WEBVTT 00:00:18.198 --> 00:00:20.240 YUFEI ZHAO: So we've been discussing graph limits 00:00:20.240 --> 00:00:21.790 for a couple of lecturers now. 00:00:21.790 --> 00:00:23.970 In the first lecture on graph limits, 00:00:23.970 --> 00:00:28.340 so two lectures ago, I stated a number of main theorems. 00:00:28.340 --> 00:00:30.830 And today, we will prove these theorems 00:00:30.830 --> 00:00:34.310 using some of the tools that we developed last time, namely 00:00:34.310 --> 00:00:35.720 the regularity lemma. 00:00:35.720 --> 00:00:40.130 And also we proved this Martingale convergence theorem, 00:00:40.130 --> 00:00:42.800 which will also come in. 00:00:42.800 --> 00:00:45.920 So let me recall what were the three main theorems that we 00:00:45.920 --> 00:00:49.160 stated at the end of two lectures ago. 00:00:49.160 --> 00:00:53.150 So one of them was the equivalence of convergence. 00:00:53.150 --> 00:00:56.570 On one hand, we defined a notion of convergence 00:00:56.570 --> 00:01:04.190 where we say that Wn approaches W, by definition, 00:01:04.190 --> 00:01:09.340 if the F densities converge. 00:01:15.390 --> 00:01:17.780 We can say convergence even without a limit in mind, 00:01:17.780 --> 00:01:22.640 where we see a sequence converges if all of these F 00:01:22.640 --> 00:01:25.400 densities converge. 00:01:25.400 --> 00:01:28.040 So the first main theorem was that the two notions 00:01:28.040 --> 00:01:34.280 of convergence are equivalent, so one notion being convergence 00:01:34.280 --> 00:01:38.120 in terms of F densities, and the second notion 00:01:38.120 --> 00:01:43.720 being convergence in the sense of the cut norm, the cut 00:01:43.720 --> 00:01:46.500 distance. 00:01:46.500 --> 00:01:48.130 There was a second term that tells us 00:01:48.130 --> 00:01:51.700 that limits always exist. 00:01:51.700 --> 00:01:53.650 If you have a convergent sequence, 00:01:53.650 --> 00:01:58.730 then you can represent a limit by a graphon. 00:01:58.730 --> 00:02:01.300 And the third statement was about compactness 00:02:01.300 --> 00:02:03.452 of the space of graphons. 00:02:03.452 --> 00:02:05.410 So we're actually going to prove these theorems 00:02:05.410 --> 00:02:06.380 in reverse order. 00:02:06.380 --> 00:02:09.590 We're going to start with the compactness and work backwards. 00:02:09.590 --> 00:02:12.100 So this is not how these theorems were originally 00:02:12.100 --> 00:02:15.610 proved, or not in that order, but it will be helpful 00:02:15.610 --> 00:02:18.640 for us to do this by first considering 00:02:18.640 --> 00:02:20.950 the compactness statement. 00:02:20.950 --> 00:02:23.480 So remember we're compactness says. 00:02:23.480 --> 00:02:31.310 I start with this space W tilda, which 00:02:31.310 --> 00:02:39.730 is the space of graphons, where I identify 00:02:39.730 --> 00:02:41.530 graphons that have distance 0. 00:02:51.610 --> 00:02:55.780 So if they have cut distance 0, then I refer to them 00:02:55.780 --> 00:02:56.680 as the same point. 00:02:56.680 --> 00:02:59.360 So this is now a metric space. 00:02:59.360 --> 00:03:02.990 And the theorem is that this space is compact. 00:03:13.890 --> 00:03:15.975 So I think this, it's a really nice theorem. 00:03:15.975 --> 00:03:18.480 It's a beautiful theorem that encapsulates 00:03:18.480 --> 00:03:21.720 a lot of what we've been talking about so far with similarities, 00:03:21.720 --> 00:03:27.150 regularity, and what not, in a qualitatively succinct way, 00:03:27.150 --> 00:03:31.080 just that this space of graphons is compact. 00:03:31.080 --> 00:03:34.410 You may not have some intuition about what the space looks 00:03:34.410 --> 00:03:36.480 like at the moment, but we'll see the proof 00:03:36.480 --> 00:03:39.090 and hopefully that will give you some more intuition 00:03:39.090 --> 00:03:41.040 I first learned about this theorem 00:03:41.040 --> 00:03:45.240 when Laszlo Lovasz, who was one of the pioneers in the subject, 00:03:45.240 --> 00:03:47.250 when he came to MIT to give a talk 00:03:47.250 --> 00:03:48.840 when I was a graduate student. 00:03:48.840 --> 00:03:52.290 And he said that analysts thought 00:03:52.290 --> 00:03:55.110 that they pretty much knew all the naturally occurring 00:03:55.110 --> 00:03:57.140 compact spaces out there. 00:03:57.140 --> 00:03:59.940 So there are lots of spaces that occur in analysis 00:03:59.940 --> 00:04:01.530 and topology that are compact. 00:04:01.530 --> 00:04:05.910 I mean, the first one you learn in analysis undergraduate 00:04:05.910 --> 00:04:08.160 is probably that an interval is compact. 00:04:08.160 --> 00:04:10.590 But there are also many other spaces. 00:04:10.590 --> 00:04:13.290 But this one here doesn't seem to be 00:04:13.290 --> 00:04:16.089 any of these classical notions of compactness. 00:04:16.089 --> 00:04:22.480 So it's, in some sense, a new compact space. 00:04:22.480 --> 00:04:23.880 So let's see how the proof goes. 00:04:27.520 --> 00:04:32.620 Now, because we are working in a metric space, 00:04:32.620 --> 00:04:42.330 it suffices to show, due to the equivalence between compactness 00:04:42.330 --> 00:04:45.340 in the sense of finite open covers 00:04:45.340 --> 00:04:54.280 and sequential compactness in the metric space, 00:04:54.280 --> 00:04:57.340 so it suffices to show sequential compactness, 00:04:57.340 --> 00:05:28.190 that every sequence of graphons has a convergent subsequence 00:05:28.190 --> 00:05:32.000 with respect to this cut metric and also will 00:05:32.000 --> 00:05:40.000 produce a limit as a convergence subsequence with a limit point. 00:05:43.390 --> 00:05:44.620 So that's what we'll do. 00:05:44.620 --> 00:05:46.720 I give you an arbitrary sequence of graphons. 00:05:46.720 --> 00:05:49.390 I want to construct by taking subsequences a convergent 00:05:49.390 --> 00:05:49.890 sequence. 00:05:49.890 --> 00:05:53.150 And I will tell you what that limit is. 00:05:53.150 --> 00:05:57.280 So here is what we're going to do, given this sequence. 00:05:57.280 --> 00:06:00.190 As I hinted before, it has to do with the regularity lemma. 00:06:00.190 --> 00:06:01.960 So we're going to apply the regularity 00:06:01.960 --> 00:06:05.600 lemma in the above form, which we did last time. 00:06:05.600 --> 00:06:16.100 So apply the weak regularity lemma, 00:06:16.100 --> 00:06:24.300 which will tell us that for each Wn, 00:06:24.300 --> 00:06:31.190 there exists a partition, in fact, a sequence of partitions, 00:06:31.190 --> 00:06:33.560 each one refining the next. 00:06:33.560 --> 00:06:37.400 So what's going to happen is, I'm going to start with Wn 00:06:37.400 --> 00:06:42.020 and starting with a trivial partition, apply that lemma, 00:06:42.020 --> 00:06:48.460 and obtain a partition P sub n, 1. 00:06:48.460 --> 00:06:51.820 And then starting with that as my P0, 00:06:51.820 --> 00:06:56.020 I'm going to apply regularity lemma again and obtain 00:06:56.020 --> 00:06:56.800 a refinement. 00:07:02.010 --> 00:07:05.530 I will have this sequence of partitions, 00:07:05.530 --> 00:07:11.470 each one refining the next. 00:07:17.640 --> 00:07:19.590 So all of these are going to be partitions 00:07:19.590 --> 00:07:21.690 of the 0, 1 interval. 00:07:21.690 --> 00:07:23.882 And as I mentioned last time, everything's 00:07:23.882 --> 00:07:24.840 going to be measurable. 00:07:24.840 --> 00:07:27.070 I'm not going to even mention measurability. 00:07:27.070 --> 00:07:31.610 Everything will be measurable such 00:07:31.610 --> 00:07:35.640 that they satisfy the following conditions. 00:07:35.640 --> 00:07:38.750 So the first one is what I mentioned 00:07:38.750 --> 00:07:41.670 earlier, is that you have a sequence of refinements. 00:07:41.670 --> 00:07:51.400 So each P sub n k plus 1 refines the previous one for all n 00:07:51.400 --> 00:07:51.900 and k. 00:07:55.560 --> 00:07:59.780 And the second condition, as given by the regularity lemma, 00:07:59.780 --> 00:08:03.410 you get to control the number of parts. 00:08:03.410 --> 00:08:07.500 So I will say in the third part what the error of approximation 00:08:07.500 --> 00:08:08.000 is. 00:08:08.000 --> 00:08:10.280 But you get to control the number of parts. 00:08:10.280 --> 00:08:16.760 So in particular, I can make sure that this number here, 00:08:16.760 --> 00:08:22.580 the number of parts in the k'th partition depends only on k. 00:08:32.299 --> 00:08:36.039 Now, you might complain somewhat, 00:08:36.039 --> 00:08:40.059 because the regularity lemma only tells you an upper bound 00:08:40.059 --> 00:08:42.150 on the number of parts. 00:08:42.150 --> 00:08:45.030 But that's OK. 00:08:45.030 --> 00:08:46.470 I can allow empty parts. 00:08:50.990 --> 00:08:58.620 So now I make sure that the k'th partition has exactly nk parts. 00:08:58.620 --> 00:09:04.380 And the third one has to do with the area of approximation. 