and understand if a certain process is operating on a
network, what does that imply for the formation process.
So most of the literature has sort of gone through this question, and this
question. And the co-determination has been looked
at a little bit, but not much. I'll say a little bit about that at the
end of the course when we sort of wrap up the last section.
Okay, so now into diffusion. And when we think about how network
structure impacts behavior what we're going to start with is the idea that some
things like simple infections, contagions, we'll think about diffusion
in a fairly mechanical way where it goes from one node to another.
when we think about richer settings where opinions are being formed, information's
being processed. there's learning going on, that's going
to be a little more complicated because it's not going to be a process where
either I've got the flu, or I don't have the flu and I've caught it.
Which we can look just at the network structure and try to understand those
things. Now we're going to actually have to think
about how do people process information? How does information flow?
What happens in terms of forming opinions?
And so forth. That will be a little more complicated.
And then when we allow people to make choices and decisions that will further
complicate things. So then they have to think about what
other people are doing and how they react to that.
So there's sort of three layers here and they'll get increasingly rich in terms of
what we're allowing people to do and what kinds of processing is going on.
Okay. So diffusion.
Let's think about, you know, we can think about things like the flu basic disease
diffusion. We can also think about ideas or basic
information, so something you just either know or don't know.
So if you're just made aware of a fact, then that's a situation where there's not
information processing and, and really complex learning going on.
It's just, you have to be told about something, okay there's a new iPhone out.
So you're just told a fact. and we can also think about you know, the
sum of applications of this basic kind of diffusion have been people adopting a
product or not. but doing so in a situation where all
they need to do is know about the product and they can make there own decision,
they don't have to worry about what other people are doing.
In situations where, whether I want to buy a product depends on what other
people are doing, that's going to be more complicated, and we'll come back to those
complementarities later. So let's just think of the simple
processes. And so we'll start with just some, some
questions and a little bit of background on this.
And then we're going to talk about the Bass model, which is sort of the simplest
and probably best known model of diffusion.
And then we'll start bringing in interaction structure and network
structure. Okay.
So first thing when we think about diffusion there's something that's just
known as sort of s shape adoption. when you look at diffusion over time and
space there are different patterns to how things spread and, and how many people
have adopted something over time or, or uh,caught the flu, et cetera.
Things start out slowly, they accelerate and eventually peak.
we can ask questions about who are initial adopters?
Are they people with high degree, low degree, what accounts for different
speeds in a diffusion process, why might there be an eventual slowdown?
There's a whole series of questions that come up.
And let's start by looking at, at one of the early studies of this.
it was a study by Coleman, Katz and Menzel in the 1960s and they were looking
at the adoption of a new drug by doctors, and in particular, there was a drug that
had just been developed. it was an antibiotic and it needed to be
prescribed by doctors. And there was a question of so their idea
was a, a doctor adopted this new drug if they prescribed it at some point in time,
okay? And so there was a question of how many
how, how much time it took for different doctors to start prescribing the drug.
And so, what they did is they, they kept track of the doctors over time and kept
track of which doctors had prescribed the drug by some time period.
So, six months out from the development of this drug, how many, or from the legal
adoption by, of this drug. Had some doctors prescribed it?
Eight months out, how many had prescribed it?
Ten months out and so forth. And what they did was they, before they
started this they surveyed the doctors. And they asked which other doctors would
you go to for advice. Okay.
And so then they kept track of, that, that gives a network of, of doctors, and
in particular, we can break subjects into three different categories, they broke
them into three different categories. They had 36 doctors that were not named
by anybody else. So nobody said that they would go to that
person for advice. 56 doctors were named by one or two
others. And 33 were named by at least three
others, three or more others. Okay, so these were different doctors, in
terms of how many other people would say they would come to this person for
advice. And then they kept track of the adoption
rates over time. And the ones that hadn't been named by
anyone else, by six months out, 31% of them had started prescribing this new
drug. By eight months out, 42%.
By ten months, 47%. by 17 months, it was 83%.
And when you look at the, the doctors that have been named by one or two
others, it started at 52% six months out, it was at two thirds, roughly, by eight
months, 70, and so forth. So, so here we're seeing higher adoption
rates at an earlier time. And then when you're named by three or
more others, it was even higher. they were picking even higher reaching
higher rates at an earlier time. And so forth.
