1 00:00:18,265 --> 00:00:19,640 YUFEI ZHAO: So today we are going 2 00:00:19,640 --> 00:00:22,320 to start a new chapter on graph limits. 3 00:00:32,020 --> 00:00:37,380 So graph limits is a relatively new subject in graph theory. 4 00:00:37,380 --> 00:00:39,190 So as the name suggests, we're looking 5 00:00:39,190 --> 00:00:44,140 at some kind of an analytic limit of graphs, which 6 00:00:44,140 --> 00:00:46,120 sounds kind of like a strange idea 7 00:00:46,120 --> 00:00:49,990 because you think of graphs as fundamentally discrete objects. 8 00:00:49,990 --> 00:00:54,390 But let me begin with an example to motivate, 9 00:00:54,390 --> 00:00:57,700 at least pure mathematical motivation for graph limits. 10 00:00:57,700 --> 00:01:01,180 There are several other ways you can motivate graphs limits, 11 00:01:01,180 --> 00:01:03,910 especially coming from more applied perspectives. 12 00:01:03,910 --> 00:01:09,070 But let me stick with the following story. 13 00:01:09,070 --> 00:01:12,880 So suppose you lived in ancient Greece 14 00:01:12,880 --> 00:01:15,280 and you only knew rational numbers. 15 00:01:15,280 --> 00:01:17,490 You didn't know about real numbers. 16 00:01:17,490 --> 00:01:20,290 But you understand perfectly rational numbers. 17 00:01:20,290 --> 00:01:22,450 And we wish to maximize. 18 00:01:26,360 --> 00:01:31,630 So we wish to then minimize the following polynomial, 19 00:01:31,630 --> 00:01:39,880 x cubed minus x, let's say for x between 0 and 1. 20 00:01:44,990 --> 00:01:47,090 So you can do this. 21 00:01:47,090 --> 00:01:49,990 And suppose also the Greeks knew calculus 22 00:01:49,990 --> 00:01:52,510 and take the derivative and all of that. 23 00:01:52,510 --> 00:01:54,370 So you find that. 24 00:01:54,370 --> 00:01:58,870 You have a problem because we know-- 25 00:01:58,870 --> 00:02:01,540 so given our advanced state of mathematics, 26 00:02:01,540 --> 00:02:04,780 we know that the maximum-- 27 00:02:04,780 --> 00:02:11,470 so the minimizer is at x equals to 1 over root 3. 28 00:02:11,470 --> 00:02:15,880 But that number doesn't exist in the real numbers. 29 00:02:15,880 --> 00:02:20,840 So how might a civilization that only knew rational numbers 30 00:02:20,840 --> 00:02:23,580 express this answer? 31 00:02:23,580 --> 00:02:29,670 They could say, the minimum occurs not in Q. So there's 32 00:02:29,670 --> 00:02:30,960 not minimized in Q-- 33 00:02:35,210 --> 00:02:39,695 but not minimized by a single number, but by a sequence. 34 00:02:47,710 --> 00:02:52,230 And this is a sequence that a more advanced civilization 35 00:02:52,230 --> 00:02:59,100 would know, a sequence that converges to 1 over root of 3. 36 00:02:59,100 --> 00:03:02,220 But I can give you this sequence through some other means. 37 00:03:02,220 --> 00:03:06,260 And this is one of the ways of defining the complete set 38 00:03:06,260 --> 00:03:07,970 of real numbers, for instance. 39 00:03:07,970 --> 00:03:10,720 But you can define explicitly a sequence of real numbers 40 00:03:10,720 --> 00:03:13,818 that converges. 41 00:03:13,818 --> 00:03:15,610 But of course, this is all quite cumbersome 42 00:03:15,610 --> 00:03:16,870 if you have to actually write down 43 00:03:16,870 --> 00:03:19,078 this sequence of real numbers to express this answer. 44 00:03:19,078 --> 00:03:21,560 It will be much better if we knew the real numbers. 45 00:03:21,560 --> 00:03:22,400 And we do. 46 00:03:22,400 --> 00:03:25,300 And the real numbers, in some sense, 47 00:03:25,300 --> 00:03:27,610 in the very rigorous sense, is a completion 48 00:03:27,610 --> 00:03:29,320 of the rational numbers. 49 00:03:33,730 --> 00:03:38,120 That's the story that we're all familiar with. 50 00:03:38,120 --> 00:03:41,090 But now let's think about graphs which 51 00:03:41,090 --> 00:03:43,760 are some kind of a discrete set of objects, 52 00:03:43,760 --> 00:03:47,050 akin to the rational numbers. 