00:09:07.110 --> 00:09:17.100 OK, so suppose we write W sub nk as the graphon obtained 00:09:17.100 --> 00:09:18.870 by applying the stepping operator. 00:09:18.870 --> 00:09:24.210 So this is the averaging operator on this partition, 00:09:24.210 --> 00:09:25.740 corresponding to the k'th partition. 00:09:25.740 --> 00:09:30.060 I apply that partition, do a stepping averaging operator 00:09:30.060 --> 00:09:32.080 on the n'th graphon. 00:09:32.080 --> 00:09:36.080 I get W sub nk. 00:09:36.080 --> 00:09:44.380 The third condition is that the k'th partition approximates-- 00:09:44.380 --> 00:09:51.780 it's a good approximation in the cut norm up to error 1 over k. 00:09:51.780 --> 00:09:54.950 So 1 over k is some arbitrary sequence going to 0 00:09:54.950 --> 00:09:57.138 as k goes to infinity. 00:10:12.100 --> 00:10:14.520 So I obtained a sequence of partitions, 00:10:14.520 --> 00:10:17.390 so by applying the regularity lemma 00:10:17.390 --> 00:10:21.480 to each to each graphon in the sequence. 00:10:24.300 --> 00:10:27.530 Now, these graphons, I mean, they 00:10:27.530 --> 00:10:29.260 each have their own vertex set. 00:10:29.260 --> 00:10:32.040 And so far, they're not related to each other. 00:10:32.040 --> 00:10:37.300 But to make the visualization easier and also in order 00:10:37.300 --> 00:10:39.620 to do the next step in the proof, 00:10:39.620 --> 00:10:46.440 I am going to do some measure-preserving bisection So 00:10:46.440 --> 00:10:49.960 think of this as permuting the vertex labels. 00:10:49.960 --> 00:11:04.760 So by replacing each Wn by some W sub 00:11:04.760 --> 00:11:15.350 n of phi, where phi is a measure-preserving bisection, 00:11:15.350 --> 00:11:24.080 we can assume that all these partitions 00:11:24.080 --> 00:11:27.940 are partitions into intervals. 00:11:36.660 --> 00:11:38.670 So initially, you might have a partition 00:11:38.670 --> 00:11:41.640 into arbitrary measurable sets. 00:11:41.640 --> 00:11:47.120 Well, what I can do is to push over the first set to the left, 00:11:47.120 --> 00:11:50.400 and so on, so do a measure-preserving bisection 00:11:50.400 --> 00:11:55.740 in a way so that I can maintain that all the partitions are 00:11:55.740 --> 00:11:58.189 visually chopping up into intervals. 00:11:58.189 --> 00:11:58.689 Yeah? 00:11:58.689 --> 00:12:01.179 AUDIENCE: So at some point, we need just one measure 00:12:01.179 --> 00:12:04.920 for the projection, like all of them be in k? 00:12:04.920 --> 00:12:07.110 YUFEI ZHAO: OK, so the question is, 00:12:07.110 --> 00:12:11.510 it may be the case that, for a given k, 00:12:11.510 --> 00:12:15.640 I can do this arrangement, but it's not 00:12:15.640 --> 00:12:18.040 clear to you at the moment why you can 00:12:18.040 --> 00:12:21.790 do this uniformly for all k. 00:12:21.790 --> 00:12:24.610 So one way to get around this is, for now, 00:12:24.610 --> 00:12:26.500 just think of for each given k. 00:12:26.500 --> 00:12:29.290 And then you'll see at the end that that's already enough. 00:12:32.898 --> 00:12:34.512 OK, any more questions? 00:12:38.170 --> 00:12:41.940 So now assume all of these P sub nk's are intervals. 00:12:41.940 --> 00:12:45.210 So in fact, what you said may be a better way to go. 00:12:45.210 --> 00:12:48.430 But to make our life a little bit easier, 00:12:48.430 --> 00:12:50.430 let's just assume for now that you can do this. 00:12:53.670 --> 00:12:56.190 OK, and what's going to happen next 00:12:56.190 --> 00:12:59.490 is some kind of a diagonalization argument. 00:12:59.490 --> 00:13:01.290 We're going to be picking subsequences. 00:13:05.760 --> 00:13:10.740 So I'm going to be picking subsequences 00:13:10.740 --> 00:13:12.810 so that they are going to have very 00:13:12.810 --> 00:13:14.850 nice convergence properties. 00:13:14.850 --> 00:13:18.150 And so I'm going to repeatedly throw out 00:13:18.150 --> 00:13:19.930 a lot of the sequence. 00:13:19.930 --> 00:13:21.930 So this is a diagonalization argument. 00:13:21.930 --> 00:13:25.390 And basically what happens is that, 00:13:25.390 --> 00:13:27.370 by passing two subsequences-- 00:13:31.564 --> 00:13:35.450 and we're going to do this repeatedly, many times-- 00:13:35.450 --> 00:13:51.420 we can assume, first, that the end points of P sub n1, 00:13:51.420 --> 00:13:59.070 they converge as n goes to infinity. 00:13:59.070 --> 00:14:14.250 So each P sub n1 is some partition of interval 00:14:14.250 --> 00:14:17.370 into some fixed number of parts. 00:14:17.370 --> 00:14:19.320 So by passing to a subsequence, I 00:14:19.320 --> 00:14:22.005 make sure that the division points all converge. 00:14:28.850 --> 00:14:34.420 And now, by passing one more time, 00:14:34.420 --> 00:14:36.860 so by passing to subsequence one more time, 00:14:36.860 --> 00:14:46.020 let's assume that also, W sub n1 converges to some function, 00:14:46.020 --> 00:14:49.080 some graphon u1, point-wise. 00:15:00.580 --> 00:15:12.100 So initially, I have these graphons. 00:15:12.100 --> 00:15:15.710 Each one of them is an m by n block. 00:15:15.710 --> 00:15:19.060 They have various division points. 00:15:19.060 --> 00:15:21.590 By passing to a subsequence, I assume 00:15:21.590 --> 00:15:24.560 that the points of division, they converge. 00:15:24.560 --> 00:15:27.130 And now by passing to an additional subsequence, 00:15:27.130 --> 00:15:32.600 I can make sure the individual values, they converge. 00:15:32.600 --> 00:15:37.700 So as a result, W sub n, 1 converges to W1-- 00:15:37.700 --> 00:15:42.382 converges to some graphon, u1, point-wise, almost everywhere. 00:15:46.030 --> 00:15:54.990 And we repeat for W sub nk for each k. 00:15:59.970 --> 00:16:01.520 So do this sequentially. 00:16:01.520 --> 00:16:03.490 So we just did it for k equals to 1. 00:16:03.490 --> 00:16:05.890 Now do it for 2, 3, 4, and so on. 00:16:08.537 --> 00:16:10.120 So this is a diagonalization argument. 00:16:10.120 --> 00:16:12.670 We do this countably many times. 00:16:12.670 --> 00:16:15.310 At the end, what do we get? 00:16:15.310 --> 00:16:20.970 We pass down to the following subsequence. 00:16:26.850 --> 00:16:29.820 And just to make my life a bit more convenient, 00:16:29.820 --> 00:16:34.900 instead of labeling the indices of the subsequence, 00:16:34.900 --> 00:16:37.380 I'm going to relabel the sequence 00:16:37.380 --> 00:16:40.560 so that it's still labeled by 1, 2, 3, 4, and so on. 00:16:40.560 --> 00:16:47.320 So we now pass to a sequence W1, W2, W3, 00:16:47.320 --> 00:16:54.630 and so on, such that if you look at the first partition, 00:16:54.630 --> 00:16:59.620 the first weak regularity partition, they produce W1,1, 00:16:59.620 --> 00:17:05.079 W2,1, W3,1, and so on. 00:17:05.079 --> 00:17:09.981 And these guys, they converge to u1, point-wise. 00:17:15.849 --> 00:17:19.695 The second level, W2,1-- 00:17:19.695 --> 00:17:32.310 sorry, W1,2, W2,2, W3,2, they converge to u2, point-wise, 00:17:32.310 --> 00:17:32.810 and so on. 00:17:46.820 --> 00:17:48.893 OK, so far so good? 00:17:48.893 --> 00:17:49.800 Question? 00:17:49.800 --> 00:17:52.310 AUDIENCE: Sorry, earlier, why did that 00:17:52.310 --> 00:17:54.110 converge to u1 point-wise? 00:17:54.110 --> 00:17:56.620 YUFEI ZHAO: OK, so the question is, why is this true? 00:17:56.620 --> 00:18:03.490 Why is Wn,1 converge to u1 point-wise? 00:18:03.490 --> 00:18:05.200 Initially, it might not. 00:18:05.200 --> 00:18:07.900 But what I'm saying is, you can pass to a subsequence. 00:18:07.900 --> 00:18:08.663 AUDIENCE: Yes. 00:18:08.663 --> 00:18:10.330 YUFEI ZHAO: You can pass to subsequence, 00:18:10.330 --> 00:18:15.210 because there are only n1 parts. 00:18:17.800 --> 00:18:24.500 So it's an n1 by m1 matrix of real numbers. 00:18:24.500 --> 00:18:27.720 And so you only have finite bounded many of them. 00:18:27.720 --> 00:18:30.080 So you can pick a subsequence so that they converge. 00:18:30.080 --> 00:18:30.580 Yeah? 00:18:30.580 --> 00:18:31.955 AUDIENCE: So how do you make sure 00:18:31.955 --> 00:18:35.420 that your subsequence is not empty at the end-- like, 00:18:35.420 --> 00:18:37.800 could you fix the first k? 00:18:37.800 --> 00:18:39.760 YUFEI ZHAO: OK, so you're asking, 00:18:39.760 --> 00:18:44.080 if we do this slightly not so carefully, 00:18:44.080 --> 00:18:46.180 we might end up with an empty sequence. 00:18:46.180 --> 00:18:48.430 So this is why I say, you have to do a diagonalization 00:18:48.430 --> 00:18:49.240 argument. 00:18:49.240 --> 00:18:53.170 Each step, you keep the first term, the sequence, 00:18:53.170 --> 00:18:56.440 so that you always maintain some sequence. 00:18:56.440 --> 00:19:00.870 You have to be slightly careful with diagonalization. 00:19:00.870 --> 00:19:01.976 Any more questions? 