Hello again folks. So we're here to talk a little bit about directed network formation. So in terms of our strategic network formation we've talked about a lot of different things in terms modeling, stability, dynamics, transfers, and so forth. And we're going to talk a little bit about directed networks and then we'll also have couple videos on fitting such models. but let's talk now about directed networks. And the, the formation game in a, in a directed network setting becomes much easier. Because people can just unilaterally choose to connect to somebody else. And we don't have to worry about consent and so forth, so it's very easy to model these games. The difference is really going to be in the application. So this has to be a setting where I can really connect to somebody without them having to want to, to allow me to connect to somebody. So it does work in settings like citing somebody's article. I can do that without their permission, or linking to a webpage. I can follow somebody on Twitter and so forth without them having to follow me. So there are setting where you can have this kind of diretic, directed setting. And so what we'll do is we'll model that as a very simple situation where people just announce their preferred set of neighbors. And then a network forms if and he based on which ever links people want to form. And here we're going to keep track of ordered pairs. Now, so the fact that ij is in this network, ij is now different from ji, so this means i is following j. this means j is following i, or, or i formed a direct link to j, and, and so forth. And then just look at the Nash set of networks, where each person is forming some links to some the links that they want. Given the links that everyone else is forming. [BLANK_AUDIO] Okay now when we think about these kinds of settings, we have to think about the flow of payoffs too. So one situation is sort of a one way flow. If I follow somebody on Twitter and they don't follow me, then I get to hear what they say but they don't hear what I say. so that's a setting where the person who pays the cost then hears information from the, the node that they're accessing. Now two way flow could be that one player forms a link and bears the cost but both benefit from it. So it could be that for instance if I add a link to another page on the Internet, then that's good for me because people can get from my page from the other. It also helps the other page get new traffic that I direct there. phone calls, you know, they, they, the, there's you could think of this, as somebody bears the cost. two people end up but phone calls actually are across to both people involved if you think about time and so forth. So there's some situations where we could think about two way flow. Now the two way flow, there's a paper by Bala and Goyal in 2000 basically did a directed version of the connections model. So the model we can think of as being the same as what we saw from the Jackson Lewinsky version. But now what's going to be different is instead of having consent to form a relationship. People can form a relationship unilateraly. And I can just direct my link to somebody else. We'll both end up benefiting, so the benefits will look just like the the connections model we had originally, but now people can form a link unilaterally. So it's a different formation process but the same payoff structure. so the person, the benefits structure, the cost structure is going to be different. It's only the person forming the link that bears the cost. So if you just want to go through in terms of efficiency. The efficiency's going to be exactly the same as before. Except now that we have half as much costs being born, so instead of having 2c per link, we're going to have 1c per link. And then in terms of what that does for the efficient calculations, is it just factors everything down by a, a factor of 2 in terms of the costs. So the efficiency in this model is exactly as it was before except for we've just divided through by 2. and so you get complete networks if costs are low enough, star networks in the middle range and the empty network when costs are high enough. One thing here is I put complete and star in quotes, because complete doesn't mean that every link in both directions is present. It just means that every two nodes are connected, and in this case, they're connected by only one link. It doesn't make sense to have links in both directions because you bear twice the cost and there's no added information flow in this situation. So, if you, if you go through this, it's, it's similar architectures. slightly different cost structure. Somebody has to bear the cost. But you only want one person bearing the cost between any two individuals, because things flow in both directions once that's formed. So efficiency's exactly the same as it was before. when we begin to look at the Nash stable networks, I'll sort of list through what you, what you can find. You can check this. so if, if cost is very low, then the sort of complete network will be Nash stable. So basically it makes sense to cut an indirect relationship to a direct relationship. And somebody, if, if the other person's not doing anything, the other, you know. If one person's not doing it, the second person will have an incentive to do it. Medium cost range, medium low, all star networks are Nash stable, plus some other networks. Whereas a star now could be formed in, in different directions, so it could be that the centre is forming some of the links. And peripheral agents are forming some of the others, so you can have a star with multiple directions on the links. again, you know, it doesn't make sense to form links in both directions because it doesn't add any benefits, and adds to cost. so you'll see stars that are going to be Nash stable, and depending on, the configuration. it, it could be that center's bearing more cost or the periphery is bearing more cost, there can be combinations of different types of stars. So any star in that network is going to be Nash stable. when you get to a higher cost so that c is bigger than delta, now let's think a little bit about comparing two stars. So if we look at a situation where we've got a, let's look at two extremes. One extreme is where the periphery formed a link. So they all link to the center. Okay. So, what's going to happen in this setting, is, the, the sensor's not going to get any cost, but they'll get benefits. And and let's think of the if another one were the sensors bearing costs, okay. And you can think of hybrids of these. Right? Well what this says is this one is the only one of the stars that's going to be Nash stable. And why is that? because if you think about it, here the center is bearing a cost by connecting to somebody, and they're only seeing one delta benefit. So that's less than the cost, they would rather sever that link. So they're in a situation where they don't want to be maintaining relationships to other peripheral agents. If the peripheral people don't bring them indirect benefits. This one is stable. Why? Because by accessing the center these people are getting all kinds of indirect benefits of paths of length too right. So they've got a bunch of indirect they get a delta plus n minus 2 delta squared by forming the one link