Hi, folks. We're back and we're talking more about diffusion and, in particular, were going to start looking at one of the best known models in the literature on diffusion, which is known as the Bass Model. And it's been used quite extensively especially in marketing and trying to understand things. So, we've gone through some questions and background. And were going to start with the Bass Model. And one thing about the Bass Model is networks are not going to make on explicit appearance here, were just going to have social interaction in the background. And after we've looked at the Bass Model, then we'll enrich the analysis by bringing interaction structure in explicitly, and try to understand how things diffuse over time. Okay, so why is the Bass Model interesting? It's a benchmark model. It gives us very stark and simple structure which can begin to produce things like the S shapes that we saw before. And there's going to be two actions, behaviors or states, so either you're not infected or you are, or you're not adopting the new product or you are. and what the Bass Model will keep track of is over time, what's the fraction of individuals who have adopted say, taken action one by time t, okay? And in this model, you move from state 0 to state 1, you don't move back. So, everybody starts at state 0, then people start moving towards state 1. Some fraction of the people can move to state 1 and that will just move over time, and we'll just keep track over time of how many people have now moved to state 1. So, they have seen the movie, or they've adopted corn, or they caught the flu etc., okay? So, it's a one-way street and we keep track of that over time. Okay, two key parameters in the Bass Model. p is going to be a rate of spontaneous innovation or adoption. So, that's going to be a rate that's independent of what else is going on in the economy or the world or the society. So, some people will just go and decide to see a movie regardless of what's going on, or they will adopt corn, regardless of whether other people are doing it, if they've heard about it, etc. And there's some other individuals who will actually do it as a function of having heard about it, or somehow imitated other individuals who were doing it. So, there's these two parameters p and q. And the Bass Model basically boils down then to a, a simple equation which keeps track of the differential of the fraction of people who have adopted at time t as, as t varies, okay? So, how is this changing with time? Well, there's two parts to it. So, at any point in time, there the people who haven't yet adopted, so these are the people who have not yet adopted yet moved to one, right, so they haven't become a one yet so they're still zeros. So, 1 minus Ft are the people who are still 0 and can possibly change. And then, the fraction of those people who change, well, some of them change directly spontaneously, that's the p. And the others change by imitating the existing population. So, you also, you have a chance of, of just spontaneously deciding to become a 1, or you also have a chance of imitating somebody in the population and that's proportional to how many in the population are already ones. So, this q parameter says that you, you have some rate of imitation. These are the ones existing. This is your imitation and so this is the probability that you end up adopting 1 due to some imitation. This is the chance that you do it spontaneously and these are the fraction of people who've not yet adopted and who could make that change. So, that gives us an expression for the differential overtime. Very simple and intuitive model in terms of its basic building blocks. Okay. So, when we look at this if you want to solve this expression, right, and then you can start with an initial condition of F of 0 equals 0. If you solve that, you get a simple equation for what F of t looks like. And it depends on, on p and q, obviously. and so higher p's and q's are going to lead to faster diffusion so more people adopting by any particular time, lower p's and ques are going to lead to lower adoption rates. And so this thing will be increasing in p and increasing in q at, at any point in time the number of people would adopt by that time. Okay, so you get a simple solution. And then, let's look at the aspects of this and, and why the model has been so well used and, and is well-known. Okay, so first thing it's, it's going to end up giving an s shape, if q is bigger than p is going to tend to 1 in the limit as t becomes large. and basically what's going on in this model is initially, only, the only way that people can adopt is, is mainly going to be through p. And then eventually, q is going to become the important parameter. and things will slow down as your, a fraction of people that have adopted eventually reaches 1. So, if we go back and look at exactly what this equation looks like, let's try and analyze this in a little more detail and try and understand why we get interesting dynamics out of this. Okay, so first of all when Ft gets close to 1, this thing is going to have to slow down. And so, what happens is that the grade at which you're gaining new people adopting as F of t gets close to 1, this thing gets close to 0, right? And so, this thing has to get close to 0. So, this thing is going to tend to 0 as Ft gets close to 1. So eventually, it has to slow down just because the fraction of people who haven't adopted yet becomes small, so even if a lot of them are adopting, there's just not many of them left and that's what gives you the last part of the curve. So that's very intuitive. And it's coming directly out of the, the limitation that this thing has to converge at most to 1. When you're initially at 0, then this part is going to go to 0, right? So, there's no imitation