00:19:06.080 --> 00:19:07.900 So by passing to a subsequence, we 00:19:07.900 --> 00:19:11.680 obtain this very nice sequence, this nice subsequence, 00:19:11.680 --> 00:19:16.960 such that each row corresponding to each level of regularization 00:19:16.960 --> 00:19:20.740 converges point-wise to some u. 00:19:20.740 --> 00:19:23.370 So what do this u's look like? 00:19:23.370 --> 00:19:24.800 So they are step graphons. 00:19:27.460 --> 00:19:29.740 So let's explore the structure of u a bit more. 00:19:36.050 --> 00:19:43.880 OK, so since we have that each-- 00:19:43.880 --> 00:19:47.070 OK, so we have the property that each partition 00:19:47.070 --> 00:19:49.190 refines the previous partition. 00:19:53.190 --> 00:20:00.680 And as a result, if you look at the k plus 1'th stepping, 00:20:00.680 --> 00:20:07.490 and I step it by the previous partition in the sequence, 00:20:07.490 --> 00:20:14.460 I should get back, I should go back one in the sequence. 00:20:14.460 --> 00:20:19.490 So this was this graphon obtained by averaging over 00:20:19.490 --> 00:20:20.870 the k'th partition. 00:20:20.870 --> 00:20:26.000 And this is the graphon obtained by averaging over the k'th plus 00:20:26.000 --> 00:20:28.070 1st partition. 00:20:28.070 --> 00:20:33.000 So if I go back one more, I should go back in the sequence. 00:20:35.640 --> 00:20:41.530 And since the u's are the point-wise limit of these 00:20:41.530 --> 00:20:50.530 W's, the same relationships should also hold 00:20:50.530 --> 00:21:06.820 for the u's, namely that u sub k should equal to u sub k plus 1 00:21:06.820 --> 00:21:17.210 if I step it with Pk, where Pk is the-- 00:21:20.020 --> 00:21:22.870 so if you look at, all these endpoints converge. 00:21:22.870 --> 00:21:28.550 And these partitions, they converge to P1. 00:21:28.550 --> 00:21:36.760 So if you look at the partitions that correspond to P1,1, P2,1, 00:21:36.760 --> 00:21:41.480 and so on, I want these partitions to converge to P1. 00:21:41.480 --> 00:21:51.460 I want these partitions to converge to P2. 00:21:51.460 --> 00:21:55.313 So all these partitions, they are partitions into intervals. 00:21:55.313 --> 00:21:56.980 So I'm just saying, if you look at where 00:21:56.980 --> 00:21:59.350 the intervals, where the divisions of intervals go, 00:21:59.350 --> 00:22:00.250 they converge. 00:22:00.250 --> 00:22:02.830 And then I'm calling the limit partition P sub k. 00:22:07.810 --> 00:22:16.240 And here we're using that P sub k plus 1 refines P sub k, 00:22:16.240 --> 00:22:18.712 because the same is true for each end. 00:22:18.712 --> 00:22:20.670 So in the limit, the same must be true as well. 00:22:25.080 --> 00:22:27.620 So you have this column of u's. 00:22:27.620 --> 00:22:32.852 So let me draw you a picture of what these u's could look like. 00:22:32.852 --> 00:22:34.810 So here is an illustration that may be helpful. 00:22:40.980 --> 00:22:43.830 So what could these u's look like? 00:22:43.830 --> 00:22:51.390 Each one of them is represented by values on the unit square. 00:22:51.390 --> 00:22:53.520 And I write this in matrix notation 00:22:53.520 --> 00:22:55.480 so that inversion is in the top left corner. 00:23:00.180 --> 00:23:05.400 Well, maybe P1 is just the trivial partition, 00:23:05.400 --> 00:23:08.910 in which case u1 is going to be a constant graphon. 00:23:08.910 --> 00:23:14.630 Let's say it has value 0.5. 00:23:14.630 --> 00:23:19.300 u2 came from u1 by some partitioning. 00:23:19.300 --> 00:23:21.290 And suppose just for the sake of illustration, 00:23:21.290 --> 00:23:23.210 there was only a partitioning into two parts. 00:23:26.950 --> 00:23:30.123 And OK, so it doesn't have to be at the origin. 00:23:30.123 --> 00:23:31.540 It doesn't have to be at midpoint. 00:23:31.540 --> 00:23:33.550 But just for illustration, suppose the division 00:23:33.550 --> 00:23:36.800 were at the midpoint. 00:23:36.800 --> 00:23:40.520 Because u1 needs to have-- 00:23:40.520 --> 00:23:44.220 so this 0.5 value should be the average value 00:23:44.220 --> 00:23:46.530 in all of these four squares. 00:23:46.530 --> 00:23:59.020 So for instance, the points may be like 0.6, 0.6, 0.4, 0.4, 00:23:59.020 --> 00:24:01.900 so for example. 00:24:01.900 --> 00:24:08.820 And in u3, the partition, the P3 partition-- so here 00:24:08.820 --> 00:24:10.120 are the partition is P1. 00:24:10.120 --> 00:24:11.740 The partition is P2. 00:24:11.740 --> 00:24:13.070 There's two parts. 00:24:13.070 --> 00:24:18.142 And suppose P3 three now has four parts. 00:24:18.142 --> 00:24:19.600 And again, for illustration's sake, 00:24:19.600 --> 00:24:22.120 suppose it is equally dividing the interval 00:24:22.120 --> 00:24:25.000 into four intervals. 00:24:25.000 --> 00:24:28.960 It could be that now each of these parts 00:24:28.960 --> 00:24:32.950 is split up into four different values in a way 00:24:32.950 --> 00:24:37.471 that so you can obtain the original numbers by averaging. 00:24:40.560 --> 00:24:44.010 So that's one possible example. 00:24:44.010 --> 00:24:50.514 Likewise, you can have something like that. 00:24:54.888 --> 00:25:07.220 Sorry, 4, 7-- so and I should maintain symmetry 00:25:07.220 --> 00:25:08.990 in this matrix. 00:25:08.990 --> 00:25:11.520 And the last one, I'm just going to be lazy and say 00:25:11.520 --> 00:25:15.960 that it's still 0.4 throughout. 00:25:15.960 --> 00:25:20.080 OK, so this is what the sequence of u's are going to look like. 00:25:20.080 --> 00:25:25.660 Each one of them splits up a box in the previous u in such way 00:25:25.660 --> 00:25:31.700 that the local averages, the step averages are preserved. 00:25:31.700 --> 00:25:32.590 Any questions so far? 00:25:35.410 --> 00:25:41.270 All right, so now we get to Martingales. 00:25:41.270 --> 00:25:45.870 So I claim that this is basically a Martingale. 00:25:45.870 --> 00:25:56.550 So and suppose you let x, y be a uniform point in the unit 00:25:56.550 --> 00:25:57.050 square. 00:25:59.930 --> 00:26:05.220 And consider this sequence. 00:26:05.220 --> 00:26:07.190 So this is now a random sequence, 00:26:07.190 --> 00:26:09.290 because x, y are random. 00:26:12.030 --> 00:26:17.752 I evaluate these u's on this uniform random point, x, y. 00:26:17.752 --> 00:26:18.960 So this is a random sequence. 00:26:23.800 --> 00:26:27.279 And the main observation is that this is a Martingale. 00:26:34.320 --> 00:26:38.450 So remember the definition of a Martingale from last time. 00:26:38.450 --> 00:26:42.990 Martingale is one where, if you look 00:26:42.990 --> 00:26:47.430 at the value of u sub k conditioned 00:26:47.430 --> 00:26:53.730 on the previous values, the expectation 00:26:53.730 --> 00:26:56.190 is just the previous term. 00:26:56.190 --> 00:26:58.493 And I claim this is true for the sequence, 00:26:58.493 --> 00:27:00.035 because of the way we constructed it, 00:27:00.035 --> 00:27:03.750 it's splitting up each box in an averaging-preserving way. 00:27:07.380 --> 00:27:09.177 A different way to see this, and for those 00:27:09.177 --> 00:27:11.010 of you who actually know what the definition 00:27:11.010 --> 00:27:13.920 of a random variable is in the sense of probability theory, 00:27:13.920 --> 00:27:21.160 is that you should view this 0, 1 squared as the probability 00:27:21.160 --> 00:27:27.115 space, in which case u itself is the random variable. 00:27:30.050 --> 00:27:34.480 And this partitioning gives you a filtration of the space. 00:27:34.480 --> 00:27:37.450 It's a sequence of sigma algebras dividing up the space 00:27:37.450 --> 00:27:39.590 into finer and finer pieces. 00:27:39.590 --> 00:27:43.280 So this is really what a martingale is. 00:27:43.280 --> 00:27:45.710 So we have a Martingale. 00:27:45.710 --> 00:27:56.770 It's bounded because the values take place in 0, 1. 00:27:56.770 --> 00:28:00.340 So by the Martingale convergence theorem, 00:28:00.340 --> 00:28:15.370 which we proved last time, we find that this sequence 00:28:15.370 --> 00:28:18.760 must converge to some limit. 00:28:21.550 --> 00:28:24.970 So this sequence of Martingale converges, 00:28:24.970 --> 00:28:29.490 which means, so if you think about the interpretation 00:28:29.490 --> 00:28:33.480 up there, so there exists a u which 00:28:33.480 --> 00:28:45.210 is a graphon such that uk converges to u point-wise 00:28:45.210 --> 00:28:51.086 almost everywhere as k goes to infinity. 00:28:57.410 --> 00:28:59.000 That's the limit. 00:28:59.000 --> 00:29:00.897 So this is the limit. 00:29:00.897 --> 00:29:02.980 And we're going to show that it is indeed a limit. 