In general when we look at the popularity of a product, a service. There are two main factors involved. One is, what we can an intrinsic value and this is, for example, when you enjoy certain music or buy a particular product just because you like it, Whether the rest of the world agrees with you or not. In contrast, the second one is called the network effect. That's because what you do and decide depend on what others do and decide. And it could also be two possibilities of that, one is, we got information dependance. Because the fact that others use something gives you some information. And this may lead to what we call information cascade coming up or in other words, the fallacy of crafts, instead of wisdom of crowds. The second type is because the fact that others use the same product gives you a higher value on this product, A typical example is the fax machine. When you are the only one with a fax machine in the world, then it's actually of no use to you. And the more people with fax machine, the higher the value of that product is to you. And similarly, for Wikipedia, exhibiting a strong positive network effect through this valuation. So one thing to be looking at different models for informationary effect, for valuationary effect, and coupling with the intrinsic value. For this lecture, we'll stay with population based models, meaning that whatever happens in a model depends only on the number of the fraction of people adopting a certain product or service. There's no topology involved and we'll be looking at two types, one is deterministic interaction in the example of information cascade. And then, in advanced material part of the video, synchronization models. We'll also be looking at random interaction models including tipping, and later in advanced material also diffusion. Then, in the next lecture, we'll look at topology based model where the graph and the matrices associated with the graph come into the picture. For any of these models, there are at least, three questions we want to formulate more precisely and then try to address. The first one is, what is the distribution of the nodes whether you know the graph or not, in different states and equilibrium? So, we have to find the states and we see binary states, we see re-state models and many state models, and this is a question about equivalent behavior. This is could be the national equilibrium through dynamic interactions of best responses, could be of equilibrium of convergence of iterated algorithms, And so on. So, the second question then is the amount of time it takes to reach in equilibrium.. That means, we have to define a time slot. We'll see that most of our models will assume a discrete time slots as usual in this course. But, there are a couple of exceptions where we also assume continuous time just to make the math in those models more attractable. The third question is often much challenging. Transient behavior, before reaching an equilibrium. Now, which model to use? That highly depends on what you want to do. Okay? Are you trying to model that fact sometimes people act at the same time? There may be synchronization models what you want, Okay? Or maybe a few early adapters changing other peoples minds. Then, maybe you want difusion model or contagion model. You want to say a few people carry the whole system over some threshold. Then, maybe a tipping model a bit more appropriate. So, each of the mathematical, models we're going to be looking at is motivated by different particle use, and we will have to be careful as to which one to adopt. So, here is a basic division of the models we'll be looking at. We will look at a population based where topology doesn't matter before we move on to topology base in the next lecture. We'll look at deterministic Interaction among the nodes, as well as probabilistic random interaction models. We will look at info cascade first in advanced material, we'll look at synchronization. And then, we'll look at tipping from random interaction model, followed by diffusion in advanced material so these two will be in the advanced material part. And for topology dependent next lecture, we'll look at contagion as well as infection, based on whether it's determinist or random interaction. And, in the advanced material also random walk which we saw a little bit from Google's PageRank early on.