[MUSIC] What if I make the second link like this? So I connect the second link differently. Now I also have four benefits, and the four arrows right, and four costs, these 4 arrows have benefits and costs, but now I have also indirect benefits and I kind of like looks like this. So, the lower two nodes also connect indirectly through one degree of separation. Now this indirect connection is there, but there's, according to our assumptions that we took, there is no cost associated with it. Because we said cost is associated with maintaining a connection with having an arrow coming into, so that is actually then the maintaining. So, we have indirect benefits, it kind of goes through here. And as a modeler, and that's my assumption in the modelling, I guess the word has benefits but it doesn't have costs, or maybe it has a lower cost that is a cost to somebody, as you can model how you want but we cannot keep it simple, very simply here, not make it so complicated. You just say it has benefits, indirect benefits. These indirect benefits, they are less than 0.8. So actually as you go through this one node, intermediating node, and I keep on going so it makes sense to multiply them. So 0.8 times 0.8, which would be 0.64, right, 8 times 8 is 64. So 0.8 times 0, yeah, because it's lower than one, so if I multiply them, gets lower very convenient for us. So I said the indirect benefit is 0.64, so if I now calculate that out, I have a direct benefit 4 times 0.8, minus a direct cost 4 times 0.2 plus an indirect benefit. And I have that actually twice, right? I have one indirect benefit going from one node to the other one and the other way around. So 2 times 0.8 squared, so 2 times 0.64 basically. Now if I sum that up, I get a higher number, 3.68. So actually my benefit social benefit then increased and it's higher than the previous case because now I take advantage of these indirect effects as well. So, the network actually will add the second link. It will tend to not maintain the second link independently on different nodes, it will add the second link rather to already connected node. Rich get richer logic, because that brings these indirect benefits with it. According to our model set up and you could set up as I said, your model differently, but we keep it very simple here. So the evolution of this network would actually tend to be pulled towards this configuration, this fourth configuration that we have here is different. Now what happens if we add a third connection? And we add the third connection and the same rich get richer logic. So the one that's connected already gets this other connection. Will we 6 benefits minus 6 costs, and then how many indirect benefits do we get now? We get 6 indirect benefits. So we have 6 direct benefits, 6 costs, direct costs, and 6 indirect benefits. One going from this node to this node, these are two, one going from this node to these nodes, these are two, and the other third one goes from this node to these node. So each one of these indirect routes, paths actually has two connections or the two benefits, the one benefit and the other benefit as well when they're connected. So, there are 6 indirect benefits, 6 times 0.8 squared. What we get here is, wow, that's quite a social benefit, 7.44, that's even higher, so the network tend towards this configuration. This is the socially most beneficial network, but is it also stable? What happens for example, if we reorganize the third link and and put it like this, so basically that's kind of like a straight line now, right? So all of them are connected and I could also pull it out it would be a straight line. What would happen then? Well, we have 6 direct benefits and 6 direct costs, how many indirect benefits do we have? Yeah, that's a tricky question actually, because we have four indirect benefits of path links 2, so of two degrees of separation, we have one here, and one here. And then we also have indirect benefits of three steps of separation, right? If you go from the first node, the first node can also go to the last node can go too. The separator I can also go all the way three steps. So this was modeled here by Jackson and his collaborators, how they model then, they then basically, Bob, multiply it again by 0.8. So each step and like reduces your benefit, the indirect benefits are not as much and as are smaller than one that's a good thing that normalizing it will be 0.8 times 0.8 times 0.8. So it would give me then also 2, two indirect benefits part of a path link of three, basically and that would be then if you calculate that out, will look like this, 0.8 at six times 0.8 minus 6 times 0.2. And then 4 indirect connection of path link two, and two in direct connections of path link three, that's the first and the last note, right? They're also connected off path link three. Now if I sum that up, I get a value that's actually smaller, 7.1, 7.1 something. So, in the previous configuration, I had 7.44, and now I have 7.1. So the network will not evolve towards that, it's just not beneficial. It might check it out just by trial and error, evolve towards that but then go back again to this previous configuration of 7.44 because that's more beneficial, it will find out it's more beneficial. And if you look at it, second to last configuration is actually a star network, right? So it's just organized a different way. There's one middle node and then these three hanging on, it's a star network with a configuration of four total nodes. And the question is why stars? Well, because the shorter path length are more valuable than the longer path length. And the direct links have an intermediate trade of between benefits and costs. That's why the star configuration is more beneficial than just having basically a straight line, does that make sense? If not, please feel free to go back and watch this segment again and forget the basic intuition behind that model, you don't have to learn this model in detail. It's just one model, but that's basically the idea of evolved networks. And we have maximum efficiency then for society as a whole.