00:29:02.980 --> 00:29:05.110 But you see, this is a construction of the limit, 00:29:05.110 --> 00:29:08.950 where we took regularity, got all these nice pieces, 00:29:08.950 --> 00:29:11.140 found convergent subsequences, and then 00:29:11.140 --> 00:29:14.140 applied the martingale convergence 00:29:14.140 --> 00:29:17.290 theorem to produce for us this candidate 00:29:17.290 --> 00:29:19.170 for the limit, this u. 00:29:19.170 --> 00:29:28.520 So now let us show that it is indeed the limit that we're 00:29:28.520 --> 00:29:33.650 looking for in the subsequence. 00:29:33.650 --> 00:29:37.500 So again, I've tossed out all the terms 00:29:37.500 --> 00:29:41.510 which we removed in passing to subsequences. 00:29:41.510 --> 00:29:43.290 So in the remaining subsequence, I 00:29:43.290 --> 00:29:48.920 want to show that the Wn's indeed converge to u. 00:29:48.920 --> 00:29:52.130 And this is now a fairly straightforward three epsilons 00:29:52.130 --> 00:29:55.910 argument, the standard analysis type argument. 00:29:55.910 --> 00:29:57.630 But OK, so let's carry it through. 00:29:57.630 --> 00:30:01.370 So for every epsilon bigger than 0, 00:30:01.370 --> 00:30:08.360 suppose you pick a sufficiently large k. 00:30:08.360 --> 00:30:11.170 There exists a sufficiently large k. 00:30:11.170 --> 00:30:17.010 And we make sure k is large enough 00:30:17.010 --> 00:30:22.880 such that u differs from u sub k in l1 norm 00:30:22.880 --> 00:30:28.400 by, at most, epsilon over 3, because the uk's, 00:30:28.400 --> 00:30:32.090 they converge to u point-wise almost everywhere. 00:30:32.090 --> 00:30:33.150 So we find this k. 00:30:33.150 --> 00:30:34.060 So let's fix this k. 00:30:39.220 --> 00:30:47.080 Then there exists an n0 such that if you look at this u sub 00:30:47.080 --> 00:30:51.930 k, it does not-- 00:30:51.930 --> 00:31:03.540 it is very close to W sub nk for all n 00:31:03.540 --> 00:31:13.150 large enough because of what happened up there. 00:31:18.300 --> 00:31:25.435 So we can now compute the difference between-- 00:31:27.950 --> 00:31:30.770 in fact, let's do it this way. 00:31:35.400 --> 00:31:42.585 So now let's compute the difference, 00:31:42.585 --> 00:31:45.810 the cut norm of the difference between the term 00:31:45.810 --> 00:31:48.060 in the sequence W sub n and u. 00:31:48.060 --> 00:31:53.510 So by triangle inequality, we have 00:31:53.510 --> 00:31:54.900 that the following is true. 00:32:11.060 --> 00:32:15.940 The cut norm is upperbounded by the l1 norm. 00:32:15.940 --> 00:32:17.332 Look at the definitions. 00:32:22.420 --> 00:32:25.710 So I'm going to replace the first couple of these cut norms 00:32:25.710 --> 00:32:28.290 by l1 norms and leave the last one in tact. 00:32:35.500 --> 00:32:41.000 The first term, I claim is, at most, epsilon over 3, 00:32:41.000 --> 00:32:44.810 because up there. 00:32:44.810 --> 00:32:47.690 The second term is going to be at least epsilon 00:32:47.690 --> 00:32:52.040 over 3, because over here. 00:32:52.040 --> 00:32:56.510 And the third term is going to be also, at most, epsilon 00:32:56.510 --> 00:33:02.660 over 3, because well, from the regularity approximation, 00:33:02.660 --> 00:33:07.010 I know that it is, at most, 1 over k. 00:33:07.010 --> 00:33:09.970 And I chose k large enough so that there is also, 00:33:09.970 --> 00:33:12.410 at most, epsilon over 3. 00:33:12.410 --> 00:33:17.285 Put everything together, we find that these two are-- 00:33:17.285 --> 00:33:21.770 they different by, at most, epsilon if n is large enough. 00:33:21.770 --> 00:33:30.380 But now, since epsilon can be arbitrarily small, 00:33:30.380 --> 00:33:38.460 we find that you indeed have convergence, as claimed. 00:33:41.750 --> 00:33:46.160 And this finishes the proof of compactness. 00:33:46.160 --> 00:33:47.700 So there are a few components. 00:33:47.700 --> 00:33:48.910 One is passing to-- 00:33:48.910 --> 00:33:53.970 so applying regularity, passing to subsequences, 00:33:53.970 --> 00:33:58.630 and obtaining this limit from the regularity approximations, 00:33:58.630 --> 00:34:00.140 these u's. 00:34:00.140 --> 00:34:03.840 And then we observe that these u's, they form a Martingale. 00:34:03.840 --> 00:34:08.340 So we can apply the Martingale convergence theorem to get 00:34:08.340 --> 00:34:12.280 us a candidate for the limit. 00:34:12.280 --> 00:34:14.449 And then the rest is fairly straightforward, 00:34:14.449 --> 00:34:17.074 because all the steps are good approximations. 00:34:17.074 --> 00:34:19.960 You put them together, you prove the limit. 00:34:23.090 --> 00:34:24.163 Any questions? 00:34:27.790 --> 00:34:30.980 All right, so you may ask, well, now we have compactness. 00:34:30.980 --> 00:34:33.179 What is compactness good for? 00:34:33.179 --> 00:34:36.022 So it may seem like a somewhat abstract concept. 00:34:36.022 --> 00:34:37.730 So in the second half of today's lecture, 00:34:37.730 --> 00:34:41.870 I want to show you how to use this compactness claim combined 00:34:41.870 --> 00:34:46.790 with the first definition of compactness that you've seen, 00:34:46.790 --> 00:34:51.110 namely every open cover contains a finite sub cover, 00:34:51.110 --> 00:34:56.179 and to use that to prove many consequences 00:34:56.179 --> 00:34:58.460 about the space of graphons. 00:34:58.460 --> 00:35:02.030 And some things that we had to work a bit hard at, 00:35:02.030 --> 00:35:06.790 but they turn out to fall from the compactness statement. 00:35:06.790 --> 00:35:09.910 So let's take a quick break. 00:35:09.910 --> 00:35:11.420 In the first part of this lecture, 00:35:11.420 --> 00:35:14.270 we proved that a space of graphons is compact. 00:35:14.270 --> 00:35:17.840 So now let me show you what we can reap as consequences 00:35:17.840 --> 00:35:20.450 from the compactness result. So I 00:35:20.450 --> 00:35:24.140 want to show you how to apply compactness 00:35:24.140 --> 00:35:25.820 and prove some consequences. 00:35:35.540 --> 00:35:38.450 As I mentioned earlier, the compactness result 00:35:38.450 --> 00:35:40.550 is related to regularity. 00:35:40.550 --> 00:35:43.040 And in fact, many of the results I'm going to state, 00:35:43.040 --> 00:35:47.870 you can prove maybe with some more work using the regularity 00:35:47.870 --> 00:35:49.010 lemma. 00:35:49.010 --> 00:35:51.320 But I also want to show you how to deduce them directly 00:35:51.320 --> 00:35:52.190 from compactness. 00:35:52.190 --> 00:35:54.650 In fact, we'll deduce the regularity lemma 00:35:54.650 --> 00:35:55.850 from compactness. 00:35:55.850 --> 00:35:58.550 So these two ideas, compactness and regularity, 00:35:58.550 --> 00:36:01.830 they go hand-in-hand. 00:36:01.830 --> 00:36:05.390 So first, more so as a warm up, but as also an interesting 00:36:05.390 --> 00:36:07.880 result, statement, an interesting statement 00:36:07.880 --> 00:36:11.548 on its own, so let me prove the following. 00:36:14.970 --> 00:36:21.580 So here is a statement that we can deduce from compactness. 00:36:21.580 --> 00:36:26.720 So for every epsilon, there exists some number N 00:36:26.720 --> 00:36:36.530 which depend only on epsilon, such that for every W graphon, 00:36:36.530 --> 00:36:49.440 there exists a graph G with N vertices, 00:36:49.440 --> 00:37:02.770 such that the cut distance between G and W 00:37:02.770 --> 00:37:05.100 is, at most, epsilon. 00:37:08.440 --> 00:37:10.540 So think about what this says. 00:37:10.540 --> 00:37:12.930 So for every epsilon, there is some 00:37:12.930 --> 00:37:17.100 bound N such that every graphon-- so a graphon is 00:37:17.100 --> 00:37:22.900 some real-value function, so taking values between 0 and 1. 00:37:22.900 --> 00:37:24.990 You can approximate it in the distance 00:37:24.990 --> 00:37:28.560 that we care about by a graph with a bounded number 00:37:28.560 --> 00:37:31.180 of vertices. 00:37:31.180 --> 00:37:33.700 This is kind of like regularity lemma. 00:37:33.700 --> 00:37:38.210 If you are allowed edge weights on this graph G, 00:37:38.210 --> 00:37:43.220 then it immediately follows from the weak regularity lemma 00:37:43.220 --> 00:37:44.570 that we already proved. 00:37:48.500 --> 00:37:50.930 And from that weak regularity lemma 00:37:50.930 --> 00:37:55.370 which allows you to get some G with edge weights, 00:37:55.370 --> 00:37:57.440 you can think about how you might 00:37:57.440 --> 00:38:03.100 turn an edge-weighted graph into an unweighted graph. 00:38:03.100 --> 00:38:04.968 So that can also be done. 00:38:04.968 --> 00:38:07.010 But I want to show you a completely different way 00:38:07.010 --> 00:38:10.700 of proving this result that follows from compactness. 00:38:13.570 --> 00:38:16.930 And so I say it's a warm up, because it's really 00:38:16.930 --> 00:38:19.918 a warm up for the next thing we're going to do. 00:38:19.918 --> 00:38:21.460 This is an easier example showing you 00:38:21.460 --> 00:38:24.690 how to use compactness. 00:38:24.690 --> 00:38:26.440 So the idea is, I have this compact space. 00:38:26.440 --> 00:38:32.380 I'm going to cover this space by open sets, by open balls. 