53 00:03:47,050 --> 00:03:51,170 And the story now is, among graphs, 54 00:03:51,170 --> 00:03:58,250 suppose I have a fixed p between 0 and 1. 55 00:03:58,250 --> 00:04:02,140 And the problem now is to minimize 56 00:04:02,140 --> 00:04:15,529 the 4-cycle density among graphs with density, with edge density 57 00:04:15,529 --> 00:04:16,029 p. 58 00:04:24,750 --> 00:04:27,450 So this is some kind of optimization problem. 59 00:04:27,450 --> 00:04:29,360 So I don't restrict the number of vertices. 60 00:04:29,360 --> 00:04:31,730 You can use as many vertices as you like. 61 00:04:31,730 --> 00:04:36,004 And I would like to minimize the 4-cycle density. 62 00:04:36,004 --> 00:04:41,500 Now, we saw a few lectures ago this inequality 63 00:04:41,500 --> 00:04:44,590 that tells us that-- 64 00:04:44,590 --> 00:04:50,730 so we saw a few lectures ago that this density is always 65 00:04:50,730 --> 00:04:53,320 at least p to the fourth. 66 00:04:53,320 --> 00:04:55,420 So in the lecture on quasirandomness, so 67 00:04:55,420 --> 00:04:57,570 we saw this inequality. 68 00:04:57,570 --> 00:05:02,140 And we also saw that this minimum 69 00:05:02,140 --> 00:05:11,860 is approached by a sequence of quasirandom graphs. 70 00:05:20,270 --> 00:05:24,260 And in some sense, that-- so the answer is p to the fourth. 71 00:05:24,260 --> 00:05:25,820 And there's not a specific graph. 72 00:05:25,820 --> 00:05:27,830 There's no one graph that minimizes. 73 00:05:27,830 --> 00:05:31,700 This 4-cycle density is minimized by a sequence. 74 00:05:31,700 --> 00:05:34,430 And just like in the story with the rational numbers 75 00:05:34,430 --> 00:05:36,920 and the real numbers, it would be nice 76 00:05:36,920 --> 00:05:39,620 if we didn't have to write out the answer 77 00:05:39,620 --> 00:05:42,050 in this cumbersome, sequential way, 78 00:05:42,050 --> 00:05:45,770 but just have a single graphical-like object that 79 00:05:45,770 --> 00:05:48,900 depicts what the minimizer should be. 80 00:05:48,900 --> 00:05:51,630 And graph limits provides a language for us to do this. 81 00:05:54,640 --> 00:05:59,160 So one of the goals of the graph limits-- 82 00:06:01,870 --> 00:06:11,890 this gives us a single object for this minimizer 83 00:06:11,890 --> 00:06:15,200 instead of taking a sequence. 84 00:06:15,200 --> 00:06:20,050 So roughly that is the idea that you have a sequence of graphs. 85 00:06:20,050 --> 00:06:22,750 And I would like some analytic object 86 00:06:22,750 --> 00:06:27,190 to capture the behavior of the sequence in the limit. 87 00:06:27,190 --> 00:06:29,680 And these graph limits can be written actually 88 00:06:29,680 --> 00:06:31,955 in a fairly concrete form. 89 00:06:31,955 --> 00:06:34,360 And so now let me begin with some definitions. 90 00:06:37,160 --> 00:06:39,250 The main object that we'll look at 91 00:06:39,250 --> 00:06:41,160 is something called a graphon. 92 00:06:45,990 --> 00:06:48,515 So it merges the two words graph, function. 93 00:06:53,273 --> 00:07:07,520 A graphon is by definition a symmetric, measurable function, 94 00:07:07,520 --> 00:07:12,560 often denoted by the letter W from the unit squared 95 00:07:12,560 --> 00:07:16,240 to the 0, 1 interval. 96 00:07:16,240 --> 00:07:23,540 And here being symmetric means that if you exchange the two 97 00:07:23,540 --> 00:07:26,390 argument variables, this function remains the same. 98 00:07:30,120 --> 00:07:30,830 So that's it. 99 00:07:30,830 --> 00:07:32,750 So that's the definition of a graphon. 100 00:07:32,750 --> 00:07:35,450 And these are the objects that will play the role of limits 101 00:07:35,450 --> 00:07:37,400 for sequences of graphs. 102 00:07:37,400 --> 00:07:40,430 And I will give you lots of examples in a second. 103 00:07:40,430 --> 00:07:42,018 So that's the definition. 104 00:07:44,710 --> 00:07:46,840 This is the form of the graphons that we'll 105 00:07:46,840 --> 00:07:48,060 be looking at mostly. 