Hi folks. So let's take a look back at what we've
seen in week five, here talking about diffusion.
So diffusion is something that's been studied in a wide variety of different
disciplines so marketing, social psychology, psychology, sociology,
economics. Anthropology.
So diffusion is something which is important in, in many different behaviors
and so there are different stylized facts, and a number of different models
that have been used. Epidemiology.
Medicine. So, so this is something which is very
fundamental to many areas. Basic thing that we've seen in some
empirics is s shaped. Adoption curves.
So things start out slowly, they pick up and then eventually asymptote.
So that kind of nice shape has some intuition to it that we talked about and,
and looked briefly at the Bass model. So the combination of imitation and
complementarities. The fact that the more other people do
seomthing the more likely a given individual is.
If you've got that kind of feature then that can give you the.
initial upswing in acceleration, the eventual saturation is going to tend to
cause the things to, to tail off eventually, so often we're going to see
these kind of curves. The Bass model's been very powerful at,
at capturing these on a macro, big scale level.
We can build tighter models that bring in the network.
And the network then can help us understand when is something going to
spread? When will we get an epidemic?
How many people are going to be reached and so forth?
So initial contagion, when we looked at those kinds of models and, and
understanding say the SIS model. high degree notdes tend to serve as hubs
and enable diffusion. So overall higher density means you're
going to tend to have more connections, more chances of diffusion or contagion.
Higher degree nodes can serve as hubs so higher variance can also increase the
possibility that you see spreading . And we saw differences then between
things like a, a power law network where we've got a scale free degree
distribution can lead to contagion in, in many cases whereas other ones might not
depending on what the degree is. Extent of diffusion.
We also looked at say, the size of the giant component.
And again that got back to our phase transitions and so forth.
And it turns out that there's a fairly rapid shift from no diffusion to a
situation where you've got diffusion starting and then very rapidly you can
hit saturation. And, and in particular between, and when
we're looking at an Erdős–Rényi kind of network, it's between degree one and
degree three, where most of the action happens below, having at least one other
friend you're not going to get diffusion, it's not really going to occur.
Things are going to die off. People tell on average less than one
other individual, you've got a decreasing system.
If they're telling more, then you've got expansion properties, you can begin to
have things spread. And by the time you hit telling three
other individuals, boom. You've got complete saturation.
homophily can begin to affect that. We didn't talk about that so much, but
there are models now that are tending to bring in more structure and breaks in the
network and so forth and that can affect exactly how this spreads.
Okay. Last thing we looked at, diffusion
modeling. Well in, in different situations we can
begin to write down explicit models of what we think is going to be happening.
And by putting those models down they're actually fairly easy to simulate to take
to data, and they can help us understand, for instance, pure effects.
So how much of something is due to information transmission?
As opposed to complementarities in the actions.
we can begin to look at things like financial contagions and understand how
the structure of the network might lead to contagion from one organization to
another. Each different situation is going to have
different kinds of properties in terms of what it takes for one node to end up
affecting another node. And so by, by changing those we can end
up with different models, but those models tend to be fairly tractable in
terms of simulating those processes and getting predictions and then taking those
to data. So, as we go forward, and look now at
learning, and games on network, some of the themes that we've seen here will
carry over there and be embellished in terms of the structures of the
interactions.
Matthew O. JacksonProfessor
