Hi. In this lecture, we're gonna talk about a model that produces a tipping point. It's a very simple model, and it comes from physics, and it's known as the percolation model. Now the idea is this: you've got, you know, ground up here, you've got water that comes down as rain and you want to ask, does it percolate through the soil or not, right? Very simple question. So how do you model something like that? Again, the essence of modeling is to simplify things, right? So what you're going to do is construct the following sort of model, just a checkerboard, like the Game of Life or cellular automata models. But the idea here is this, that you've got a bunch of squares and they can either be, sort of, filled in like this or they can be left open. Now the idea is this, you can only jump from a filled-in square to a filled-in square. So thinking of a... Think of a frog trying to cross a river. So it can jump along here, along here, along here. But it gets stuck, doesn't make it. And it can go here and it gets stuck. So this is a case where it wouldn't percolate because you can't get from here all the way down to the bottom. So here's the model. Really simple model. You ask the following question: Let P equal the probability that I fill in a square. So for each square I flip a coin and if P is 1/2, then half the time I fill in a square, half the time I don't. If it's 1/3, a third of the time I fill in a square and two thirds of the time, I don't. And then we ask, does it percolate? So a really simple question. Well, here's what the graph looks like. As long as P is less that 59.2%, it doesn't percolate. This is for a big graph, right? But once you get above that, the system tips, right? Right here there's a tip. And then it becomes likely that it does percolate. So you get this really abrupt change, and I notice this is a non-linear function, right? A linear function looks sort of like this, and this thing goes sort of flat and then makes this tip right at that point. So what's causing the tip? Well, what's causing the tip is that, if you look back at the picture, right, for P less than 59%, there just aren't enough things filled in. So if you're at 30% up to 31%, if I filled in, like, one more square, it's still not likely that I percolate. But once I get to about 59%, then so many squares are filled in that it becomes suddenly really likely that I'm gonna be able to make it to the bottom. So it should be pretty clear that going from 20 to 21% isn't gonna have much of an effect, and going from 21 to 22% isn't gonna have much of an effect. But going from 58 to 59 to 60 suddenly has a huge effect. Now, what's great about this model is that it can be applied to all sorts of stuff. So we're gonna first apply it to forest fires, and then we're gonna apply it to banks. So we're gonna use a NetLogo model here to try and make sense of this. And what we have is a model here, a simple model of forest fires. And what we have is we have one button that shows the density, that's right up here. And then we just set this thing up, and that's gonna fill in trees with this density. So currently, it's at 57%. And then I'm gonna, across this left edge, you'll see, if you look really closely, you can see red. I'm gonna start fire along that edge, and we're just gonna see what happens if I let it go. So if I let that go, you see that -- oh, it comes closer and it almost makes it. Let's try it again. Let's set it up again. Oh it comes close and almost makes it. Doesn't quite do it. One more time. Just about, you know, pretty good. But let's move this thing up to then 61% which is above that 59% threshold and look what happens here. At 61%, it makes it. And let's do it again. 61% again, look what happens, it makes it, right. 61% again, it makes it. So what we see in this very simple model is that if we go to 57%, right? and set it up, we're not likely to make it. But if we just increase it a little bit, let's just make it even 60%, right, which is barely above the threshold, then what happens is the fire spreads throughout the whole space. So we see this phase transition, right, right at 59%. From the fire not spreading, to there not being a fire over the whole space, to there being a fire over the whole space. Okay, so we can think about that forest fire model in the following way: think of the yield. So suppose we had a forest and we wanted to get as much wood as we could from the forest but we knew there was a chance of fires. What would our yield curve look like? Well, there's that critical value, right? At 59%. Right? And what would happen is, if we planted more and more trees, we'd get, a nice linear yield. We'd get more and more wood. But once we got out of the critical threshold, suddenly our yield would fall off really fast and there'd also be a tip in terms of the yield. So not only is there a tip in terms of the likelihood of fire, there's also a tip in terms of the yield. So what's nice is -- remember we talked about fertility of models. We have a model that was used to explain percolation. 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, ignoring things like wind speed, terrain and things like that -- there seems to be some density at which a fire's really likely to spread, and a density of trees at which it's not likely to spread. But let's push it even further. Let's imagine we have a model of percolating banks. But what would that look like? What do I mean? Well, let's again have this checkerboard thing, and let's suppose that, you know, here's a bank. Here's bank 1. Here's bank 2. Here's bank 3. Here's bank 4. And here's bank 5. Now, what we can imagine, suppose this bank fails. It makes a bunch of bad loans. Well, suppose that this bank then has borrowed money from these banks. Right, so these banks have all given money to bank 1. But when bank 1 fails, it then can't pay this money back to these other banks. And if it's loaned enough money, then these banks may fail. If they loaned enough money to bank 1, they may fail because they don't get their money back, and so the failure can spread. So, remember we talked about that before, about how the IMF has constructed these models of banks. We have... Here's a bank failing, and it spreads to other banks failing, and that leads to other failings and so on, right? Now these models are more sophisticated than just bank fails and moves to the next bank. What you do is you write down sophisticated accounting equations where banks have assets, capital and liabilities. And you've got loans of different durations and those loans fail. And you can sort of ask: If we put stress on the system having a bank fail, how