going on because there's nobody to imitate. And everything is just happening from the spontaneous adopters. So, initially, this thing is going to, this thing is going to look like 1, everybodycan still adopt, but all of it's going to happen through the p. So, what starts out is you start out with a slope of p, right, so you start out with some initial adoption rate from 0, you're going to start out at a slope of p, and then eventually, the q's going to start kicking in. So, as you get more and more people adopting here, then the acute can kick in, and that can begin to, to give you the S shape. So, the idea can be that the S shape can start going up, as q begins to kick in, as more people start imitating, okay? But there's, there's a competing factor, which is, as more people are adopting, this thing is also going down, right, so you have fewer people left to adopt. So, whether or not you get this S shape is going to be a race between the increase in q and the decrease in the population that's left. And eventually, we know it has to, to, to asymptote and, and be concave. And so, the question is whether it's going to be initially convex. And so, when we can, we can analyze that by looking at this process close to 0. We know that the initial slope is, is p, so let's look at, at what happens at some small epsilon. So, we've, we've just started moving out. And we'll see whether we're going to start accelerating or not. Is it accelerating or is it going to be already decreasing in speed? So, what does dF dt look like at, at some small epsilon? It's going to look like p plus q epsilon times 1 minus epsilon. So, if you just plug in a small epsilon for this, you get this. To a need to get the initial convexity, you need this thing to be bigger than p, right? We have to be accelerating, the, the slope started out at p, now we have to be getting a slightly larger slope. So initially, to get that S shape, you're going to have to have this be bigger than p, and what does that tell us? that tells us basically that q is going to have to be bigger than p. So, for very small epsilon, the only way that you can have this thing be bigger than p is to have q be bigger than p. So, if q is bigger than p then you'll get it effectively the q, q epsilon is going to be bigger than the p epsilon. and so we end up with a situation where you get the convex initial condition. So, q bigger than p gives us the initial growth where we get a convexity at the beginning, okay? So, we get this S shape here if p is bigger than q, we get initially the, the slope is p, then it starts accelerating. And eventually, when F of t gets very large, it's going to have to slow down because now there's just not many of the adopters left, and then things slow down and we eventually get the asymptote towards to 1, okay? So, that's the Bass Model. very compact, simple, easy model. the the, this model has been used quite extensively. Why it has been used so extensively? it's, it's very compact, so if you can begin, you know, suppose that we've, we're just here at some time period, that's already enough to estimate what p was and to begin to see, to estimate what q is. So, as this process takes off, you don't need much data to begin to analyze and form estimates of p and q. Once you've got the estimates of p and q, you can get estimates of what the rest of the process is going to look like. So, this model has been used extensively in forecasting by trying to estimate from initial take up. So, if you look, say, at the box office of a new movie, how many people go see the movie in the first week? How many people go see the movie in the second week? Based on that first week and second week, you've already got a piece of two pieces of this curve. You can begin to project how long is it going to take for this thing to keep you know, to slow down and, and where will it eventually reach in terms of its asymptote. So, the Bass Model has been enriched a lot by adding in extra moving parts. Maybe some people wouldn't see the movie, so here this assumes it goes up to 100%, maybe some people would never see a certain movie. you can enrich it by, you know, adding in different kinds of heterogeneity and so forth. So, there's different ways to enrich this model. And people use richer versions of the Bass Model for forecasting. But this, this basic simple part gives us that S shape in a very simple way. And we realize that, you know, it, it's this combination of social imitation, which grows with time, which gives us that convexity, which is important here. Okay, beyond the, the Bass Structure what we're going to look at next is component structures and start bringing the network in. So here, you know, more generally, the reach of diffusion is going to be bounded by some component structure. So, if, if there's certain groups of people that don't interact with others, then you know, things aren't going to move across one group. it could be that some players or, or nodes are immune. So, if we're talking about the flu, people could be, have a vaccine against it and maybe not catch it. Or it could be that, you know, there's certain individuals who just wouldn't go see a certain kind of movies, that will never see a horror movie and, and so you could think of them as being immune to certain kinds of diffusion. it might be that some relationships between individuals that are transmitting with higher probability than others. So, if we start looking at the network structure, we're going to see that, that there could be some probabilistic transmission. so the answers of questions of when we get diffusion and its extent aren't answered so easily by the simple Bass Model. And we'll have to enrich these kinds of diffusion models in order to answer those, and so that's what we'll take a look at next.
