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Hi. In this lecture, we're gonna talk about a model that produces a tipping

Â point. It's a very simple model, and it comes from physics, and it's known as

Â the percolation model. Now the idea is this: you've got, you know, ground up

Â here, you've got water that comes down as rain and you want to ask, does it

Â percolate through the soil or not, right? Very simple question. So how do you model

Â something like that? Again, the essence of modeling is to simplify things,

Â right? So what you're going to do is construct the following sort of model,

Â just a checkerboard, like the Game of Life or cellular automata models. But the idea

Â here is this, that you've got a bunch of squares and they can either be, sort of,

Â filled in like this or they can be left open. Now the idea is this, you can only

Â jump from a filled-in square to a filled-in square. So thinking of a... Think of a

Â frog trying to cross a river. So it can jump along here, along here, along here.

Â But it gets stuck, doesn't make it. And it can go here and it gets stuck. So this is

Â a case where it wouldn't percolate because you can't get from here all the way down

Â to the bottom. So here's the model. Really simple model. You ask the following

Â question: Let P equal the probability that I fill in a square. So for each

Â square I flip a coin and if P is 1/2, then half the time I fill in a square,

Â half the time I don't. If it's 1/3, a third of the time I fill in a square and

Â two thirds of the time, I don't. And then we ask, does it percolate? So a really

Â simple question. Well, here's what the graph looks like. As long as P is less

Â that 59.2%, it doesn't percolate. This is for a big graph, right?

Â But once you get above that, the system tips, right? Right here there's a tip. And

Â then it becomes likely that it does percolate. So you get this really abrupt

Â change, and I notice this is a non-linear function, right? A linear

Â function looks sort of like this, and this thing goes sort of flat and then makes

Â this tip right at that point. So what's causing the tip? Well, what's causing the

Â tip is that, if you look back at the picture, right, for P less than 59%, there

Â just aren't enough things filled in. So if you're at 30% up to 31%,

Â if I filled in, like, one more square, it's still not likely that I

Â percolate. But once I get to about 59%, then so many squares are filled in that

Â it becomes suddenly really likely that I'm gonna be able to make it to the bottom. So

Â it should be pretty clear that going from 20 to 21% isn't gonna have much

Â of an effect, and going from 21 to 22% isn't gonna have much of an

Â effect. But going from 58 to 59 to 60 suddenly has a huge effect. Now, what's

Â great about this model is that it can be applied to all sorts of stuff. So we're

Â gonna first apply it to forest fires, and then we're gonna apply it to banks. So

Â we're gonna use a NetLogo model here to try and make sense of this. And what we

Â have is a model here, a simple model of forest fires. And what we have is we have

Â one button that shows the density, that's right up here. And then we just set this

Â thing up, and that's gonna fill in trees with this density. So currently, it's at

Â 57%. And then I'm gonna, across this left edge, you'll see, if you look really

Â closely, you can see red. I'm gonna start fire along that edge, and we're just gonna

Â see what happens if I let it go. So if I let that go, you see that -- oh, it comes closer

Â and it almost makes it. Let's try it again. Let's set it up again. Oh it comes

Â close and almost makes it. Doesn't quite do it. One more time. Just about, you

Â know, pretty good. But let's move this thing up to then 61% which is above

Â that 59% threshold and look what happens here. At 61%, it makes it. And

Â let's do it again. 61% again, look what happens, it makes it, right. 61%

Â again, it makes it. So what we see in this very simple model is that if we go

Â to 57%, right? and set it up, we're not likely to make it. But if we just increase

Â it a little bit, let's just make it even 60%, right, which is barely above the

Â threshold, then what happens is the fire spreads throughout the whole space. So we

Â see this phase transition, right, right at 59%. From the fire not spreading, to

Â there not being a fire over the whole space, to there being a fire over the

Â whole space. Okay, so we can think about that forest fire model in the following way:

Â think of the yield. So suppose we had a forest and we wanted to get as much wood

Â as we could from the forest but we knew there was a chance of fires. What would

Â our yield curve look like? Well, there's that critical value, right? At 59%. Right?

Â And what would happen is, if we planted more and more trees, we'd get,

Â a nice linear yield. We'd get more and more wood. But once we got out of the critical threshold,

Â suddenly our yield would fall off really fast and there'd also be a tip in terms of the yield.

Â So not only is there a tip in terms of the likelihood of fire, there's also a tip in terms of the yield.

Â So what's nice is -- remember we talked about fertility of models. We have a model that was used to explain percolation.

Â Like, why does there seem to be soil that percolates or doesn't percolate? And we can use that for forest fires, and what we see is that,

Â ignoring things like wind speed, terrain and things like that -- there seems to be

Â some density at which a fire's really likely to spread, and a density of trees

Â at which it's not likely to spread. But let's push it even further. Let's imagine

Â we have a model of percolating banks. But what would that look like? What do I mean?

Â Well, let's again have this checkerboard thing, and let's suppose that, you know,

Â here's a bank. Here's bank 1. Here's bank 2. Here's bank 3.

Â Here's bank 4. And here's bank 5. Now, what we can imagine, suppose this

Â bank fails. It makes a bunch of bad loans. Well, suppose that this bank then

Â has borrowed money from these banks. Right, so these banks have all given

Â money to bank 1. But when bank 1 fails, it then can't pay this money

Â back to these other banks. And if it's loaned enough money, then these

Â banks may fail. If they loaned enough money to bank 1, they may fail because

Â they don't get their money back, and so the failure can spread. So, remember we

Â talked about that before, about how the IMF has constructed these models of banks.