minus the cost. So that peripheral agents are willing to keep those things because they're they're getting this extra cost. when we look at a situation where the, when the cost becomes high enough, , then we end up with a situation where if things can be, have complete networks, be efficient, but not equalibria. So,Nash stable networks are going to depend on the exact cost structure. What, what's a little bit different here then wha, was before is that the, the relative costs are going to select out who might be forming the links. And so you can have some prediction about links going in one direction or the other, even though flow might go in both directions. Okay? So let's talk briefly about an, another version of a directed model, and this is a one way flow model. So now we have to keep track of directed flow. And this is when I form a link that I can access this person, I can listen to them and I can listen, I hear things indirectly that they listen to or here. So if I connect to somebody and they're connected to somebody else and so forth. Then the benefits flow back to me from these connections. But the fact that I connect it to them doesn't give them this person any benefits unless they've also connected back to me or have some indirect connection back to me. Okay, so thing flow in the direction and so the person who bares the cost also gets the benefits. The other person does not get the benefits. Okay? So one example of this is a directed connections model with no decay, where the delta is basically 1. And in that situation then, the utility becomes just the number of people that you can access via directed paths, minus however many links I formed out, outward. So what's my outward degree in the network times the cost, right. So I'm bearing a number. So if I have three different links then I'm going to have an out degree of 3 and how many people I reach is going to depend on how many people they're connected to. Right? So in this case, it might be that I can reach, 1, 2, 3 directly, plus 4, 5, 6, 7, 8 in total. And, so my reach would be eight. Okay, so in that situation, then you've got some number of people that you access for your, your links. Minus your degree times the cost for degree is now measured in outward degree. Okay, so, very simple payoff structure. And efficient networks in this world, as long as c is less than n minus 1, are going to be wheels. In particular, wheels that have a particular direction to them. You should be careful because those wheels also mean something else in, in graph theory. But Bala and Goyal used to work wheels so, basically this is a directed graph where each node points to a subsequent node which points back. And so, each one here can access all of the others. So if you want to think what's the best way to do this in so that each node can access every other node. So we got maximum reach, everybody's reaching n minus 1. We're doing this with only n links, so everybody's sponsoring one link and we're getting the maximum reach possible in this world. Right? So, and each, each link is responsible then for connecting a given individual to n minus 1 other individuals. Right? So we, we've got a situation where we have the most efficient architecture. possible being a wheel. Or if the cost goes above this, then it doesn't make sense to have any links. And, and then you're better off with the empty network. [BLANK_AUDIO] no stable networks when cost is, is low enough. then the wheels turn out to be the only Nash stable networks that are strict Nash equilibria. So that everybody has a strict incentive to keep the the nodes that they links that they have. if, if the cost gets to a higher range then, wheels and empty networks are the only strict, Nash stable networks. So then you get stuck at the empty network, because you, nobody wants to form a ring to somebody else unless, they bring in direct benefits. so, so basically what we've got here is a situation where again we have some conflict between stability and efficiency. You can go through and convince yourself of, of why these things are true. the strictness is important here in getting these things. There's lots of, of networks which are Nash equilibria that are not strict Nash equilibria. So for instance here we've got a situation where this is Nash stable. You can check that nobody wants to delete any link that they have, so for instance 2 accesses 1 and also accesses 3 and 4 via this link, so they're happy. 3 can reach everybody via the links, but they have to have 2 links because they have to reach 4 as well as, as 2 and 1, and, and so forth. You can go through and check this. But, 1 is actually indifferent between where they place this last link, right? They're indifferent between having it here. They could also put it to 4 instead. And they could access everybody through either connecting to 3 or connecting to 4 since there's direct links in both directions between 3 and 4. Okay? So this is Nash stable, but not strictly Nash stable. So the if you check on the wheel, if we go back to the wheel then that's a a situation where nobody. So if we did this one in terms of a wheel, now anybody who changes the the links of severed this one and put it somewhere else. You would actually lose access to somebody. So they would no longer be able to be given the direction of the network now, the one way flow part they would no longer be able to access that node. So strictness is important in, in making those results come out. Okay, so that's just a glimpse at directed network formation. Theres going to be different applications, and I think that's one thing that's important to emphasize here is that which. Model that we should be using has nothing to do with whether we like unilateral or, or mutual consent formation as a, as a model. What's important is, what does the application actually demand? So, if we're dealing with situations with alliances, or friendships, or social relationships. A whole series of things you really need to have mutual consent and, and two way formation processes. If you're doing things like citing an article, forming a link on a web page. then you can deal with unilateral network formation, and so it's not a question of which is a better model. there are different models and they fit different applications. And some network settings are ones that are naturally directed and unilateral. others are ones where mutual consent is really needed, and a lot, a lot of social settings are going to fit into that category. And so which model you use really depends on the application. It's not an issue of which one's you know, sort of a, a nicer model; they're, they're a different models. Okay. so that sort of takes us through a lot of modeling. Of, of, of strategic network formation and some basic looks at different issues. And we'll have a couple of looks at models for fitting these things in in dealing with data. And then we're going to turn to diffusion as the next major topic. So we'll start working. This has been, so far, we've been looking at network formation, and the next major subject is now, given the network, what's actually happening on that network? How do we understand behavior on that network? And what the consequences of different network formations for the actual behaviors that result.