106 00:07:48,060 --> 00:07:51,730 But just to mention a few remarks, 107 00:07:51,730 --> 00:07:59,440 that the domain can be instead any product 108 00:07:59,440 --> 00:08:03,130 of any square of a probability measure space-- 109 00:08:10,300 --> 00:08:12,240 so instead of taking the 0, 1 interval, 110 00:08:12,240 --> 00:08:14,560 I could also use any probability measure space. 111 00:08:14,560 --> 00:08:16,590 So it's only slightly more general. 112 00:08:16,590 --> 00:08:19,380 So there are some general theorems in measure theory 113 00:08:19,380 --> 00:08:22,680 that tells us that most probability measure 114 00:08:22,680 --> 00:08:24,420 spaces, if they're nice enough, they 115 00:08:24,420 --> 00:08:26,160 are in some sense equivalent or can 116 00:08:26,160 --> 00:08:28,480 be captured by this interval. 117 00:08:28,480 --> 00:08:32,007 So I don't want you to worry too much about all the measure 118 00:08:32,007 --> 00:08:33,299 where there are technicalities. 119 00:08:33,299 --> 00:08:36,210 I think they are not so important for the discussion 120 00:08:36,210 --> 00:08:37,140 of graph limits. 121 00:08:37,140 --> 00:08:43,620 But there are some subtle issues like that just lurking behind. 122 00:08:43,620 --> 00:08:46,017 But I just don't want to really talk about them. 123 00:08:46,017 --> 00:08:47,600 So for the most part, we'll be looking 124 00:08:47,600 --> 00:08:49,280 at graphons of this one. 125 00:08:49,280 --> 00:08:57,060 And also the-- so instead of the domain, so the values-- 126 00:08:57,060 --> 00:09:04,830 so instead of 0, 1 interval, you could also 127 00:09:04,830 --> 00:09:09,750 take a more general space, for example, the real numbers 128 00:09:09,750 --> 00:09:12,000 or even the complex numbers. 129 00:09:12,000 --> 00:09:13,680 I'm going to use the word graphon 130 00:09:13,680 --> 00:09:21,220 to reserve this word for when the values are between 0 and 1. 131 00:09:21,220 --> 00:09:25,950 And if it's in R, let me call this just a kernel, 132 00:09:25,950 --> 00:09:29,100 although that will not come up so much. 133 00:09:29,100 --> 00:09:31,320 So when I say graphon, I just mean 134 00:09:31,320 --> 00:09:32,850 the values between 0 and 1. 135 00:09:32,850 --> 00:09:36,450 Although if you do look up papers in the literature, 136 00:09:36,450 --> 00:09:39,180 sometimes they don't use these words so consistently. 137 00:09:39,180 --> 00:09:41,480 So be careful what they mean by a graphon. 138 00:09:44,678 --> 00:09:45,720 So that's the definition. 139 00:09:45,720 --> 00:09:47,555 But now let me give you some examples 140 00:09:47,555 --> 00:09:50,310 on how do we think of graphons and what do they 141 00:09:50,310 --> 00:09:52,780 have to do with graphs. 142 00:09:52,780 --> 00:09:56,820 So if we start with a graph, I want to show you 143 00:09:56,820 --> 00:09:58,730 how to turn it into a graphon. 144 00:10:03,390 --> 00:10:06,590 So let's start with this graph, which you've seen before. 145 00:10:06,590 --> 00:10:07,720 This is the half graph. 146 00:10:16,480 --> 00:10:22,650 So from this graph, I can label the vertices 147 00:10:22,650 --> 00:10:35,570 and form an adjacency matrix of this graph, where 148 00:10:35,570 --> 00:10:41,940 I label the rows and columns by the vertices 149 00:10:41,940 --> 00:10:44,670 and put in zeros and ones according 150 00:10:44,670 --> 00:10:48,810 to whether the edges are adjacent. 151 00:10:53,100 --> 00:10:54,860 So that's the adjacency matrix. 152 00:10:58,230 --> 00:11:01,650 And now I want you to view this matrix 153 00:11:01,650 --> 00:11:03,900 as a black and white picture. 154 00:11:03,900 --> 00:11:07,410 So think one of these pixelated images, where I 155 00:11:07,410 --> 00:11:12,435 turn the ones into black boxes. 156 00:11:21,100 --> 00:11:23,620 Of course, on the blackboard, black is white and white 157 00:11:23,620 --> 00:11:26,310 is black. 