So we just discussed the basic metrics of a graph and how they can make certain nodes more, central than others. And, that was the analysis part of the problem. So, if you remember back from the first video, we talked about analysis. And with this in mind, we're going to now move to the synthesis part. In which we're going to try to apply what we've learned. And we're going to continue our discussion of influence models. But this time, taking a social graph into consideration. So social graph meaning topology. And more specifically, we'll try to understand how a person social relationships can influence her to adopt a product or good. Depending upon how many of our neighbors have done so. So the question is how does social relationships influence adoption of products or goods? So we consider to start off a Star network. A star network is one that has a one center node which emanates out to all these other nodes All the terminal nodes. And none of these terminal nodes are connected to one another either. That's how you get the picture of a star. So each of the nodes here is in one of two states. State 0, which is, has an n. Which, intuitively, or, the interpretation there, means that they have not adopted a given product or good, whatever that product may be. The, the people with n. These guys have not adopted it. And those in state one, are with y for yes. And that means that they have adopted it. So these guys have adopted a product. So in this case, I'm half of these nodes have adopted it and half of them have not. So the question that we want to ask is do the four nodes that are in state 1, so those four nodes, constitute enough social influence in order for the center node to flip? Meaning for the center to adopt the product as well. So is that going to cause the center node to become a yes or will the center node not and stay as a no. So in this next example suppose you're the person in the middle. So this is you in the middle and the product that we're considering is an iPad. So some of your friends have recently purchased the newest brand of iPad. Meaning they're in state 1, so these guys all have this, the new iPad. And the other ones don't, which are in state 0. So you can imagine if the more people you see that have the product, that have the iPad. I'm assuming that they like it and they're satisfied and positive with it. The assumption, I guess, would be that if they don't like it they're going to discard it, in this case. The more influence that would have over you and the more likely you would be to buy the iPad 2. So is there a way for use to tell whether or not you will follow suit and buy the such an iPad? So in order to answer this question, we have to take the approach of finding a flipping threshold for each node. The flipping threshold is the fraction of the node's neighbors that must have already flipped before that node will flip as well. So, in this figure, 50% of the center node's neighbors have flipped, right because four out of eight equals 50%. So 50% or one half of them have already flipped. So, if your threshold, whatever value is between 0 and 1, is less than 50%, that means that you will flip. And again, by flip, we mean go from state 0 to state 1. On the other hand, if your thresholds was greater than 50%, you don't flip. So no flip in this case. So it's going to depend up what your threshold is. So for instance, if your threshold was 80% then 50% is not enough. So, that's a no go. That means you're not going to flip. You're going to stay in state 0. And if, if the threshold was that high, then that would mean you'd need at least, 80% of your neighbors, to have flipped. 50% as we said is not enough. Since you have eight neighbors you would need seven of eight of them to flip because that, will be greater than 0.8. And 6 8ths on the other hand is 0.75, which is not. So, in order to determine the number of neighbors you need, you could just kind of count up sequentially. And find out the first number of the smallest number that's greater than the, the threshold. So that's, that's the number of chambers you need to have flipped. Now, realistically the flipping threshold is hard to estimate. Because it's going to depend upon a number of things like the product you're considering. Right. So, some products are cheaper than others and some are naturally more attractive than others. So, cheaper more attractive. And those two things, as it gets cheaper and as it gets more attractive, it's intuitively going to lower the threshold for a given person. Now typically, it's going to depend on the person, herself. Some people are easy to sway and others are not. And so I might be easy to sway and I might be, if I see one of my friends buy something, I have to have it. Whereas you, may be harder to budge. You may not want to buy something, in that case. And there's other factors, as well. There going to influence this like the strength of social ties. So we can say here, also the strength of social ties, strength of social ties, right. So some peop, some of your friends may influence you more than others depending upon how close they are to you. And how we define a link always is a consideration, among other things. But for our purposes, we're going to assume that we know the flipping threshold. And additionally, we're going to assume that it's the same for each node. So we're going to assume that, basically we're going to say that it's homogenous. Across the node. So everyone has the same flipping thresholds and we know what that flipping threshold is. [BLANK_AUDIO]