So what this shows is that the diffusion process actually differed based on the
position of the doctors in the network. there's been follow-up studies that
there's some difficulties in doing these kinds of studies and making sure that
you've got a clean test. Because it could be that whether you're
named by three or more others also correlates with other things which could
correlate with whether or not you heard about this drug from advertising or other
sources. You might be pressured directly by the
drug company. So, there's a, a series of other things
that might be accounting for these kinds of results.
There's been a series of follow up studies that have tried to make sure
that. The, the kind of finding here is actually
robust and not something that's just the spurious correlation.
But in any case what we do see is differences here based on the the
connectant and, and that seems to hold up if you, if you look at the, the, the data
with more scrutiny. so here, I just plotted based on how many
months out, what those rates were. And what you see is the ones that were
named by nobody else, had lower rates at each point in time.
Eventually they reached the ones by one or two others.
and the one or two others are below the ones named by three or more others, but
you're, so you're seeing adoption rates different adoption rates based on, on the
degree of the individuals involved. And that's one thing we can pay attention
to in trying to model diffusion and understand diffusion, why is it that it
might vary by the degree of the individual?
And there'll be fairly intuitive explanations for that, as you might
expect. let's take a look at another example
which is another fairly famous one. this is data analyzed by V Griliches in a
paper in 1957 and It goes back to data that had been
collected earlier in the 30s and 40s on the adoption of hybrid corn among farmers
in different areas of the United States. So basically hybrid corn where you had
mixed the genetic material of different corn species was being developed
actually, you know, this kind of husbandry of corn has gone on for a
millenia. But it was being marketed and developed
in a new way in the 1930s. And the yields of the corn that was being
produced in this way were somewhere between 15 to 25% higher.
Than the existing corn strains. So you could, you could get much higher
yields from the corn seed that they were being developed at this point and time.
And this type of corn eventually just completely replaced the old single.
corn types that existed before that. And Griliches sort of analyzes why it
took, took different paths in different places.
So if we look at these here are three different states that are corn producers,
Iowa is a, is a state which is basically is mostly corn.
there's a lot of corn in Iowa and it's, a, a very good climate for growing corn.
And, it was the state which adopted this earliest on.
then in Wisconsin you see a different curve.
a little later adoption. there's more veh, variety of, of things
grown in Wisconsin. And in Kentucky eh, eh, even less
hospitable to corn, and, and has other things going on in terms of what it's
growing. And indeed, and that comes in even later
than Wisconsin. And he's got other, you know, Texas and
Arkansas and a whole series of states. but what we're see here that was sort of
important was the fact that this had an s shape to it.
So in particular we see the adoptions start out fairly slowly where it takes,
you know, until 1935 or so. Before it's even hit sort of 10% even in,
in exceeded 10% even in Iowa. And so, it took quite awhile before it
really starts hitting. Then, it accelerates.
So, it starts out slowly, it accelerates and then, it comes back.
So, we get this very nice s shape which is actually observed in a number of
different. applications.
So a lot of, of diffusion processes will have this shape.
And we can try and understand exactly what it is that might lead to this shape.
Why does it start out slowly? The idea that it eventaully has to
asymptote and slow down, that's obvious. I mean it can't go above 100% so it's got
to slow down eventually. It can't just keep going forever.
so that part's easy. The difficulty is going to be in figuring
out why does it start accelerating in this manner.
And word of mouth and, and the social interaction parts are going to be
important in sort of explaining that acceleration.
Okay, so questions that come up. You know, what's the extent of a
diffusion? How does it depend on the particulars of
a network structure? Can we say something about these time
shapes, where does the s shape come in? Can we say something about welfare
analyses? If you want to accelerate a, a diffusion.
How would you want to do it? So suppose you want to make sure that
corn actually diffuses quickly, how would you do that?
If you want to make sure a flu doesn't diffuse, how do you prevent one from
diffusing? Who would you want to vaccinate?
How would you go about doing that? There's a whole series of questions that
we can begin to analyze using diffusion. And in particular modeling the network
process which is going to be important in, in answering a lot of these
questions.
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