158 00:11:26,310 --> 00:11:29,370 So I turn the ones into black boxes. 159 00:11:29,370 --> 00:11:34,750 And I leave the zeros as empty white space. 160 00:11:34,750 --> 00:11:36,790 So I get this image. 161 00:11:36,790 --> 00:11:41,670 And I think of this image as a function. 162 00:11:41,670 --> 00:11:53,190 And this is the function going from 0, 1, squared to 0, 163 00:11:53,190 --> 00:11:55,920 1, interval, taking only 0 and 1 values. 164 00:12:05,820 --> 00:12:10,120 So that's a function on the square. 165 00:12:10,120 --> 00:12:12,770 But now, so this is a single graph. 166 00:12:12,770 --> 00:12:14,530 So for any specific graph, I can turn it 167 00:12:14,530 --> 00:12:16,950 into a graphon like this. 168 00:12:16,950 --> 00:12:18,880 But now imagine you have a sequence of graphs. 169 00:12:18,880 --> 00:12:21,940 And in particular, consider a sequence of half graphs. 170 00:12:28,480 --> 00:12:31,520 So here is H3. 171 00:12:31,520 --> 00:12:34,540 So Hn is the general half graph. 172 00:12:34,540 --> 00:12:38,320 And you can imagine that, as n gets large, 173 00:12:38,320 --> 00:12:40,690 this picture looks like-- 174 00:12:44,280 --> 00:12:46,080 instead of the staircase you just 175 00:12:46,080 --> 00:12:49,170 have a straight line connecting the two ends. 176 00:13:00,800 --> 00:13:04,510 And indeed, this function here, this graphon, 177 00:13:04,510 --> 00:13:11,400 is the limit of the sequence of half graphs 178 00:13:11,400 --> 00:13:13,553 as n goes to infinity. 179 00:13:16,730 --> 00:13:18,790 So one way you can think about graphons 180 00:13:18,790 --> 00:13:20,710 is you have a sequence of graphs. 181 00:13:20,710 --> 00:13:23,440 You look at their adjacency matrix. 182 00:13:23,440 --> 00:13:25,832 You view it as a picture, a pixelated image, 183 00:13:25,832 --> 00:13:27,790 black and white according to the zeros and ones 184 00:13:27,790 --> 00:13:30,160 in its adjacency matrix. 185 00:13:30,160 --> 00:13:35,140 And as you take a sequence, you make your eyes a little bit 186 00:13:35,140 --> 00:13:35,750 blurry. 187 00:13:35,750 --> 00:13:38,260 And then you think about what the sequence of images 188 00:13:38,260 --> 00:13:40,120 converges to. 189 00:13:40,120 --> 00:13:44,110 So the resulting limit is the limit 190 00:13:44,110 --> 00:13:47,120 of this sequence of graphs. 191 00:13:47,120 --> 00:13:50,440 So that's an informal explanation. 192 00:13:50,440 --> 00:13:52,408 So I haven't done anything precisely. 193 00:13:52,408 --> 00:13:53,950 And in fact, one needs to be somewhat 194 00:13:53,950 --> 00:13:56,620 careful with this depiction because let 195 00:13:56,620 --> 00:13:57,925 me give you another example. 196 00:14:01,150 --> 00:14:16,080 Suppose I have a sequence of random or quasirandom graphs 197 00:14:16,080 --> 00:14:18,860 with edge density 1/2. 198 00:14:25,270 --> 00:14:28,230 So what does this look like? 199 00:14:28,230 --> 00:14:29,860 And I have this picture here. 200 00:14:29,860 --> 00:14:31,630 And I have a lot of-- 201 00:14:34,280 --> 00:14:35,450 so I have a lot of-- 202 00:14:37,990 --> 00:14:40,810 one-half of the pixels are black. 203 00:14:40,810 --> 00:14:44,010 And the other half pixels are white. 204 00:14:44,010 --> 00:14:45,930 And you can think, from far away, 205 00:14:45,930 --> 00:14:48,570 I cannot distinguish necessarily which ones are black and which 206 00:14:48,570 --> 00:14:49,610 ones are white. 207 00:14:49,610 --> 00:14:52,470 And in the limit, it looks like a grayscale image, 208 00:14:52,470 --> 00:14:57,030 with a grayscale being one-half density. 209 00:14:57,030 --> 00:15:01,330 And indeed, it converges to the constant function, 1/2. 210 00:15:10,760 --> 00:15:15,190 So the limits represented by this problem up here 211 00:15:15,190 --> 00:15:20,120 is the constant graphon with the constant value p. 212 00:15:20,120 --> 00:15:22,370 But now let me give you a different example. 