00:38:32.380 --> 00:38:38.230 So the open balls are going to be this B sub epsilon 00:38:38.230 --> 00:38:41.200 G. So for each graph, G, I'm going 00:38:41.200 --> 00:38:53.005 to consider the set of graphons that are within epsilon of G. 00:38:53.005 --> 00:38:54.755 So this is you have some topological space 00:38:54.755 --> 00:38:55.810 or some metric space. 00:38:55.810 --> 00:38:58.260 I have a point G. And I look at its open ball. 00:39:07.930 --> 00:39:09.580 This is the ball. 00:39:09.580 --> 00:39:22.753 So I claim that these open balls, they form an open cover, 00:39:22.753 --> 00:39:25.620 of the space. 00:39:30.300 --> 00:39:31.410 Where is that? 00:39:31.410 --> 00:39:35.010 So I want to show every point W is covered. 00:39:39.220 --> 00:39:55.930 So this follows from the claim that every W is the limit 00:39:55.930 --> 00:39:57.340 of some sequence of graphs. 00:40:04.940 --> 00:40:08.600 So we didn't technically actually prove this claim. 00:40:08.600 --> 00:40:11.895 I said that if you take W random graphs, you get this. 00:40:11.895 --> 00:40:15.320 So we didn't technically prove that. 00:40:15.320 --> 00:40:16.870 But OK, so it turns out to be true. 00:40:16.870 --> 00:40:18.780 There are easier ways to establish it as well 00:40:18.780 --> 00:40:20.790 by taking l1 approximations. 00:40:20.790 --> 00:40:24.780 But the point is that, if you use this claim here, 00:40:24.780 --> 00:40:28.460 you do not get a bound on the number of vertices. 00:40:28.460 --> 00:40:32.660 It could be that for very bizarre-looking W's, you 00:40:32.660 --> 00:40:37.690 might require much more number of vertices. 00:40:37.690 --> 00:40:40.320 And a priori, you do not know that it is bounded 00:40:40.320 --> 00:40:43.510 as a function of epsilon. 00:40:43.510 --> 00:40:47.110 But now we have this open cover. 00:40:47.110 --> 00:40:55.870 So by compactness of this space of graphons, 00:40:55.870 --> 00:41:03.340 we can find an open cover using a finite subset 00:41:03.340 --> 00:41:10.510 of these graphs, so G1 to Gk, so a finite subset 00:41:10.510 --> 00:41:13.740 to do an open cover. 00:41:13.740 --> 00:41:22.330 And now we let N to be the least-common multiple of all 00:41:22.330 --> 00:41:25.105 of these vertex set sizes. 00:41:49.130 --> 00:41:59.140 So all of these graphs, they are within-- 00:42:03.010 --> 00:42:05.270 so for each of these graphs, I can replace it 00:42:05.270 --> 00:42:07.550 by a graph on exactly N vertices. 00:42:16.300 --> 00:42:28.440 There exists a graph Gi prime of exactly N vertices, 00:42:28.440 --> 00:42:34.208 such that they represent the same point 00:42:34.208 --> 00:42:35.250 in the space of graphons. 00:42:38.436 --> 00:42:40.740 So why is this? 00:42:40.740 --> 00:42:43.950 Think about the representation of a graphon using 00:42:43.950 --> 00:42:45.630 from a graph. 00:42:45.630 --> 00:43:00.650 If I start with G and I blow up each vertex into some k 00:43:00.650 --> 00:43:04.530 vertices, then it turns out-- 00:43:04.530 --> 00:43:06.530 I mean, you should think about why this is true. 00:43:06.530 --> 00:43:09.580 But it's really not hard to see if you draw the picture. 00:43:09.580 --> 00:43:12.662 So remember, this black and white picture, that actually, 00:43:12.662 --> 00:43:13.620 they're the same point. 00:43:13.620 --> 00:43:17.360 They are represented by the same graphon. 00:43:17.360 --> 00:43:18.890 OK, and that's it. 00:43:18.890 --> 00:43:23.790 So we found these G's. 00:43:23.790 --> 00:43:27.540 All of them have exactly N vertices such 00:43:27.540 --> 00:43:32.700 that their epsilon open balls form an open cover 00:43:32.700 --> 00:43:33.870 of the space of graphons. 00:43:33.870 --> 00:43:36.000 So every graphon can be approximated 00:43:36.000 --> 00:43:37.172 by one of these graphs. 00:43:40.070 --> 00:43:44.030 So you get that from compactness. 00:43:44.030 --> 00:43:46.760 The statement says, for every epsilon, their exists an N. 00:43:46.760 --> 00:43:50.650 So N is a function of epsilon.l What's the function? 00:43:54.590 --> 00:43:57.090 This proof doesn't tell you anything about that. 00:43:57.090 --> 00:44:06.860 So this proof gives no information 00:44:06.860 --> 00:44:18.750 about the dependence of N on epsilon. 00:44:18.750 --> 00:44:21.420 So in some sense, it's even worse than some of the things 00:44:21.420 --> 00:44:23.700 we've seen in the earlier discussion 00:44:23.700 --> 00:44:25.950 on Szemerédi's regularity lemma where there were tower 00:44:25.950 --> 00:44:27.030 or Wowzer-types. 00:44:27.030 --> 00:44:29.340 Here there is no information, because it comes 00:44:29.340 --> 00:44:32.850 from a compactness statement. 00:44:32.850 --> 00:44:39.316 So you just know there exists a finite open cover, no bounds. 00:44:39.316 --> 00:44:43.970 OK, any questions about this warm-up application? 00:44:43.970 --> 00:44:45.160 So it feels a bit magical. 00:44:45.160 --> 00:44:46.160 So you have compactness. 00:44:46.160 --> 00:44:51.216 And then you have all of these consequences. 00:44:51.216 --> 00:44:54.080 So now let me show you how you can deduce the regularity 00:44:54.080 --> 00:44:56.900 lemma itself from compactness. 00:44:56.900 --> 00:45:03.010 In fact, in the proof of the existence, 00:45:03.010 --> 00:45:06.840 in the proof of compactness, we only used weak regularity. 00:45:06.840 --> 00:45:09.590 And now let me show you how you can use the weak regularity 00:45:09.590 --> 00:45:11.780 consequence of namely compactness 00:45:11.780 --> 00:45:16.650 to bootstrap itself to strong regularity. 00:45:16.650 --> 00:45:20.150 So we saw a version of strong regularity in the earlier 00:45:20.150 --> 00:45:23.510 chapter when we discussed Szemerédi's regularity lemma. 00:45:23.510 --> 00:45:27.770 So let me state it in a somewhat different-looking form, 00:45:27.770 --> 00:45:32.640 but that turns out to be morally equivalent. 00:45:32.640 --> 00:45:35.590 Suppose I have a vector of epsilons. 00:45:41.740 --> 00:45:44.090 So all of these are positive real numbers. 00:45:49.680 --> 00:45:55.470 The claim is that there exists an M which 00:45:55.470 --> 00:46:10.310 only depends on this vector such that for every graphon W, 00:46:10.310 --> 00:46:10.950 one can-- 00:46:10.950 --> 00:46:25.010 so every graphon W can be written as the following, 00:46:25.010 --> 00:46:26.260 decomposing the following way. 00:46:26.260 --> 00:46:31.970 We write W as a sum of a structured part, 00:46:31.970 --> 00:46:38.540 a pseudo-random part, and a small part, 00:46:38.540 --> 00:46:51.060 where the structured part is a step function with k parts, 00:46:51.060 --> 00:46:57.450 but k is, at most, M, this claimed bound M. 00:46:57.450 --> 00:47:03.640 The pseudo-random part has a very small cut norm, 00:47:03.640 --> 00:47:08.520 so its cut norm, very small, even compared 00:47:08.520 --> 00:47:10.870 to the number of parts. 00:47:10.870 --> 00:47:22.117 And finally, the small part has l1 norm bounded by epsilon 1. 00:47:22.117 --> 00:47:22.950 So that's the claim. 00:47:22.950 --> 00:47:25.530 You can always-- there exists some 00:47:25.530 --> 00:47:28.200 bound M in terms of these error parameters 00:47:28.200 --> 00:47:30.667 so that you have this decomposition. 00:47:30.667 --> 00:47:32.250 So we saw some version of this earlier 00:47:32.250 --> 00:47:35.280 when we discussed the spectral proof of regularity lemma. 00:47:35.280 --> 00:47:38.520 And I don't want to go into details of how these two 00:47:38.520 --> 00:47:41.880 things are related, but just to comment that depending 00:47:41.880 --> 00:47:44.010 on your choice of the epsilon parameters, 00:47:44.010 --> 00:47:46.110 it relates to some of the different versions 00:47:46.110 --> 00:47:48.820 of regularity lemma that we've seen before. 00:47:48.820 --> 00:47:54.600 So for example, if epsilon k is roughly epsilon, 00:47:54.600 --> 00:47:57.240 some fixed epsilon over k squared, 00:47:57.240 --> 00:48:08.400 then this is basically the same as Szemerédi's regularity 00:48:08.400 --> 00:48:16.920 lemma, whereas if all the k's are the same epsilon, 00:48:16.920 --> 00:48:20.236 then this is roughly the same as the weak regularity lemma. 00:48:27.540 --> 00:48:30.820 All right, so how to prove this claim? 00:48:30.820 --> 00:48:32.430 We're going to use compactness again. 00:48:38.070 --> 00:48:43.670 So first, there always exists an l1 approximation 00:48:43.670 --> 00:48:55.870 so that every W has some step function u associated 00:48:55.870 --> 00:49:02.010 to it such that the l1 distance between W and u 00:49:02.010 --> 00:49:06.060 is, at most, epsilon 1. 00:49:06.060 --> 00:49:08.730 So again, this is one of these more measured theoretic 00:49:08.730 --> 00:49:10.610 technicalities I don't want to get into, 00:49:10.610 --> 00:49:12.567 but so it's not hard to prove. 