So, now suppose we're on a marketing campaign and to run a maximize the number of people who are going to buy a product. So the question that they ask is, how can we increase the number of people who are adopting? And what can we do to make this happen? Well given that we know the social graph, we have a few possibilities in theory. One is to lower the threshold. So if we can somehow convince people um,to lower their thresholds and be more open our products by improving it somehow. Even for just some of the nodes naturally, we're going to have a better chance to having everyone end up in state 1. Or increasing the number of people in state 1. So one way to do this could be through improving the product. Another way would be the break clusters. Right? So, if we can cut some social ties with the in-clusters that we see this would also work. So basically make them have less friendships with other clusters somehow. And, the last one would be to seed some of the nodes in the cluster. So, it would be trying to influence them directly. So, the last one, seeding is the most plausible one, here. Especially for a marketing strategy. Because we could presumably pay one of the nodes in the cluster to adopt our product. And in the case that we had before, where we saw this cluster of Eve, George, Frank and Hannah. Turns out that seeding any one of them with a product, so if we seeded Eve with the product, for instance. It would cost any, all the other ones to flip, right? So now that Eve is seeded with the product, Frank has 50% exactly, so he'll flip. And, once Frank is flipped now, George has 2 3rds of his flip so he will also flip. And Hannah also flipped at the same time. And that would have worked in any case. We could have seeded Hannah and then caused, Eve and Frank to flip and, then George to flip too in the next iteration. So more generally, the question we want to ask is. Which node or nodes should we seed in order to guarantee the maximum number will flip? And that we have the minimum time to reach new equilibrium? Well, of course, you know, in theory, we'd want to seed everyone, if we could, because that would, you know, have everything happen instantaneously. But realistically, you're going to be under some sort of a budget constraint. So which nodes you would seed, in order to maximize the extent of flipping equilibrium, that's the question. And the thing is that more influential nodes are probably going to cost more than others. For instance, celebrities. Like designer brands may want to pay some celebrity in order to adopt their products. Because that will cause the most people to be able to purchase it. So the question I want to ask is, under a total budget constraint. Which nodes do you want to seed, meaning do we want to pay to change their states. In order to maximise the extent of flipping at equilibrium. So maximize the number that will flip and minimize the time that it takes to reach that new equilibrium. So while this question is hard to answer, there's some intuitions that are clear. If you can see just one node, you want it to be the most important one. And how do we define the most important node? Well, by the centrality measures that we discussed before. [BLANK_AUDIO]. But what we can seed in more than one node is a little counter intuitive because you might on one hand think. That you should just choose the two most important nodes. But it turns out that you don't just want the two most central. You have to consider their combined influential power and make sure their combines powers are enough. Because by seeing the two most important nodes, you may not be having any additional effect over just seeding the most important node. And you may have to distribute the influence a little more. As an example consider again our original social graph of six nodes between Anna, Ben, Cara, Dana, Even and Frank. So we can draw that up. Anna to Ben and to Cara and Dana up here. And Evan down here. And they're connected, and then Frank over here. So if we can only seed one node, we're intuitively going to want it to be Cara. Because if you recall, Cara's the most important node, both in terms of closeness and between node centrality. And expect, and hence we expect her to have the most social influence. Most influence. So let's see what happens if we seed Cara. So we'll put our seed here. At time one, we see, nothing happening on this side. The time one that causes Anna to flip. And then, at time two. So, at time one Anna flips because, now, 50% of her links are flipped. And we're going to assume the threshold again. We're assuming it's 50% here. And at time two. It's going to cause Ben to flip. So, now nothing else is going to happen. We, we haven't penetrated this cluster over here. But basically what's happened is we, we have three nodes flipped after two durations. But still, we haven't penetrated this question. In order to so that we'd have to put some influence in there, somehow. The other hand, if we tried to see any of the other nodes, we would not have gotten as many and ended up flips. For instance, if we had seeded Anna. We would've only flipped Ben. If we had seeded Ben, we would have only flipped Anna. If we had seeded Frank, in fact we wouldn't have flipped anyone. If we had seeded Dana or Evan, we would've only flipped Frank. So, the maximum we would've gotten by flipping anyone else would have been one, but Cara's getting us three. So could we that was the best choice, for just one node, is to choose the most influential person. But now let's consider the case of two nodes. So you may think on the one hand, let's choose Cara and Anna because those were the two highest in terms of between centrality. But if you think about it here, Anna's not doing anything for us in terms of this other cluster. So if we seed Anna, all that this is going to do for us is lower the number of iterations down to 1 in order to see Ben. because she'll cause Ben to flip immediately and adopt the product. But that's all it's doing. It's not causing us to increase the number of nodes that are flipped. So it would be kind of be a waste to seed Anna and Cara. Especially if, if we consider the fact that Anna and Cara may cost the most money if they realize how influential they are. So this may not be the best strategy. For instance, let's consider picking Cara and Frank. So let's draw this again with Anna, Ben, Cara, Dana, Evan and Frank. So if we seed Cara and Frank. Seed, seed. If we seed Cara and Frank, then after one iteration. We see that Anna will be flipped, obviously. Dana will also be flipped because she has 2 3rds, and same thing with Evan. Evan will also be flipped. So, time one, we have Anna, Dana and Evan flipping. And at time two, we have Ben flipping. So we get everyone to flip after two iterations. So, this is all nodes after two iterations. So clearly that's a much better choice than choosing Cara and Anna. And we chose Frank who was the least influential in terms of between the centrality. And we got that to happen. But, wonder, do we even need to choose Cara at all. Maybe not. It turns out that wed don't have to. At present if we seed Anna and Dana, watch what happens. So you have Anna and Ben, this is Cara, Dana, Evan, and Frank. So if we seed Anna, this would be seed, and we seed Dana. And, of course, there's a connection here. This seed will cause Ben to flip, time one. And it will also cause Cara to flip because she has 2 3rds now, and it will also cause Frank to flip on the first iteration. So time one, we would have Ben, Kara, Frank. And then at time two, write that up here, we would get Evan. So summary here is that we have all nodes again and again at two iterations. And here we didn't even make use of Cara, the most influential person. So, there's many different ways of getting the same results in terms of seeding. the, the key point is that you have to make sure that the combined influence among the people or the collective influence is maximized. And that's not necessarily an easy thing to do, it's it's not a clear problem. For instance if you consider, a really really large graph. It would be very difficult to decide if you're putting multiple seeds in where are the optimal places to put them. Would be in order to maximize the number of nodes and to minimize the number of iterations it takes to get there. So here, as we said, seeding Cara and Frank, or Anna and Dana, will give best result. And in particular, we don't need the most influential node of Cara. [BLANK_AUDIO]