213 00:15:25,350 --> 00:15:27,001 Consider a checkerboard. 214 00:15:36,340 --> 00:15:53,280 So here is a checkerboard, where I color the squares according 215 00:15:53,280 --> 00:15:56,070 to, in this alternating black and white manner, 216 00:15:56,070 --> 00:15:58,162 according to a usual checkerboard. 217 00:16:02,890 --> 00:16:07,690 And as the number of squares goes to infinity, 218 00:16:07,690 --> 00:16:10,330 what should this converge to? 219 00:16:13,210 --> 00:16:14,960 By the story I just told you, you 220 00:16:14,960 --> 00:16:18,020 might think that if you zoom out, 221 00:16:18,020 --> 00:16:21,030 everything looks density 1/2. 222 00:16:21,030 --> 00:16:24,210 And so you might guess that the image, the limit, 223 00:16:24,210 --> 00:16:27,820 is the 1/2 constant. 224 00:16:27,820 --> 00:16:28,870 But what is this graph? 225 00:16:32,570 --> 00:16:34,090 It's a complete bipartite graph. 226 00:16:38,390 --> 00:16:44,053 It is a complete bipartite graph between all the even rows. 227 00:16:44,053 --> 00:16:46,470 And there's a different way to draw the complete bipartite 228 00:16:46,470 --> 00:16:48,590 graph-- 229 00:16:48,590 --> 00:16:59,740 namely, that picture, just by permuting the rows and columns. 230 00:16:59,740 --> 00:17:04,000 And it's much more reasonable that this 231 00:17:04,000 --> 00:17:07,030 is the limit of the sequence of complete bipartite 232 00:17:07,030 --> 00:17:08,470 graphs with equal parts. 233 00:17:11,010 --> 00:17:12,569 So one needs to be very careful. 234 00:17:12,569 --> 00:17:17,810 And so it's not necessarily an intuitive definition. 235 00:17:17,810 --> 00:17:20,520 The idea that you just squint your eyes 236 00:17:20,520 --> 00:17:22,770 and think about what the image becomes, 237 00:17:22,770 --> 00:17:26,010 that works fine for intuition for some examples, 238 00:17:26,010 --> 00:17:27,147 but not for others. 239 00:17:27,147 --> 00:17:28,980 So we do really need to be careful in giving 240 00:17:28,980 --> 00:17:30,390 a precise definition. 241 00:17:30,390 --> 00:17:33,750 And here the rearrangement of the rows and columns 242 00:17:33,750 --> 00:17:34,950 needs to be taken care of. 243 00:17:46,777 --> 00:17:47,860 So let me be more precise. 244 00:17:50,700 --> 00:17:57,680 Starting with a graph G, I can-- 245 00:17:57,680 --> 00:18:03,280 so let me label the vertices by 1 through n. 246 00:18:03,280 --> 00:18:15,490 I can denote by W sub G this function, this graphon, 247 00:18:15,490 --> 00:18:20,292 obtained by the following procedure. 248 00:18:24,630 --> 00:18:30,820 First, you partition the interval 249 00:18:30,820 --> 00:18:39,090 into intervals of length exactly 1 over n. 250 00:18:44,680 --> 00:18:54,330 And you set W of x comma y to be basically 251 00:18:54,330 --> 00:18:57,780 what happened in the procedure above. 252 00:18:57,780 --> 00:19:04,530 If x and y lie in the box I sub I cross I sub J, 253 00:19:04,530 --> 00:19:11,300 then I put in 1 if I is adjacent to J and 0 otherwise-- 254 00:19:15,800 --> 00:19:18,390 so this picture, where we obtained 255 00:19:18,390 --> 00:19:21,720 by taking the adjacency matrix and transforming it 256 00:19:21,720 --> 00:19:22,980 into a pixelated image. 257 00:19:28,342 --> 00:19:29,800 What are some of the things that we 258 00:19:29,800 --> 00:19:32,437 would like to do with graph limits or graphs in general? 259 00:19:32,437 --> 00:19:32,937 Yeah? 260 00:19:32,937 --> 00:19:37,273 AUDIENCE: Is the range also 0, 1, squared or 0, 1? 261 00:19:37,273 --> 00:19:38,190 YUFEI ZHAO: Thank you. 262 00:19:38,190 --> 00:19:39,060 The range is 0, 1. 263 00:19:42,730 --> 00:19:45,850 So here are some quantities we are interested in when 264 00:19:45,850 --> 00:19:48,170 considering graph limits. 