00:49:12.567 --> 00:49:14.400 So roughly speaking, you have some function. 00:49:14.400 --> 00:49:16.820 You can approximate it using steps. 00:49:26.860 --> 00:49:29.950 So similar to what we did just now, 00:49:29.950 --> 00:49:33.220 if you just do that, the number of steps 00:49:33.220 --> 00:49:36.840 might not be a function of epsilon, 00:49:36.840 --> 00:49:40.480 so you might need much more steps just doing 00:49:40.480 --> 00:49:45.570 that if your W looks more pathological. 00:49:45.570 --> 00:49:48.120 So now what we're going to do is consider 00:49:48.120 --> 00:49:52.050 the following function, k of W. And I 00:49:52.050 --> 00:49:57.390 define it to be the minimum k such 00:49:57.390 --> 00:50:10.660 that there exists a k step graphon u such that u you 00:50:10.660 --> 00:50:20.290 minus W is, at most, epsilon 1. 00:50:20.290 --> 00:50:26.030 So among all the step function approximations, 00:50:26.030 --> 00:50:29.330 pick one that has the minimum number of steps 00:50:29.330 --> 00:50:34.755 and call the number of steps k of W. So now, 00:50:34.755 --> 00:50:36.130 as before, we're going to come up 00:50:36.130 --> 00:50:38.780 with an open cover of the space of graphons. 00:50:41.380 --> 00:50:44.770 So the open cover is going to be consisting 00:50:44.770 --> 00:50:50.722 of the cut norm balls of-- 00:50:50.722 --> 00:50:52.750 actually, what notation did I use over there? 00:51:01.200 --> 00:51:03.990 So this is a ball centered around W 00:51:03.990 --> 00:51:08.660 with radius epsilon sub kW. 00:51:08.660 --> 00:51:15.972 This is an open cover of the space of graphons 00:51:15.972 --> 00:51:19.490 as W ranges over all graphons. 00:51:23.980 --> 00:51:26.490 So I'm literally looking at every point in the space 00:51:26.490 --> 00:51:28.020 and putting an open ball around it. 00:51:28.020 --> 00:51:31.750 So obviously, this is an open cover. 00:51:31.750 --> 00:51:33.850 And because of compactness, there 00:51:33.850 --> 00:51:35.340 exists a finite sub cover. 00:51:41.080 --> 00:51:46.910 So there exists a finite set, we write 00:51:46.910 --> 00:52:06.940 curly s, of graphons such that these balls, as I range over W 00:52:06.940 --> 00:52:13.010 and curly s, they cover the space of graphons. 00:52:22.550 --> 00:52:25.430 Now the goal is, given the W, I want 00:52:25.430 --> 00:52:26.870 to approximate it in some way. 00:52:26.870 --> 00:52:31.580 So having a finite set of things to work with 00:52:31.580 --> 00:52:36.010 allows us to do some kind of approximations. 00:52:36.010 --> 00:52:45.950 So thus, for every W graphon, there exists a W prime in s 00:52:45.950 --> 00:52:53.420 whose ball in that collection covers the point W, such that W 00:52:53.420 --> 00:53:06.400 is contained in this ball. 00:53:06.400 --> 00:53:16.670 And OK, so given this W prime, because of this definition 00:53:16.670 --> 00:53:22.460 over here, so there exists a u which 00:53:22.460 --> 00:53:35.570 is a k step graphon with k, at most, 00:53:35.570 --> 00:53:48.450 the maximum over all such possible number of steps, 00:53:48.450 --> 00:53:59.830 such that W and W prime, they are close in cut norm 00:53:59.830 --> 00:54:03.250 because you have this open cover. 00:54:03.250 --> 00:54:11.380 And furthermore, W prime is close to a graphon 00:54:11.380 --> 00:54:13.180 with a small number of steps. 00:54:21.970 --> 00:54:32.860 So suppose we now write W as u plus W minus W prime and then 00:54:32.860 --> 00:54:37.870 plus W prime minus u. 00:54:37.870 --> 00:54:39.850 We find that this is the decomposition that we 00:54:39.850 --> 00:54:44.290 are looking for because the u-- 00:54:44.290 --> 00:54:46.380 so this is the structural component-- 00:54:46.380 --> 00:54:58.635 has k steps, where k is less than this quantity here. 00:54:58.635 --> 00:55:03.890 And that quantity there is just some function of epsilons. 00:55:03.890 --> 00:55:05.900 So it's, at most, some function of the epsilons. 00:55:12.030 --> 00:55:18.700 It doesn't depend on the specific choice of W. 00:55:18.700 --> 00:55:22.570 The second term, this is this pseudo-random piece, 00:55:22.570 --> 00:55:35.586 because it's cut norm is small, so what we have here. 00:55:35.586 --> 00:55:38.508 Yeah, so this entire thing should be subscript. 00:55:45.690 --> 00:55:52.620 And finally, the third term here is the small term, 00:55:52.620 --> 00:55:54.810 because it's l1 norm is small. 00:56:01.307 --> 00:56:03.557 So putting them together, we get the regularity lemma. 00:56:06.480 --> 00:56:10.140 So again, the proof gives you no information whatsoever 00:56:10.140 --> 00:56:13.230 about the bound M as a function of the input 00:56:13.230 --> 00:56:16.180 parameters, the epsilons. 00:56:16.180 --> 00:56:18.890 So it turns out you can use a different method 00:56:18.890 --> 00:56:20.030 to get the bounds. 00:56:20.030 --> 00:56:22.700 Namely, we actually more or less did this proof 00:56:22.700 --> 00:56:25.830 when we discussed regularity lemma, the strong regularity 00:56:25.830 --> 00:56:26.330 lemma. 00:56:26.330 --> 00:56:29.120 So we did a different proof where we iterated an energy 00:56:29.120 --> 00:56:30.780 increment argument. 00:56:30.780 --> 00:56:33.740 And that gave you some concrete bounds, some bounds which 00:56:33.740 --> 00:56:36.020 iterates on these epsilons. 00:56:36.020 --> 00:56:37.565 But here is a different proof. 00:56:37.565 --> 00:56:41.150 It gives you less information, but it elegantly 00:56:41.150 --> 00:56:43.690 uses this compactness feature of the space of graphons. 00:56:47.440 --> 00:56:48.660 Any questions? 00:56:54.310 --> 00:56:56.040 OK, so we proved compactness. 00:56:56.040 --> 00:56:59.130 So now let's go on to the other two claims, 00:56:59.130 --> 00:57:01.210 namely the existence of the limit 00:57:01.210 --> 00:57:04.080 and that equivalences of convergence. 00:57:04.080 --> 00:57:05.800 The existence of the limit more or less 00:57:05.800 --> 00:57:07.410 is a consequence of compactness. 00:57:15.380 --> 00:57:23.690 So you have this sequence of graphons, W1, W2, and so on. 00:57:23.690 --> 00:57:34.260 And the claim is that, if this sequence of F densities 00:57:34.260 --> 00:57:47.750 converges for each F, then there exists 00:57:47.750 --> 00:57:57.680 some limit W such that all of these sequences of F densities 00:57:57.680 --> 00:58:01.140 converge to the limit density. 00:58:01.140 --> 00:58:05.810 So that was the claim, so nothing about cut norms 00:58:05.810 --> 00:58:10.040 in at least as far as the statement goes. 00:58:10.040 --> 00:58:15.540 Well OK, from compactness, we know 00:58:15.540 --> 00:58:19.680 that you can produce always a subsequential limit. 00:58:19.680 --> 00:58:28.030 So by compactness or sequential compactness, 00:58:28.030 --> 00:58:36.810 there exists some limit point which we call W. 00:58:36.810 --> 00:58:40.950 And this W has the property that, 00:58:40.950 --> 00:58:49.230 for some subsequence, the cut distance 00:58:49.230 --> 00:58:54.120 from the subsequence converges to W. 00:58:54.120 --> 00:59:07.660 So for some subsequence n0 as ni going to infinity. 00:59:07.660 --> 00:59:20.785 But now, by the counting lemma, the sequence of F densities-- 00:59:20.785 --> 00:59:22.410 so the counting Lemma tells you, if you 00:59:22.410 --> 00:59:26.040 have cut distance going to 0, then the F density 00:59:26.040 --> 00:59:28.120 should also go to 0. 00:59:28.120 --> 00:59:29.620 So indeed, that's what we have here. 00:59:38.450 --> 00:59:40.840 So this is so far just for the subsequence. 00:59:40.840 --> 00:59:45.340 But we assumed already that the entire sequence 00:59:45.340 --> 00:59:49.550 converges in respect to every F densities. 00:59:56.940 --> 00:59:59.310 So it must be the same limit. 01:00:04.130 --> 01:00:08.660 And that finishes the proof of convergence, 01:00:08.660 --> 01:00:11.830 so proof of the existence of the limit. 01:00:11.830 --> 01:00:16.260 So we obtain this limit from compactness. 01:00:16.260 --> 01:00:19.590 Next, let's prove the equivalence of convergence. 01:00:19.590 --> 01:00:21.310 And this one is somewhat trickier. 01:00:21.310 --> 01:00:24.150 So what happens here is that we would 01:00:24.150 --> 01:00:27.840 like to show that these two notions of convergence, 01:00:27.840 --> 01:00:30.750 one having to do with F densities 01:00:30.750 --> 01:00:33.320 and another having to do with cut distance, 01:00:33.320 --> 01:00:36.100 that these two notions are equivalent to each other. 01:00:41.250 --> 01:00:51.700 So the goal here is to show that this F density convergence is 01:00:51.700 --> 01:01:01.390 equivalent to the statement that W sub n is Cauchy with respect 01:01:01.390 --> 01:01:04.630 to the cut distance. 