[MUSIC] So what I wanted to tell you about smaller word networks, which leads us to our last kind of network formation that I want to talk to you about. And actually, I want to connect that to an entire branch of research that has to do with a very important question of how we can grow a certain kind of network. So can we actually grow a network with particular characteristics? Of we said, well, the small world and the scale-free, the preferential attachment networks, they just we find them often society so it's useful to grow them and to study them. What the random networks are kind of like a contrast and it was a remind fight. But there's an entire another question that is, can we grow networks with certain characteristics? For example, in a very efficient network or a very stable network and for the social side and that's of course a very important, because they said the social is actually is a network, right? The network is what makes the total more than the sum of its part it, it makes a society out of individual. So can we grow networks with this specific characteristics? And I want to expose you a little bit to that idea now in this examples. So one of the logics that we can go about it is to take a logic from economic cost and gains. Because having and maintaining connections has benefits. You gain from having friends, but it's also costly. It takes takes your time, might take resources for company to connect to many other companies, collaborators, clients, if you work in a team. If we have a team that's too big, there's so many people you don't have time for everybody. So there are costs and there's benefits, the benefit might be that a big team gives you a lot of diversity of skills, but it's also cost. So we go with this setup. So basically with the basic assumption that just a very reasonable assumption that making connection has benefits, but also costs. And then we kind of like grow and rearrange and form the network. We add for example links, you can also add nodes, but let's go by adding links that have certain cost and gains until we are an equilibrium. Equilibrium is when nothing changes anymore. So in physics, there's a definition of nothing changes on average things are in an equilibrium, or if you happen to be from economics, an equilibrium, the economy is pretty much stable. So sometimes people have this example of if you roll a ball in a sink, the ball keeps rolling until it stops, until it's at equilibrium. Then nothing changes anymore. Often, in social negotiations you can think about it that nobody can actually improve their position at the point of a stable equilibrium. Think about it like tic-tac-toe. So here you play tic-tac-toe and then the stable equilibrium would be either one of the party wins or that you have a tie. For example, just like right now, you have a tie. And then you're in equilibrium, basically in deadlock in that. But the equilibrium is also useful, because it makes, it's stable. If one person could still benefit, change something in benefit while the network will always change and always changing, always change. So if nothing changes, if nobody can get better or worse than we are in equilibrium basically, right? And then we have a stable network configuration. So that's our goal, to make the network stable, that's one goal. And no link or node can be added and deleted anymore, it's called a pairwise-stable equilibrium. And now, we also want to make the network efficient. Stable as one and these are complimentary goal. As we can see there's a certain trade off as well of making it work efficient and making it stable. Let's work with both of them. And we use a symmetric connections model, that's what it's called. And we look for efficiency. And while we go about it is we say, a given a certain cost and a benefit of links, what's the most efficient network structure? So for example, you could assume that well theirs cause and benefits, monetarily or timewise. You can normalize that and say 80% you get a benefit of making a connection and 20% is the cost of having the connection. So it might be in dollars or in times. You save 80% of the time, but it costs you 20% of the time i if you keep the connection with this collaborator or in financial. So we work with this kind of trade off. And these values are defined by you, by your assumption, by what you find an empirical reality. So you're looking at bigger reality and you calibrate your model that's it. Now what we find is if we have a very low cost of making a connection, if it's basically free to make a connection but we have a lot of gains. Then the best network we can build is a completely connected network. If only benefits and no downside then the best thing is I just connect with everybody. Now on the other hand if I have a very high cost, if my cost of maintaining a connection is much higher actually than the benefit that I get from it, then I just cut ties, right? If all your friends are extremely annoying, and just steal your time and your money is, you better just stay at home by yourself. So the cost is much higher than the benefit then the empty network, [LAUGH] will be uniquely efficient. Otherwise if if this very low cost completely. Connected network, what happens in between them? Usually, we have something in between, some costs and some benefits. And it turns out that there, in general, you can come up with a star network. So the star network would be the result of being uniquely efficient Internet. And why the star network started work. We've mentioned that before, for example, happens a lot and you see that a lot. For example, in airlines, that's the reason why you always fly to some cities like Denver or Dallas and change planes, right? We said that's because it has a small average path length if you go through the star. You can get to everybody quite quickly, still. And it has to do with the small average path length. But let's become a little more concrete and do it with a cost-benefit analysis to show that in the immediate level of cost and benefit, the star network, the star configuration of the network is actually the uniquely efficient, stable network configuration. All right, so we work with four nodes, very simple, and we have an empty network to begin with. No benefits of links, because there are no links and no cost. Now we make the first connection, two degrees. So two nodes benefit, right? In that case, the left upper node benefits and the left lower node benefits because they have a connection, and both of them also have costs. So we have two benefits and two costs. Let's go with this example that I mentioned before, so my benefit would be 80%, my costs would be 20%. And we normalize that between zero and one. It could also be, I don't know, a different number in time of dollars or time and calculate this way. So we have 2 times the benefit of 0.8 minus 2 times the cost of 0.2. So two times 80% benefit minus two times 20% cost. If you put that together your net benefit would be 60, 0.6, right, 0.8 minus 2 level be your net benefit. Now if you calculate that out, you get 1.2, 0.6 plus 0.6 is 1.2. And that's your net benefit, your left over benefit, you have a benefit of making this connection. So your network would go from the empty network. It would make this first connection. It's more beneficial for this network to have one connection than not having anybody. So this network would evolve towards making one connection, it is beneficial. What happens to the second connection? Well, if I add a second connection, I have four benefits and four costs, right? So if you calculate that out, four times the benefits minus four times the cost, what do you get? Do we get a higher or a lower value, and what number do you get? So 4 times 0.8 minus 4 times 0.2, this is not like we don't do math in this course because most of our math will be done by computers. That's why we do computational social science, but it gives you the intuition. I mean, this is basic, use your phone to type it in. To calculate it or do behavior. That's basically but that's counting. That's what we do. Okay, so four times 0.8 minus four times 0.2 is 2.4, that's twice as much benefits. So the network will be drawn towards this which is more beneficial to society, right? It's more beneficial than this, in the empty network which is zero. The one connection with one length is 1.2 and with two connections, has 2.4. So it's more beneficial to add the second link. Our society of eventually will add the second link, right? So that's where the network will evolve.