All right, so this is one of the shorter lectures that we will have in this course. And, but we still did cover quite a bit of material in general. And so we looked at first, the idea of search engines. We had a specific focus on Google Search, but we looked at how Google came to be. We looked at the difference between relevance and importance, right, so relevance is how close your query matches to the given page, or how close a given page matches to your query. Like how many keywords on that page are the same as in your query. And the important score is independent upon what's actually searched. It's, it's just dependent upon the webgraph. And hyperlink connectivity among the webpages. Then we looked at the idea of a webgraph, and graphs in general. And so, we looked at how we'd shoot the webpages as nodes and hyperlinks as the links in that graph and how that was, that's really important when you're dealing with important scores specifically. Then we looked at PageRank, we had a really high emphasis on PageRank here. And we saw how they calculate their importance score terms, in terms of those recursive equations, recursive relationships, and simultaneous equations that we use to, to spread importance score, right? So, we calculate the equations, we wrote one out for each node, right? So you only had the number of equations being equal to the number of unknowns. again, we saw a recursive equations, and then we looked at two fixes that we need to make to that dangling nodes and disconnected graphs. Dangling nodes are nodes that simply don't point to anyone else, and they cause us to have no solution to the PageRank computation, and disconnected graphs, are just graphs with more than one connected component, meaning there's more than one subgraph. Or two different portions of the graph that don't connect to each other at all. And so some major themes we saw here. First, we looked at graphs and graphs are huge. They're a huge thing mathematical notion of graphs. And we're going to use them again. especially, as we move towards the Internet and discussing the Internet, we'll talk about graphics routers and that'll be another fun topic. And we looked again at randomization. And we've seen randomization before. We saw that with, in the WI-FI chapter, we, we looked at choosing a random contension window size and here that was in terms of just randomly selecting and entering URL, adding some randomization to that. So, that's a recurring theme throughout many network topics and different networking algorithms, the idea of randomization. And then the idea, again, of consensus, right? So, here we had to find the unique consensus, among the webpages. To determine what the rank order should be in order to make the output as useful as possible. But consensus, as we said, it's hard because first of all, it's not that easy to solve a huge, huge system of equations, right? And we have to guarantee that that system of equations is going to have a unique solution. So, coming up with that consensus is not the easiest task in the world, and even beyond this in adwords we said Google selling ad space. We won't have time to cover this in this course, but they have to come up with consensus among bidders, and bidders for those ad spaces and determine how to charge accordingly. And again, there's no right answer to that question. It's just what they think works the best. And then and Wikipedia has to follow the process of the rough consensus when they, whenever there's edits on a page and there's conflicting opinions on the edits and which ones are correct. they may establish some sort of a voting grounds in order to do that. And we look at we look at those in much more detail in the optional reading associated with this course. Unfortunately, we won't have time to cover them, even though they are very interesting topics. but yeah, so, we'll leave you with that and I hope you enjoyed the lecture.