01:01:07.410 --> 01:01:11.360 All right, claim one of the directions is easy. 01:01:11.360 --> 01:01:12.360 Which direction is that? 01:01:22.070 --> 01:01:24.298 So which direction is the easy direction? 01:01:28.590 --> 01:01:31.230 So which way, left, going left, going right? 01:01:34.520 --> 01:01:36.240 OK, so going left? 01:01:36.240 --> 01:01:40.620 So I claim that this is easy, because it 01:01:40.620 --> 01:01:41.830 follows from counting lemma. 01:01:50.633 --> 01:01:53.050 Counting lemma, remember the spirit of the counting lemma, 01:01:53.050 --> 01:01:56.130 at least qualitatively, is that if you have two graphons that 01:01:56.130 --> 01:01:59.760 are close in cut distance, then they are close in F densities. 01:01:59.760 --> 01:02:03.030 So if you have Cauchy with respect to cut distance, 01:02:03.030 --> 01:02:06.764 then they are Cauchy, and hence, convergent in F densities. 01:02:09.370 --> 01:02:12.760 And it's the other direction that will require some work. 01:02:12.760 --> 01:02:16.130 And this one is actually genuinely tricky. 01:02:16.130 --> 01:02:21.160 So and it's almost kind of a miraculous statement, 01:02:21.160 --> 01:02:25.630 that somehow if you only knew the F densities-- 01:02:25.630 --> 01:02:28.480 so somebody gives you this very large sequence of graphs 01:02:28.480 --> 01:02:30.760 and only tells you that the triangle densities, 01:02:30.760 --> 01:02:34.090 the C4 densities, all of these graph densities, they converge. 01:02:34.090 --> 01:02:37.600 Somehow from these small statistics, 01:02:37.600 --> 01:02:41.140 you conclude that the graphs globally look 01:02:41.140 --> 01:02:43.370 very similar to each other. 01:02:43.370 --> 01:02:44.870 That's actually, if you think about, 01:02:44.870 --> 01:02:49.035 this is an amazing statement. 01:02:49.035 --> 01:02:50.160 OK, so let's see the proof. 01:02:54.030 --> 01:02:56.100 The proof method here is somewhat 01:02:56.100 --> 01:03:00.450 representative of these graph-limit-type arguments. 01:03:00.450 --> 01:03:03.870 So it's worth paying attention to see how this one goes. 01:03:03.870 --> 01:03:07.380 So by compactness, if-- 01:03:07.380 --> 01:03:11.550 OK, we're going to set up by contradiction. 01:03:11.550 --> 01:03:24.290 If the sequence is not Cauchy, then there 01:03:24.290 --> 01:03:26.380 exists two limit points. 01:03:31.930 --> 01:03:34.830 So there exists at least two distinct limit points. 01:03:34.830 --> 01:03:40.200 And call them u and W, and such that-- 01:03:46.370 --> 01:03:50.110 so because you have two separate limit points, 01:03:50.110 --> 01:03:54.590 you must have that this sequence, 01:03:54.590 --> 01:03:57.990 at least along a subsequence that converges to W, 01:03:57.990 --> 01:04:05.970 converges in F densities to W. So initially, this 01:04:05.970 --> 01:04:07.260 is true along subsequence. 01:04:07.260 --> 01:04:09.160 But the left-hand side is convergent, 01:04:09.160 --> 01:04:12.790 so this is true along the sequence. 01:04:12.790 --> 01:04:15.370 But u is also a limit point. 01:04:15.370 --> 01:04:21.990 So the same is true for u. 01:04:26.337 --> 01:04:34.670 And therefore, the F density in W must equal to the F density 01:04:34.670 --> 01:04:45.670 in u for all F. 01:04:45.670 --> 01:04:49.420 So we would be done if we can prove that the F densities, 01:04:49.420 --> 01:04:51.670 the collection of all these F densities, 01:04:51.670 --> 01:04:53.705 they determine the graphon. 01:04:53.705 --> 01:04:54.830 And that's indeed the case. 01:04:54.830 --> 01:04:56.540 And so this is the next claim. 01:04:56.540 --> 01:05:00.140 So it's what I will call a moment 01:05:00.140 --> 01:05:17.450 lemma, is that if u and W are graphons such that the F 01:05:17.450 --> 01:05:27.380 densities agree for all F, then the cut distance between u 01:05:27.380 --> 01:05:31.910 and W is equal to 0. 01:05:31.910 --> 01:05:34.880 Somehow the local statistics tells you globally 01:05:34.880 --> 01:05:39.420 that these two graphons must agree with each other. 01:05:39.420 --> 01:05:42.360 Does anyone know why I call it a moment lemma? 01:05:42.360 --> 01:05:46.730 There is something else which this should remind you of. 01:05:46.730 --> 01:05:50.040 So there are some classical results in probability that 01:05:50.040 --> 01:05:53.240 tells you, if you have two probability distributions, 01:05:53.240 --> 01:05:57.830 both, assume are nice enough, then if they have the same 01:05:57.830 --> 01:06:01.490 k'th moment for every k, so first moment, second moment, 01:06:01.490 --> 01:06:03.770 third moment, if all the moments agree, 01:06:03.770 --> 01:06:07.260 that these two probability distributions should agree. 01:06:07.260 --> 01:06:10.130 And this is some graphical version of that. 01:06:10.130 --> 01:06:12.600 So instead of looking at the probability distribution, 01:06:12.600 --> 01:06:15.590 we're looking at graphons, which are two-dimensional. 01:06:15.590 --> 01:06:17.135 These are two-dimensional objects. 01:06:17.135 --> 01:06:19.760 And this moments lemma tells you that in these two-dimensional, 01:06:19.760 --> 01:06:21.927 in the corresponding two-dimensional moments, namely 01:06:21.927 --> 01:06:25.040 these F moments, if they agree, then 01:06:25.040 --> 01:06:27.380 the two graphons must agree. 01:06:30.240 --> 01:06:33.810 So it's the analog of the probability theory 01:06:33.810 --> 01:06:37.860 statement about moments. 01:06:37.860 --> 01:06:39.540 The proof is actually somewhat tricky. 01:06:39.540 --> 01:06:41.190 So I'm only going to give you a sketch. 01:06:44.880 --> 01:06:49.590 And the key here is to consider the W random graph, which 01:06:49.590 --> 01:06:51.090 we saw last lecture. 01:06:57.326 --> 01:07:00.320 So this is W random graph with k vertices 01:07:00.320 --> 01:07:06.070 sampled using the graphon W. 01:07:06.070 --> 01:07:09.010 So a key observation here is that, 01:07:09.010 --> 01:07:19.140 for every F, the probability that the sampled W 01:07:19.140 --> 01:07:26.560 random graph agrees with F-- 01:07:26.560 --> 01:07:29.960 and here, there is a bit of a technicality. 01:07:29.960 --> 01:07:31.910 I want them to agree as labeled graphs. 01:07:35.880 --> 01:07:39.340 So the vertices of W are a priori labeled 1 through k. 01:07:39.340 --> 01:07:43.330 And this kW random graph is generated with vertices 01:07:43.330 --> 01:07:44.890 labeled 1 through k. 01:07:44.890 --> 01:07:48.640 They agree with some probability that is completely 01:07:48.640 --> 01:07:54.170 determined by the F densities. 01:07:54.170 --> 01:07:54.670 Yes? 01:07:54.670 --> 01:07:56.740 AUDIENCE: Is k the number of vertices of F? 01:07:56.740 --> 01:08:02.390 PROFESSOR: Yeah, so k is the number of vertices of F. 01:08:02.390 --> 01:08:04.867 And the specific formula is not so important. 01:08:04.867 --> 01:08:05.950 Let me just write it down. 01:08:05.950 --> 01:08:09.470 But the point is that, if you know all the F densities, then 01:08:09.470 --> 01:08:12.500 you have all the information about the distribution 01:08:12.500 --> 01:08:15.110 of this W random graph. 01:08:15.110 --> 01:08:18.810 And the way you can calculate the actual probability 01:08:18.810 --> 01:08:21.173 is via an inclusion exclusion. 01:08:23.840 --> 01:08:26.090 And the reason we have to do this inclusion exclusion 01:08:26.090 --> 01:08:28.340 is just because this is more like counting 01:08:28.340 --> 01:08:29.760 induced subgraphs. 01:08:29.760 --> 01:08:31.939 And this is counting actual subgraphs. 01:08:31.939 --> 01:08:33.770 So there is an extra step. 01:08:33.770 --> 01:08:36.880 But the point is that, if you knew this data, the moment's 01:08:36.880 --> 01:08:40.026 data then you immediately know the distribution of the W 01:08:40.026 --> 01:08:40.609 random graphs. 01:08:51.800 --> 01:08:57.520 OK, so if I have two graphons for which I know that their F 01:08:57.520 --> 01:09:00.350 densities agree, then I should be 01:09:00.350 --> 01:09:05.660 able to conclude that the corresponding W 01:09:05.660 --> 01:09:19.580 random graphs also have the same distribution, in particular, 01:09:19.580 --> 01:09:25.340 this, the W random graph and the u random graph 01:09:25.340 --> 01:09:27.324 have the same distribution. 01:09:34.010 --> 01:09:37.939 I am going to create a variant of the W random graph 01:09:37.939 --> 01:09:42.979 which is something called an H random graph. 01:09:42.979 --> 01:09:44.720 It's kind of like the W random graph 01:09:44.720 --> 01:09:46.800 except I forget the very last step. 01:09:46.800 --> 01:09:51.319 So I only keep a weighted-- 01:09:51.319 --> 01:10:06.310 think of it as, so think this is an edge-weighted graph where 01:10:06.310 --> 01:10:16.860 you sample x1 through xk uniformly between 0 and 1. 