Hi, folks. So, let's just wrap up our discussion and summarize what we've learned from our diffusion models. So, in terms of you know, basic lessons, the first thing we saw is that there are phase transitions, sort of, that can be very abrupt in terms of whether or not diffusion occurs or not. And so, you know, when we end up with very low densities of interactions we basically get no contagion. And we move to middle density and we begin to get a lot of diffusion and very rapidly in, in the model we looked at when we were computing things for something like the diffusion model of, on an Erdos–Renyi random graph or as a simple one word and component sized calculations going from an average degree of just one to three led us to almost no diffusion to almost complete diffusion. And then once we get to these higher densities we get sure infection, and, and essentially all that are susceptible to being infected. And decree, degree correlates with who ends up being infected both, you know, in, in most of the models you've looked at having higher rates of interaction are going to lead to higher infection rates. So, we've seen these kind of, of basic points. And, you know, more generally, in terms of the modeling diffusion modeling's going to be important. It's, it's going to be important in a lot of applications that go beyond just something like the flu, to model both information and pure effects. We saw that in the last analysis we just did. so you want to keep, you know, track of things that are not just simply a infe, infection models but people communicate, and that distinguishes such models from epidemiology. we need a lot more models and studies that identify the fine deals, details of what matters in interactions. So when we think about diffusion, am I, am I just processing information? Am I learning from other people? do I care about how they, what, what they're doing? Do is there peer pressure? So, you know, if we're looking at something even, even something as simple as the flu there's decisions. Do I get vaccinated? Does it depend on how many of my friends are getting vaccinated. do I not take a trip if there's an epidemic of flu in the city that I might be going to. There's all kinds of things that really enrich the setting and make the diffusion models much more complicated. And these are areas where epidemiologists are starting to do studies as we just saw. I mean there's studies in ecnomics. there's studies in different areas which are, which, you know, are enriching these things. And a lot of that's going to be done by, once we complicate these models, we can do these things by simulation. Take the simulated diffusion models and try and match them to data and actually learn quite a bit that way. So, network structure matters. sometimes it's retractable, otherwise simulations go a long ways towards making predictions. we can experiment with, with changes in network structure and begin to see, you know, how these things affect overall behavior. So, in terms of, of outlines, what we've done is we've gone through a lot of the course. we've talked about networks influencing behavior. And now, we're getting to sort of the home stretch where we're going to talk about enriching things. So, diffusion so far, we've treated until the last, video just a, we just talked about we were treating the process of diffusion as being fairly mechanical. And now we'll more explicitly try and model learning and model effects. And then we'll come to modeling games on network where we actually have strategic interactions between individuals, which will sort of be an enrichment of the last application we saw and trying to understand those. So, next up is learning and then we'll go to Games on Networks and understand interactions and behavior.