Â We have... Here's a bank failing, and it spreads to other banks failing, and that

Â leads to other failings and so on, right? Now these models are more sophisticated

Â than just bank fails and moves to the next bank. What you do is you write down

Â sophisticated accounting equations where banks have assets, capital and liabilities.

Â And you've got loans of different durations and those loans fail. And you

Â can sort of ask: If we put stress on the system having a bank fail, how far, you

Â know, how fast does that spread. So the basic premise is the same, this notion of

Â peculation. But what you do is you add more detail, rich, you know, accurate

Â detail about exactly what those loans look like. And then you could ask the question,

Â is there a tipping point in the case of these banks? So, is there, sort of, a s-,

Â a state at which, the entire system is poised to suddenly have all sorts of banks

Â failures? Right? And again that's a question you can ask in the context of

Â that richer model, so maybe the insight of from the simple percolation model holds in the bank case

Â and maybe it won't, that's something that we're only gonna understand by writing down that

Â richer model. Now you can do the same thing in the context of country failures.

Â I put this graph up in one of the first lectures for this course. This is a

Â case where we have England fail first, right? And then what happens is I think

Â that spreads to Ireland and a couple of other countries and then here it spreads

Â to Germany and then it spreads down to France, so what happens is you can ask if

Â one country fails, will that percolate to all the other countries or not? And you

Â can get a sense of, is this whole system sort of poised for some giant failure? In

Â other words you can ask, is there a tipping point in the system? The sort of,

Â you know, country finance, country level financial systems. Alright. Let's go even

Â one further. We could also take the same model and think about information

Â percolating. What do we mean by that? Well imagine there's some network of people,

Â right? So here's a network of people and you can estimate if there's some

Â probability that if I hear some rumor, get some piece of information or know

Â something, that I'm going to tell it to my friends, which is now instead if there's

Â probability things are going to move across links, I can ask, as a function of that

Â probability, what's the likelihood that the information percolates, that

Â everybody hears about it? And I can use the same model. And so, what would the

Â model tell me? The model would say, well, if a rumor is juicy enough, or if a piece

Â of information is important enough, then it's really likely to spread. And

Â everybody's likely to hear it. If it's not important enough, then it may not spread.

Â So here's what's really interesting. Let's go back and think about

Â information percolation. You might think: Here's the value of a piece of

Â information. Here's how juicy a piece of gossip is. And here's, sort of,

Â how many people hear. So here's the number of people here. Now you might think, well,

Â that should be linear. The more valuable the information, the more juicy the

Â gossip, the more people should hear. But if you actually construct a network model

Â that looks something like this, and assume that there's some probability of

Â people telling people across links, what you're likely to get is something that

Â maybe has a bit of a tip, right? That nothing happens if it's not very juicy or

Â the value of information is pretty low, it doesn't spread. But then once it gets

Â above some critical threshold like right here, it takes off and people are, almost

Â everybody's likely to hear. And so this says that we should expect the

Â distribution of information, the distribution of rumors, not to be, sort

Â of, varied in a linear way with how interesting information or how valuable it

Â is, but instead to possibly have this kink, to possibly have this tipping point

Â and the reason why is because information spreads through networks, and

Â because it spreads through networks, you get this same kind of percolation

Â phenomenon. Okay, so, interestingly, we took this percolation model from physics,

Â we saw how it gave us insight into forest fires, and here it gives us insight into

Â possibly how information, you know, spreads through a system, you know,

Â spreads through a graph. Okay, let's have just a little bit of fun here, let's

Â really take the shackles off. We've got this model and we've got this

Â model about sort of percolation from one thing to another. Well, we can

Â apply this to the following idea: that sometimes problems in mathematics, or

Â problems in engineering, or scientific problems or innovations, people have been

Â working on them for years. And then suddenly a whole bunch of people figure

Â it out at approximately the same time. And this can be a bit of a puzzle.

Â Nobody knows how to make a steam engine, and suddenly everybody's making a steam engine.

Â Everybody's trying to figure out some way to identify

Â DNA and a whole bunch of people -- Crick and Watson found it first, and

Â other people on the heels of finding it, right? Why is it that we see these bursts

Â of scientific activity in a particular area? Why do we see many people come to

Â the same innovations, the same scientific breakthrough at the same time? Well, you

Â could use the percolational model to basically say, well, this could be the

Â logic of it. Think about constructing, let's say, a mathematical proof:

Â oftentimes it is a matter of getting from A to B, to putting together bits of logic.

Â Think about innovation, oftentimes it's not giving all the parts to work. So

Â to have a car you need an engine that works, you need a braking system, you need

Â steering mechanisms and all these parts. As information and knowledge accumulates,

Â we fill in more squares. So initially, we can't get from A to B. But what

Â happens is: Technology. And information. Start filling in squares and then

Â eventually somebody can find a path. So, what's interesting is not

Â only can they find that path. Somebody else can find a path because there's

Â multiple paths because once we get above the threshold that doesn't mean that

Â there's one path, it could be that there's many paths. And so it's at least plausible

Â that a percolation type model might explain why we suddenly see bursts of

Â activity in particular areas as the knowledge base increases, again

Â speculative, but that's what's fun in our models. Once you have a model you can be

Â like, "I wonder if this applies in another setting," and gives us an insight as to

Â why we suddenly see bursts of things like scientific activity. So that's the

Â percolation model, right? Very simple... Checkerboard and you basically just ask:

Â "Can the frog jump from the top to the bottom?" You can use it to understand the

Â percolation of water, you can use it to understand forest fires, you can use it to

Â construct a richer model of bank failure, you can use it to understand how

Â information percolates through a system, through a social system, and you can

Â even speculate on whether this might explain why we see bursts of scientific

Â activity in particular areas, and bursts of innovations. All right. Thanks.

Â