265 00:19:48,170 --> 00:19:58,954 So given two graphs, G and H, we say that a graph homomorphism-- 266 00:20:04,770 --> 00:20:13,390 so a graph homomorphism between from G to H 267 00:20:13,390 --> 00:20:26,137 is a map of their vertexes such that the edges are preserved. 268 00:20:30,910 --> 00:20:40,865 So you have-- and so whenever uv is an edge of H, 269 00:20:40,865 --> 00:20:48,880 your image vertices get mapped to an edge of G. 270 00:20:48,880 --> 00:20:52,810 And we are interested in the number of graph homomorphisms. 271 00:20:52,810 --> 00:21:04,400 So often I use uppercase to denote a set of homomorphisms G 272 00:21:04,400 --> 00:21:05,390 to H-- 273 00:21:05,390 --> 00:21:07,580 and lowercase to denote the number. 274 00:21:16,690 --> 00:21:21,480 So for example, the number of homomorphisms 275 00:21:21,480 --> 00:21:24,570 from a single vertex-- 276 00:21:24,570 --> 00:21:30,093 so a single vertex with no edge to a graph G, that's just the-- 277 00:21:30,093 --> 00:21:31,010 what is this quantity? 278 00:21:34,572 --> 00:21:36,530 So some number of vertices of G-- 279 00:21:40,730 --> 00:21:48,740 what about homomorphisms from an edge to G? 280 00:21:48,740 --> 00:21:51,180 AUDIENCE: The number of edges? 281 00:21:51,180 --> 00:21:53,790 YUFEI ZHAO: Not quite the number of edges, but twice 282 00:21:53,790 --> 00:21:54,690 the number of edges. 283 00:21:59,730 --> 00:22:02,810 What about the number of homomorphisms 284 00:22:02,810 --> 00:22:04,310 from a triangle to G? 285 00:22:10,420 --> 00:22:12,170 AUDIENCE: 6 times the number of triangles. 286 00:22:12,170 --> 00:22:14,570 YUFEI ZHAO: So yeah, you got the idea-- so the 6 times 287 00:22:14,570 --> 00:22:16,088 the number of triangles. 288 00:22:22,172 --> 00:22:26,740 So now let me ask a slightly more interesting question. 289 00:22:26,740 --> 00:22:28,890 What about the number of homomorphisms from H 290 00:22:28,890 --> 00:22:29,665 to a triangle? 291 00:22:34,990 --> 00:22:37,010 What's a different name for this quantity here? 292 00:22:45,630 --> 00:22:47,550 It's the number of proper three colorings. 293 00:22:53,820 --> 00:22:57,380 So it's the number of proper three colorings, 294 00:22:57,380 --> 00:23:05,160 the number of proper colorings of H with three 295 00:23:05,160 --> 00:23:07,020 labeled colors, red, green, and blue. 296 00:23:09,940 --> 00:23:11,640 So think about the three vertices. 297 00:23:11,640 --> 00:23:13,510 That's red, green, and blue. 298 00:23:13,510 --> 00:23:16,790 And whichever vertex of H can map to red, 299 00:23:16,790 --> 00:23:18,610 color that vertex red. 300 00:23:18,610 --> 00:23:21,150 So you see that there is a one-to-one correspondence 301 00:23:21,150 --> 00:23:25,680 between such homomorphisms and proper colorings. 302 00:23:25,680 --> 00:23:28,740 So many important graph parameters, graph quantities, 303 00:23:28,740 --> 00:23:33,420 can be encoded in terms of graph homomorphisms. 304 00:23:33,420 --> 00:23:35,100 And these are the ones that we're 305 00:23:35,100 --> 00:23:36,920 going to be looking at most of the time. 306 00:23:39,780 --> 00:23:42,490 When we're thinking about very large graphs, 307 00:23:42,490 --> 00:23:45,490 often it's not the number of homomorphisms that concern us, 308 00:23:45,490 --> 00:23:48,980 but the density of homomorphisms. 309 00:23:48,980 --> 00:23:52,730 And the difference between homomorphisms 310 00:23:52,730 --> 00:23:59,450 on one hand and subgraphs is that the homomorphisms are not 311 00:23:59,450 --> 00:24:03,320 quite the same as subgraphs, other 312 00:24:03,320 --> 00:24:06,620 than this multiplicity, because you might have 313 00:24:06,620 --> 00:24:12,220 non-injective homomorphisms. 314 00:24:12,220 --> 00:24:14,770 But these non-injective homomorphisms 315 00:24:14,770 --> 00:24:18,700 do not end up contributing very much because they only 316 00:24:18,700 --> 00:24:25,540 have n to the number of vertices of H minus 1 317 00:24:25,540 --> 00:24:30,070 on that border where I think of n as the number of vertices 318 00:24:30,070 --> 00:24:33,690 of G. n is supposed to be large. 