Okay. so let's just wrap-up our discussion of, learning. And, in terms of a summary, in terms of what we saw in the Degroup Model, we found convergence and consensus if and only if we had aperiodicity. the limiting influence was related to iten vector, concepts and gives a nice foundation for that and in fact. Has this nice intuition that how important how influential a given individual is depends on how much weight they get in which depends on how much weight they're the indirect weight that they're getting from their neighbors and so forth. getting wisdom out in terms of a overall society converging nobody's retains too much influence that's going to be important. now in terms of the learning models that we've gone through. We saw that the Bayesian model was computationally demanding in a number of setting, of ways, in terms of the kinds of calculations that people might have to do and the gaming that might go on in that kind of setting. So those models can be quite set difficult. The restricted Bayesian version we saw gives a consensus, but the, the network didn't play much of a roll. Understanding this sort of limits in the Bayesian setting is something that is still being studied there's a number of paper's looking at this the DeGroot model is a very attractable alternative model it, it is much more naive in terms of the way that people update non the less it can be accurate so that's an interesting aspect of it as long as it's somewhat balanced in terms of the way that people are getting weights in. And nobody's retaining too much influence, it still be a very accurate way of updating. the nice thing is that it, it, it, the mathematics behind it are simple, it has some intuitive feels to it and it can be useful in terms of working with data. Now, in terms of the to-do list there's a lot that, that is missing from these models and you know, there, there are models being developed of, they're sort of between myopic and rational combine some of these things. There's also richer settings that we might want to think about and so, obviously there's a lot of the world where consensus is not reached. And this model, the models we were looking at were very simple ones in the sense that first of all we're running time to infinite. we're assuming some stationarity in terms of what is going on. It's not as if the world was changing and, and, as we were going along. And the there wasn't any strategy involved. So wasn't just if people had different preferences for what other people want to believe. So if you think about voting in election and you want certain, you have a bias in terms of what kinds of programs you'd like to be enacted then you might want to convince somebody there's somebody is a better politician than somebody else and so strategic communication could be very different than the kind of. Of settings that we were looking at where everybody's just trying to estimate more or less the same thing and there wasn't any strong preference involved. So you know, we could enrich things and, and begin to look at that and there's some work being done on that. there's also a lot that we can do that it, that I didn't talk about here but that has been done in terms of beginning you know so for instance beginning to the group models very attractable in understanding how the speed of learning depends on say the segregation structure of the network. So if we bring the homophily into play. And try to understand how things work. You can do that. you can begin to, to understand more about how speed depends on the, the structure of the network and how it relates to different properties of the network. So there's a lot of tractable things that can be done here. And so this is sort of an important area of research in, in understanding how society Forms opinions and another thing that's sort of missing here is, you know, the, the in terms of the network, we've a little bit agnostic about what nodes are. And you could imagine that some nodes are news outlets, some are individuals, so, so we get our news and, and information in different ways. You could begin to enrich the models by taking into account different types of nodes and things. So there's a lot to be done and a lot that we need to still understand about how learning works. The nice thing is that there are things that we, one can do to make things reasonably tractable and to work with models where we can begin to say something systematic about how the network structure and influence the eventual beliefs And there's sort of an active area for things to be looked at. Okay, so that wraps up our discussino of learning. And the next thing we're going to be taking a look at is looking at games and networks. So now we're going to be looking at situations where people's decisions of whether or not they're going to take an action really depends on what they think their friends are doing, and That will give us a game structure. And then we can try and analyze what happens in that setting.