So let's briefly summarize what we've covered in this lecture. We talked about social networking sites, and that's what we talked about in the beginning, how we motivated it, specifically Facebook and Twitter, which are the two dominant social networking sites today. And we looked at differences between the two, especially in terms of the graph and the way the graph is set up. Because on Facebook, the links are bidirectional, whereas in Twitter they are unidirectional, because just because I follow you doesn't mean that you have to follow me. We looked at charting influential power on social networking sites, and how certain companies are trying to reconstruct the graphs of these sites in order to figure out who the influential people are. And then we also talked about marketing campaigns and how you could stage some campaigns to try to seed some people with a given product in order to be able to influence others. And the ones that you want to choose are going to be the ones that are the most influential or the most important. We've talked in a significant portion of this lecture about measuring influence, which is the analysis portion of the problem. How do we quantify that? And specifically we talked about many centrality measures. We talked about three of them in particular. And there's many more than three. The three that we talked about were degree centrality which is simply counting the degree of the node. We've talked about closeness centrality which is finding the average shortest path distance from the node to all the other nodes in the graph and then taking the reciprocal of that, so it's looking at how close the given node is to all of its, other, nodes in the network. And we also looked at between-ness centrality which is looking at the, number of shortest paths that each node is lying on. So that's looking more at the connectivity of the graph. So there's no clear answer as to which of the centrality measures is the best. To a certain extent, when doing any of these problems you should look at all of the centrality measures, but clearly measures such as closeness and between the centrality are going to be more informative than those such as degree centrality. We also then turn our attention to the synthesis problem. So once we can figure out who the important people are, how do we use that to be able to influence adoption and adoption fractions within a population. And we looked at one particular model of contagion, and in order to do contagion, we said that every node needed to have a flipping threshold, and the nodes in this case are people. Different people will not necessarily have the same threshold, but we made that assumption that thresholds is certainly going to be influenced by the product. So, given the product if it's cheaper or attractive it's going to have a lower threshold, meaning that you're quicker to adopt it the more of your peers that you see adopting it. Then we turn that to say, how would we use the influential people in the network in order to seed a contagion and seed people with the given products in order to maximize the fraction of people at the end who had actually adopted the product. So major themes that we've covered here, which are present throughout other lectures as well, the first is importance scores. So centrality measures really are a measure of importance of a node in a different way. And indeed we could have looked at another measure of importance here, we could have used the, the page rank computations scores as being another centrality measure. So, figuring out which nodes in a graph are important is definitely present in many of the lectures. We looked at topology models as opposed to population-based models. We looked at population-based models under the context of YouTube, previously. And this was, topology-based, actually leveraging the social graphic connectivity among users to do influence. And finally again, that crowds are not so wise, so under certain circumstances we can have the wisdom of crowds work out if we have independence of assumptions, if you remember from back when we discussed Amazon and Netflix, but in these conditions where the network is now in place Last time we talked about information cascade and the fallacy of the crowds. And this time it's really about, the social graph and, having influence from your peers and having that type of a network effect go on. Thanks, and see you in the next lecture.