319 00:24:33,690 --> 00:24:37,170 So in terms of graph limits when n gets large, 320 00:24:37,170 --> 00:24:40,350 I don't need to distinguish so much between homomorphisms 321 00:24:40,350 --> 00:24:44,020 and subgraphs. 322 00:24:44,020 --> 00:24:57,380 We define the homomorphism density, denoted by the letter 323 00:24:57,380 --> 00:25:04,060 t, from H to G, by-- 324 00:25:04,060 --> 00:25:08,810 define it to be the fraction of all vertex maps 325 00:25:08,810 --> 00:25:11,690 that are homomorphisms. 326 00:25:17,890 --> 00:25:22,720 So this is also equivalent to be defined as the probability 327 00:25:22,720 --> 00:25:31,780 that a uniform random map from the vertex set 328 00:25:31,780 --> 00:25:40,930 of H to the vertex set of G is a homomorphism from H to G. 329 00:25:40,930 --> 00:25:43,460 So it's a graph homomorphism. 330 00:25:43,460 --> 00:25:45,680 And this quantity turns out to be quite important. 331 00:25:45,680 --> 00:25:48,740 So we're going to be seeing this a lot. 332 00:25:48,740 --> 00:25:55,170 And because of this remark over here, in the limit, 333 00:25:55,170 --> 00:26:02,600 this quantity of graph homomorphism densities 334 00:26:02,600 --> 00:26:06,960 in the limit as the number of vertices G goes to infinity 335 00:26:06,960 --> 00:26:16,470 and H fixed, the homomorphism densities 336 00:26:16,470 --> 00:26:20,467 approaches the same limit as subgraph densities. 337 00:26:29,132 --> 00:26:30,840 So you should regard these two quantities 338 00:26:30,840 --> 00:26:32,070 as basically the same thing. 339 00:26:35,575 --> 00:26:36,450 Any questions so far? 340 00:26:41,100 --> 00:26:45,480 So all of these quantities so far defined are for-- 341 00:26:45,480 --> 00:26:49,290 so everything is defined so far for graphs, so what happens 342 00:26:49,290 --> 00:26:50,820 between graphs and graphs. 343 00:26:50,820 --> 00:26:53,590 So what about for graphons? 344 00:26:53,590 --> 00:26:56,730 I'll give you this limit object, this analytic object. 345 00:26:56,730 --> 00:27:01,500 I can still define densities by integrals now. 346 00:27:01,500 --> 00:27:06,560 So suppose I start with a symmetric measurable function. 347 00:27:14,480 --> 00:27:19,250 So tell me, for example, a graphon. 348 00:27:19,250 --> 00:27:23,780 But I can let my range be even more generous. 349 00:27:23,780 --> 00:27:29,390 Starting with such a function, I define the graph homomorphism 350 00:27:29,390 --> 00:27:35,030 density from a fixed graph H to this graphon or kernel, 351 00:27:35,030 --> 00:27:45,223 more generally, to be the following integral, where I'm-- 352 00:27:45,223 --> 00:27:46,640 before writing down the full form, 353 00:27:46,640 --> 00:27:48,015 let me first give you an example. 354 00:27:48,015 --> 00:27:49,530 I think it will be more helpful. 355 00:27:49,530 --> 00:27:57,690 So if I'm looking at a triangle going to W, what I would like 356 00:27:57,690 --> 00:28:01,725 is the integral that captures the triangle density. 357 00:28:08,410 --> 00:28:19,120 So this quantity here, if I let x, y, and z vary over 0 and 1, 358 00:28:19,120 --> 00:28:21,670 0 through 1, independently and uniformly, 359 00:28:21,670 --> 00:28:27,540 then this quantity here captures the triangle density in W. 360 00:28:27,540 --> 00:28:30,660 In fact, and I'll state this more precisely in a second-- 361 00:28:30,660 --> 00:28:33,180 if you look at the translation from graph 362 00:28:33,180 --> 00:28:37,160 to graphon and combine that translation 363 00:28:37,160 --> 00:28:41,240 with this definition here, you recover the triangle density. 