Hi. In this next set of lectures, we're gonna look at some models of tipping points. Now these are gonna be highly non-linear models in the sense that what happens is gonna be that a small change is gonna lead to a big effect. So instead of being sort of a nice linear thing like we've seen before, we're gonna see abrupt changes, right, where the system tips at some point. Now our goal here is to really try and understand in a deep way, how tipping points occur, and what is a tipping point and what isn't a tipping point. Because one of the things we're going to learn is a lot of things that look like tipping points, really aren't tipping points. So let's get started. What is a tipping point? Well, imagine the proverbial case of a tipping point: the straw that broke the camel's back. Right, so, suppose this is a graph of the weight of a camel, and here I'm adding pieces of straw. Right, so I add pieces of straw, the weight of the camel goes up in a nice linear way. Rather than weight of the camel, I should put weight of camel plus straw, right. That goes up in a nice linear way. But, if I were to grab the height of the camel, what would happen, is the height of the camel would be pretty much constant as I add more and more straw, until eventually, the camel falls down. Its back breaks and then the camel become much, much, shorter, right. So this point right here is where the tip occurs. So that's what we mean by tipping point. One more piece of straw has a huge effect on some variable interest, namely the height of the camel. Also the speed of the camel as well. That would also tip. That would go from a positive number to zero. So people talk a lot about tipping points. Malcolm Gladwell a few years back wrote a book called The Tipping Point which sold millions of copies. And when people talk about tipping point, they often mean kinks and curves. So if you look at housing starts in the United States, right, they're going up really nicely from the beginning of the new century up until about 2006 and then there was this big kink right here. So people tend to say this point is a tipping point. And I'll show you that yeah, that probably was a tipping point. Then people look at things like -- this is users on Facebook. This is a graph of Facebook users graph and you see this thing goes up here. And there will be people that say oh, right here. 2008. That's the tipping point. I'm going to argue that's not true, okay. Same is true if you look at the world population. You might say oh my gosh, look at that. In the 1940s the world population took this big kink. It did kink, but I'm going to argue that's also not a tipping point. Same's true with Wikipedia articles. People say oh boy, Wikipedia in 2004, this was really the tipping point. These are all cases of the graph having a big kink in it, but they're not necessarily a tipping point. A tipping point is when a small change leads to a large effect. These processes that we've just looked at, Facebook, world population, Wikipedia entries, these are all cases of just exponential growth. When you have exponential growth, you're gonna get a curve that takes off, just like if I draw, like, you know X squared or X cubed or X to the fourth, right? I'm gonna get something that sort of, you know, it goes like that, and you can convince yourself. Ooh, right there the slope seems to be changing really fast. Nothing tipped. That's just an inevitable process of growth. So one reason we want these models is to be able to make sense of what's a tip and what's not a tip, and we also want to ask "what kind of models produce tips, right, how can we get tips and when are we likely to see them." So here's what we're gonna do: we're gonna construct two really famous models, one from physics, one from epidemiology. The physics model's known as a percolation model, and many of you maybe own houses and you had to dig wells, and there's the question "Did the ground perc?". Did it percolate? Did the water sort of filter down through the ground, or was it too clay-like and would hold the water in. So percolation literally refers to you know, Can sort of water make its way through the system, or like coffee percolating through the coffee grounds. Then we are going to study the SIS model, where S stands for Susceptible, I stands for Infected, and S stands for Susceptible again. This is a simple model of a disease, where you don't have the disease, you get infected and then you don't have the disease again. Both of these models will have tipping points. Then we're going to make a distinction between types of tips. I'm going to talk about direct tips. Direct tips is a situation where a particular action, tips that same dimension, that same variable. So for example in kind of a state of war, a battle might tip a war you know from one side winning to another side winning right. So it tips the same exact entity. Alternately, there's a contextual tip. A contextual tip means: something changes in the environment that makes it possible for something to happen. 'Kay. So direct tip is where the variable itself changes, it causes itself to tip, so it could be, you know, people buying a particular product. It could be battles in a war. A contextual tip is, something in the environment changing in such a way that it then causes the system to move from one state to another. I'm gonna make a second distinction between types of tips. Remember we talked about these four classes of things that the world can fall into, right? It can be stable, right, like an equilibrium, it can be periodic. Right? It can be random. Right? Or it can be complex. So these are the four types of states the system could be in. Well, the system could tip from one to the other. So it can be stable. Then tip to periodic. And then it could be periodic. And tip to random. So we can talk about not only direct tips and contextual tips, we can talk about tips between class, so it goes from one class to another -- and tips from within class. So for example, if a system tips from one equilibrium to another equilibrium, that would be a within class tip, right, cause the overall structure of the system hasn't changed in terms of how we categorize the types of outcomes we see. Alright, so that's the plan. We're gonna construct some basic fundamental models that produce tipping points. We'll show that a lot of things that people call tipping points really aren't tipping points. They're just exponential growth. And then we'll classify types of tips, right? We'll talk about between-class tips, and within-class tips. We'll talk about direct tips, and we'll talk about contextual tips. Then we'll conclude with just a very short lecture on how you might measure tippiness. And this is some work I've actually done with a colleague of mine, PJ Lamberson. Coming up with some measures of how tippy a system is. Okay, let's get started, thank you.