far, you know, how fast does that spread. So the basic premise is the same, this notion of peculation. But what you do is you add more detail, rich, you know, accurate detail about exactly what those loans look like. And then you could ask the question, is there a tipping point in the case of these banks? So, is there, sort of, a s-, a state at which, the entire system is poised to suddenly have all sorts of banks failures? Right? And again that's a question you can ask in the context of that richer model, so maybe the insight of from the simple percolation model holds in the bank case and maybe it won't, that's something that we're only gonna understand by writing down that richer model. Now you can do the same thing in the context of country failures. I put this graph up in one of the first lectures for this course. This is a case where we have England fail first, right? And then what happens is I think that spreads to Ireland and a couple of other countries and then here it spreads to Germany and then it spreads down to France, so what happens is you can ask if one country fails, will that percolate to all the other countries or not? And you can get a sense of, is this whole system sort of poised for some giant failure? In other words you can ask, is there a tipping point in the system? The sort of, you know, country finance, country level financial systems. Alright. Let's go even one further. We could also take the same model and think about information percolating. What do we mean by that? Well imagine there's some network of people, right? So here's a network of people and you can estimate if there's some probability that if I hear some rumor, get some piece of information or know something, that I'm going to tell it to my friends, which is now instead if there's probability things are going to move across links, I can ask, as a function of that probability, what's the likelihood that the information percolates, that everybody hears about it? And I can use the same model. And so, what would the model tell me? The model would say, well, if a rumor is juicy enough, or if a piece of information is important enough, then it's really likely to spread. And everybody's likely to hear it. If it's not important enough, then it may not spread. So here's what's really interesting. Let's go back and think about information percolation. You might think: Here's the value of a piece of information. Here's how juicy a piece of gossip is. And here's, sort of, how many people hear. So here's the number of people here. Now you might think, well, that should be linear. The more valuable the information, the more juicy the gossip, the more people should hear. But if you actually construct a network model that looks something like this, and assume that there's some probability of people telling people across links, what you're likely to get is something that maybe has a bit of a tip, right? That nothing happens if it's not very juicy or the value of information is pretty low, it doesn't spread. But then once it gets above some critical threshold like right here, it takes off and people are, almost everybody's likely to hear. And so this says that we should expect the distribution of information, the distribution of rumors, not to be, sort of, varied in a linear way with how interesting information or how valuable it is, but instead to possibly have this kink, to possibly have this tipping point and the reason why is because information spreads through networks, and because it spreads through networks, you get this same kind of percolation phenomenon. Okay, so, interestingly, we took this percolation model from physics, we saw how it gave us insight into forest fires, and here it gives us insight into possibly how information, you know, spreads through a system, you know, spreads through a graph. Okay, let's have just a little bit of fun here, let's really take the shackles off. We've got this model and we've got this model about sort of percolation from one thing to another. Well, we can apply this to the following idea: that sometimes problems in mathematics, or problems in engineering, or scientific problems or innovations, people have been working on them for years. And then suddenly a whole bunch of people figure it out at approximately the same time. And this can be a bit of a puzzle. Nobody knows how to make a steam engine, and suddenly everybody's making a steam engine. Everybody's trying to figure out some way to identify DNA and a whole bunch of people -- Crick and Watson found it first, and other people on the heels of finding it, right? Why is it that we see these bursts of scientific activity in a particular area? Why do we see many people come to the same innovations, the same scientific breakthrough at the same time? Well, you could use the percolational model to basically say, well, this could be the logic of it. Think about constructing, let's say, a mathematical proof: oftentimes it is a matter of getting from A to B, to putting together bits of logic. Think about innovation, oftentimes it's not giving all the parts to work. So to have a car you need an engine that works, you need a braking system, you need steering mechanisms and all these parts. As information and knowledge accumulates, we fill in more squares. So initially, we can't get from A to B. But what happens is: Technology. And information. Start filling in squares and then eventually somebody can find a path. So, what's interesting is not only can they find that path. Somebody else can find a path because there's multiple paths because once we get above the threshold that doesn't mean that there's one path, it could be that there's many paths. And so it's at least plausible that a percolation type model might explain why we suddenly see bursts of activity in particular areas as the knowledge base increases, again speculative, but that's what's fun in our models. Once you have a model you can be like, "I wonder if this applies in another setting," and gives us an insight as to why we suddenly see bursts of things like scientific activity. So that's the percolation model, right? Very simple... Checkerboard and you basically just ask: "Can the frog jump from the top to the bottom?" You can use it to understand the percolation of water, you can use it to understand forest fires, you can use it to construct a richer model of bank failure, you can use it to understand how information percolates through a system, through a social system, and you can even speculate on whether this might explain why we see bursts of scientific activity in particular areas, and bursts of innovations. All right. Thanks.