Why do all random graphs end up identical?

[00:00] Imagine you and a friend both want to draw a 

[00:07] in. Hey. Hey. How you doing? You know I do know 

[00:14] Look. Here's what you do. Okay. Both people draw 

[00:21] of vertices they flip a coin to decide whether 

[00:28] Initially, the two networks look very different, 

[00:35] countably infinite number of vertices, they would 

[00:43] just a network that seems similar, the identical 

[00:53] folks. I want you both to draw a new random 

[01:00] my plan to make it different is instead of 

[01:05] like on two different axes to make it random. 

[01:10] I would then join the next two edges together, 

[01:16] I just did that one. That's not going to be drawn. 

[01:22] a very different network. So my plan is I've 

[01:30] pair up two of them I'm going to get the smallest 

[01:35] in its binary representation and if it's a one I'm 

[01:41] first position in two is a zero. So I don't link 

[01:47] going to link those. First position in four is a 

[01:53] So I link those. Now I do all the second 

[02:00] that's all well and good folks, but it turns 

[02:06] same network. Unbelievable. Yes, it turns out the 

[02:17] Even though random graphs may start off looking 

[02:24] the more similar they become. And should you have 

[02:32] it will always converge to be exactly the 

[02:40] It's inevitable. But no matter what you do, 

[02:45] mathematician Richard Rado or Rado. Anyway, people 

[02:51] after him have and will continue to because it 

[02:57] graph and you won't hear graph theorists really 

[03:04] imagine the random graph. There's only one and 

[03:11] behind you and it is the creator of all other 

[03:18] the Peterson graph. Big fan. You can think of your 

[03:24] this set of vertices exists exactly like this as 

[03:31] graph that's in there too. Any finite or indeed 

[03:39] there somewhere. It's inevitable and it contains 

[03:47] the radioraph. And that could be it. That could 

[03:54] that phenomenal? So if that's all you're here for, 

[03:58] James 3 is sponsoring the video and I want you to 

[04:03] office are now open. I'll talk about more about 

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[04:11] while we could end the video here, wouldn't it 

[04:18] just said? Because with one additional fact 

[04:23] you can both convince yourself that the radioraph 

[04:30] yourself that it's inevitable. So, we're 

[04:37] it's just us people taking it seriously and we 

[04:42] loose on calling things either networks or graphs. 

[04:49] a collection of nodes joined together by edges a 

[04:57] going to call this a graph. So that's why I flip 

[05:03] but I want to use graph. So if you look things 

[05:07] um terminology. And we're talking a simple graph. 

[05:16] often call nodes because then if we call them 

[05:21] making this equivalent to shapes like polyhedrrons 

[05:27] have them linked together or you don't. That's 

[05:34] I mean in other situations this is fine but in 

[05:42] going out from one vertex and coming back in. 

[05:48] an edge or there is not an edge. And I've been 

[05:54] I strictly mean is a countably infinite graph of 

[06:01] of all for every whole number all the countable 

[06:06] vertex for that. So there's more than seven there 

[06:11] correspondence to the counting numbers and they're 

[06:17] they're just temporary labels to help us keep 

[06:22] that vertex. At any point we could re label them. 

[06:28] you cannot distinguish between the vertices 

[06:33] are not. And that's just it. Um the board can 

[06:39] foot. Uh so with that in mind, as long as you're 

[06:45] which can be a little annoying, but that's okay. 

[06:50] and you're okay following the logical steps of 

[06:57] We've got everything we need. We can prove these 

[07:02] said is true. We can achieve all of our hopes 

[07:09] using a single property of an infinite graph. 

[07:14] And what the extension property means is you can 

[07:20] infinite graph. And let's call them collection 

[07:25] other vertices you want and call them collection 

[07:32] vertex that is in A and B. So you pick your own A 

[07:38] want. The extension property means that out 

[07:43] which is not in A or B and it has the interesting 

[07:50] with every single verticy in A but absolutely none 

[07:58] vertices in A that you do want to all be connected 

[08:04] don't want to be connected and you're guaranteed 

[08:09] wishes. And that's it. If we can show that a 

[08:15] that mean that it definitely contains every other 

[08:22] that any two graphs that both have the extension 

[08:28] with the coin and the dice graphs in a moment, 

[08:33] the binary digits to decide if we're going to link 

[08:38] A and B and they're numbered. And again, the 

[08:44] the vertices in some order. Doesn't matter what. 

[08:52] You have two vertices. You pick whichever one has 

[08:58] will be our place position value. We then look at 

[09:05] other vertex's bigger number. And if that position 

[09:12] two vertices together with an edge. And we don't 

[09:18] for these two sets, we can always find by a method 

[09:26] fulfills the extension property requirements. And 

[09:32] okay, there's going to be some other vertex out 

[09:40] So, first of all, we just guarantee n's going to 

[09:44] bigger than six. Six has three digits in binary. 