01:10:16.860 --> 01:10:26.680 And I put edge weight between i and j to be W of xi xj. 01:10:26.680 --> 01:10:29.580 So the difference between this H version and the G version 01:10:29.580 --> 01:10:31.110 is that the G version is obtained 01:10:31.110 --> 01:10:34.120 by turning this weight into an actual edge 01:10:34.120 --> 01:10:37.320 with that probability, but if I don't do the last step, 01:10:37.320 --> 01:10:41.150 I obtain this intermediate object. 01:10:41.150 --> 01:10:46.920 So the following are true. 01:10:46.920 --> 01:10:49.430 And this is where I'm going to skip the proofs. 01:10:52.700 --> 01:11:02.430 If I look at this H random graph and the G random graph, 01:11:02.430 --> 01:11:05.663 they are very close in cut distance. 01:11:05.663 --> 01:11:07.080 You can think of this as the claim 01:11:07.080 --> 01:11:11.250 that G and P is very close to G in cut distance. 01:11:11.250 --> 01:11:12.930 So they are very close in cut distance. 01:11:18.160 --> 01:11:23.550 As k going to infinity with probability 1-- 01:11:23.550 --> 01:11:25.050 so now I'm going to do the proof. 01:11:25.050 --> 01:11:27.720 But it's some kind of a concentration argument. 01:11:33.990 --> 01:11:41.930 And the second claim is that the H random graph is actually 01:11:41.930 --> 01:11:46.660 very close to the original graphon W, as well. 01:11:51.260 --> 01:11:57.770 This is also little l1 in distance as k goes to infinity. 01:11:57.770 --> 01:12:01.180 So this one is, again, not so obvious. 01:12:01.180 --> 01:12:08.870 But it's easier in the case when W 01:12:08.870 --> 01:12:16.520 itself is a step function, in which case, the produced 01:12:16.520 --> 01:12:20.180 H is almost the same as W, except the boundaries are 01:12:20.180 --> 01:12:24.150 slightly shifted, perhaps. 01:12:24.150 --> 01:12:30.890 And so you first approximate W by a step function, 01:12:30.890 --> 01:12:33.530 and prove this up to an epsilon approximation, 01:12:33.530 --> 01:12:37.470 and then let the steps go to infinity. 01:12:37.470 --> 01:12:41.060 So if you have these two claims, so then 01:12:41.060 --> 01:12:58.640 we see that this one here is identically distributed as ku. 01:12:58.640 --> 01:13:06.200 So it should follow that the corresponding H random graph 01:13:06.200 --> 01:13:14.765 for u, if you place the same inequalities by the u versions, 01:13:14.765 --> 01:13:16.550 it should also be true. 01:13:16.550 --> 01:13:21.730 So because these two are the same distribution, 01:13:21.730 --> 01:13:24.220 if you follow this chain, you obtain 01:13:24.220 --> 01:13:30.590 that the cut distance between u and w is equal to 0. 01:13:38.410 --> 01:13:45.790 I want to close by mentioning, in some sense-- 01:13:45.790 --> 01:13:50.580 so here, you have two graphons that have exactly the same F 01:13:50.580 --> 01:13:51.770 moments. 01:13:51.770 --> 01:13:54.790 But what if I give you two graphons which have very 01:13:54.790 --> 01:13:56.500 similar moments to each other? 01:13:56.500 --> 01:13:59.140 Can you conclude that the two graphons 01:13:59.140 --> 01:14:01.370 are close to each other? 01:14:01.370 --> 01:14:04.510 And that will be some kind of an inverse counting lemma. 01:14:04.510 --> 01:14:06.670 And in fact, it does follow as a corollary. 01:14:17.070 --> 01:14:20.340 And the statement is that, for every epsilon, 01:14:20.340 --> 01:14:32.330 there exists k and eta such that if the two graphons u and W are 01:14:32.330 --> 01:14:42.080 such that the F densities do not differ by more than eta 01:14:42.080 --> 01:14:51.460 for every F on, at most, k vertices, 01:14:51.460 --> 01:15:01.120 then the cut distance between u and W is, at most, epsilon. 01:15:01.120 --> 01:15:03.910 So the counting lemma tells you, if the cut distance is small, 01:15:03.910 --> 01:15:07.120 then all the F moments are close to each other. 01:15:07.120 --> 01:15:09.430 And the inverse tells you this converse. 01:15:09.430 --> 01:15:13.090 So it tells you this, if you have similar F moments, 01:15:13.090 --> 01:15:18.690 up to a certain point, then this is small. 01:15:18.690 --> 01:15:20.720 You can deduce the inverse counting lemma 01:15:20.720 --> 01:15:23.730 from the moments lemma via a compactness argument 01:15:23.730 --> 01:15:27.420 similar to the one that we did in class today. 01:15:27.420 --> 01:15:30.240 And I want to give you a chance to practice with that argument. 01:15:30.240 --> 01:15:33.120 So this will be on the homework, for the next homework. 01:15:33.120 --> 01:15:34.860 I'll give you some practice with using 01:15:34.860 --> 01:15:37.780 these compactness arguments. 01:15:37.780 --> 01:15:42.670 But you see, just with the other compactness statements, 01:15:42.670 --> 01:15:46.860 it doesn't tell you anything about the k and the epsilon 01:15:46.860 --> 01:15:48.210 as a function of-- 01:15:48.210 --> 01:15:51.600 the k and eta as a function of epsilon. 01:15:51.600 --> 01:15:54.400 So there are other proofs that gives you concrete bounds, 01:15:54.400 --> 01:15:58.270 but this proof here is much simpler 01:15:58.270 --> 01:16:04.370 if you assume the corresponding results about compactness. 01:16:04.370 --> 01:16:11.780 And finally, I want to mention that in the moments lemma, 01:16:11.780 --> 01:16:15.980 in order to deduce that u and w have the same-- 01:16:15.980 --> 01:16:18.320 that they are basically the same graphon, 01:16:18.320 --> 01:16:23.060 we need to consider F moments for all F's. 01:16:23.060 --> 01:16:25.640 So you might ask, could it be the case 01:16:25.640 --> 01:16:31.800 that we only need some finite set of F's to deduce-- 01:16:31.800 --> 01:16:34.220 to recover the graphon? 01:16:34.220 --> 01:16:40.400 Is it the case that you can recover W 01:16:40.400 --> 01:16:44.410 from only a finite number of F moments? 01:16:44.410 --> 01:16:47.000 And this is, it's actually a very interesting problem 01:16:47.000 --> 01:16:50.660 for which we already saw one instance. 01:16:50.660 --> 01:16:54.710 Namely, when we discussed quasi-random graphs, 01:16:54.710 --> 01:17:01.910 we saw that if you know that the k2 moment is p 01:17:01.910 --> 01:17:12.860 and also the C4 moment is p to the 4, then 01:17:12.860 --> 01:17:16.880 we can deduce that the graphon must 01:17:16.880 --> 01:17:18.942 be the constant graphon, p. 01:17:18.942 --> 01:17:20.930 OK, so we didn't do it in this language, 01:17:20.930 --> 01:17:23.060 but that's what the proof does. 01:17:23.060 --> 01:17:27.050 And likewise, you can use this to deduce a qualitative version 01:17:27.050 --> 01:17:40.670 where you have an extra slack and an extra slack over here. 01:17:40.670 --> 01:17:45.410 So you might ask, except for the constant graphons, 01:17:45.410 --> 01:17:48.050 are there other graphons for which you can similarly 01:17:48.050 --> 01:17:49.880 deduce-- 01:17:49.880 --> 01:17:53.840 recover this graphon from just a finite amount of moments data? 01:17:53.840 --> 01:17:56.510 And such graphons are known as finitely forcible. 01:18:03.350 --> 01:18:08.000 So finitely forcible graphons W such 01:18:08.000 --> 01:18:20.690 that a finite number of moments can uniquely recover-- 01:18:25.350 --> 01:18:28.620 can uniquely identify this graphon, W. 01:18:28.620 --> 01:18:30.510 And a very interesting question is, 01:18:30.510 --> 01:18:34.130 what is the set of all finitely forcible graphons? 01:18:34.130 --> 01:18:36.290 And it turns out, this is not at all obvious. 01:18:36.290 --> 01:18:38.790 And let me just give you some examples, highly non-trivial, 01:18:38.790 --> 01:18:42.240 that turned out to be finitely forcible. 01:18:42.240 --> 01:18:47.040 For example, anything which is a step graphon 01:18:47.040 --> 01:18:47.915 is finitely forcible. 01:18:52.120 --> 01:18:55.350 The half graphon which corresponds 01:18:55.350 --> 01:18:58.870 to the limit of a sequence of half graphs 01:18:58.870 --> 01:19:00.450 is finitely forcible. 01:19:00.450 --> 01:19:03.630 I mean, already, I think neither of these two examples 01:19:03.630 --> 01:19:06.570 are easy at all. 01:19:06.570 --> 01:19:10.140 And this example here can be generalized where 01:19:10.140 --> 01:19:12.000 you have any polynomial curve. 01:19:20.820 --> 01:19:22.905 I think this has to be-- 01:19:22.905 --> 01:19:28.660 so if it's a polynomial curve, it's also finitely forcible. 01:19:28.660 --> 01:19:30.370 But turns out finitely forcible graphons 01:19:30.370 --> 01:19:32.920 can get quite complicated. 01:19:32.920 --> 01:19:38.150 And there is still rather quite a bit of mystery around them. 01:19:38.150 --> 01:19:43.810 OK, so next time, I want to discuss some inequalities that 01:19:43.810 --> 01:19:45.640 come out of-- 01:19:45.640 --> 01:19:49.034 you can state between different F densities. 01:19:49.034 --> 01:19:49.940 OK, great. 01:19:49.940 --> 01:19:51.820 That's all for today.