[09:50] binary, it's going to be bigger. And then we go 

[09:55] what? I'm actually going to give it six digits in 

[10:01] numbers to make it do what we want it to do. So we 

[10:07] want it to link to three and four, so one's there 

[10:12] put a zero there. We don't want it link to six, 

[10:18] we want. So we happen to not care about the fifth 

[10:24] be a little bit lazy. which means now uh we've 

[10:33] Vertex number 13 if we were joining vertices up 

[10:41] extension property requirements. And that would 

[10:49] You could definitely find another numbered verticy 

[10:54] and you're doing that by construction. The 

[11:01] for the coin flip graph, we're going to start with 

[11:07] just going to pick some vertex out here somewhere. 

[11:14] a probability vertex I is the one we're after. 

[11:20] not you. We could say actually the probability 

[11:31] to some value X and X is between zero and one. It 

[11:41] one we're after. It's just probably not. So we'll 

[11:47] call that vertex J. There's a probability that's 

[11:53] that it's not I or J equals X to the 2 now because 

[12:02] that X squ is going to be smaller than X because 

[12:07] vertex at random K and then check is it not I J 

[12:14] if we had n different vertices that we've looked 

[12:22] we're after yet is x to the n. And as n approaches 

[12:32] zero. So the probability after we've checked 

[12:38] b that we haven't found the one that we're after 

[12:44] So the probability that vertex does exist is one. 

[12:53] guaranteed. I don't want to get into probability 

[12:59] some malicious graph like what if there were 

[13:03] point is if you've got infinitely many vertices 

[13:09] certain with probability one that the extension 

[13:15] x that's anything between zero and one. So this 

[13:21] a dice. As long as there's a nonzero chance you've 

[13:27] will have the extension property almost certainly 

[13:34] than that? The point is all random graphs converge 

[13:44] We now know that all our graphs have the extension 

[13:49] all the same graph in a moment. It's easier to 

[13:54] implies all other graphs are subgraphs. So as 

[14:02] graph. I have the diamond graph and I've 

[14:08] that's arbitrary. You can number them however you 

[14:13] other graph is somewhere in the radar. And we do 

[14:20] the lowest numbered vertex one, just pick a vertex 

[14:27] Then to fill in the second vertex, we work out 

[14:34] one. So we do want it connected to one. So we're 

[14:42] it connected to well, we haven't got anything else 

[14:48] kind of null set over here somewhere. Uh that is 

[14:54] by the extension property, we can definitely find 

[14:58] nothing in B. So now we've got two. Now three has 

[15:06] already got. So now we redefine A to not just be 

[15:15] B is still the empty category. And we know by the 

[15:23] three add it to the graph and then we just keep 

[15:29] we define our categories A and B to force that 

[15:36] property it will. So four four has to be connected 

[15:51] All right. I've defined three and four to be in 

[15:57] property we can find a vertex that's exactly 

[16:02] to both of these but not that one. And then we 

[16:07] and B. It guarantees we can find the next vertex 

[16:14] As long as we have the extension property, we 

[16:20] our choosing. Finally, I have two infinite graphs 

[16:27] binary, your choice. The point is they both have 

[16:32] they're exactly the same graph. And we're going 

[16:37] a region within the graph. And we'll keep the 

[16:44] both of the graphs, showing that the whole graphs 

[16:49] how we're going to do it. First of all, we'll use 

[16:56] Doesn't matter how you number them. Just assign a 

[17:01] one and vertex one in a region. Doesn't matter 

[17:05] else in the region. We're now on the first graph 

[17:13] which may or may not be connected to the first 

[17:17] same thing on this side. So using the extension 

[17:22] to the first one may not be the second one and we 

[17:27] vertex outside the region in this graph. Check 

[17:33] the region. Move it in. And at the same time, 

[17:39] one over here and move it in as well. And then 

[17:43] Find the matching one. Move it in. And then we 

[17:48] whichever one we move in on one, we can find a 

[17:53] other one. And because we're going backwards and 

[17:59] gradually match up all the vertices in both of 

[18:06] are identical vertices. And that's it. That's 

[18:15] identical. Now again, this does in involve not 

[18:22] these kind of matching things up. But if you've 

[18:26] and and matching stuff with countably infinite 

[18:32] this kind of matching approach. But it does work. 

[18:38] are exactly the same graph. All infinite random 

[18:46] that is the rando graph. And isn't it satisfying 

[18:52] the logic, the proof, and we took what were 

[18:58] made them well almost obvious. In fact, the longer 

[19:03] the whole extension property is just kind of 

[19:10] Any sufficiently well-explained mathematics is 

[19:18] in some cases a lot of explanation. But the point 

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