364 00:28:45,830 --> 00:28:51,500 More generally, for H instead of a triangle, 365 00:28:51,500 --> 00:28:57,110 the H density in a graphon is defined to be the integral of-- 366 00:28:57,110 --> 00:29:01,010 instead of this product here, I take 367 00:29:01,010 --> 00:29:05,880 a product corresponding to the graph structure of H 368 00:29:05,880 --> 00:29:14,360 with one factor for each edge of H. 369 00:29:14,360 --> 00:29:28,910 And the variables go over the vertex set of H. 370 00:29:28,910 --> 00:29:33,000 So this is the definition of homomorphism densities, 371 00:29:33,000 --> 00:29:37,250 not for graphs, but for symmetric measurable functions, 372 00:29:37,250 --> 00:29:39,300 in particular, for graphons. 373 00:29:39,300 --> 00:29:40,970 And we define it this way because-- 374 00:29:40,970 --> 00:29:43,900 and we use the same symbols because these two definitions 375 00:29:43,900 --> 00:29:44,400 agree. 376 00:29:48,970 --> 00:29:52,780 If you start with a graph and look at the H density in G, 377 00:29:52,780 --> 00:29:58,780 then this quantity here is equal to the H density 378 00:29:58,780 --> 00:30:02,590 in the graphon associated to the graph G constructed 379 00:30:02,590 --> 00:30:03,520 as we did just now. 380 00:30:06,710 --> 00:30:09,110 So make sure you understand why this is true 381 00:30:09,110 --> 00:30:11,030 and why we defined the densities this way. 382 00:30:13,905 --> 00:30:14,780 Any questions so far? 383 00:30:24,990 --> 00:30:31,310 So we've given the definition of graph homomorphism density. 384 00:30:31,310 --> 00:30:34,260 And we've defined these objects, these graphons. 385 00:30:40,550 --> 00:30:43,800 And I mentioned even something about the idea of a limit. 386 00:30:43,800 --> 00:30:46,980 But in what sense can we have a limit of graphs? 387 00:31:04,430 --> 00:31:06,250 So here is an important definition 388 00:31:06,250 --> 00:31:08,650 on the convergence of graphs. 389 00:31:08,650 --> 00:31:14,140 So in what sense can we say that a sequence of graphs converge? 390 00:31:14,140 --> 00:31:28,680 So we say that a sequence of graphs G sub n-- 391 00:31:32,800 --> 00:31:36,310 graphs or graphons, so these two definitions 392 00:31:36,310 --> 00:31:42,610 are interchangeable for what I'm about to say regarding limits 393 00:31:42,610 --> 00:31:44,470 for graphons, in which case I'm going 394 00:31:44,470 --> 00:31:46,450 to denote them by W sub n. 395 00:31:49,830 --> 00:31:57,370 So we say the sequence is convergent 396 00:31:57,370 --> 00:32:06,590 if the sequence of subgraph densities-- 397 00:32:06,590 --> 00:32:08,390 of course, if you are looking at graphons, 398 00:32:08,390 --> 00:32:14,600 then you should look at the graphon, the subgraph density 399 00:32:14,600 --> 00:32:15,790 in-- 400 00:32:15,790 --> 00:32:20,570 homomorphism density in graphons if this sequence 401 00:32:20,570 --> 00:32:41,540 converges as n goes to infinity for every graph H. 402 00:32:41,540 --> 00:32:43,460 So that's the definition of what it 403 00:32:43,460 --> 00:32:46,220 means for a sequence of graphs to converge, 404 00:32:46,220 --> 00:32:48,860 which so far looks actually quite different from what 405 00:32:48,860 --> 00:32:50,810 we discussed intuitively. 406 00:32:50,810 --> 00:32:52,790 But I will state some theorems towards the end 407 00:32:52,790 --> 00:32:56,450 of this lecture explaining what the connections are. 408 00:32:56,450 --> 00:32:58,538 So intuitively what I said earlier 409 00:32:58,538 --> 00:33:00,080 is that you have a sequence of graphs 410 00:33:00,080 --> 00:33:05,600 that are convergent if you have some vague notion of one 411 00:33:05,600 --> 00:33:09,260 image morphing into a sequence of images morphing 412 00:33:09,260 --> 00:33:11,288 into this final image. 413 00:33:11,288 --> 00:33:12,830 Still hold that thought in your mind. 414 00:33:12,830 --> 00:33:15,050 But that's not a rigorous definition yet. 415 00:33:15,050 --> 00:33:18,470 The definition we will use for convergence 416 00:33:18,470 --> 00:33:20,780 is if all the subgraph-- 417 00:33:20,780 --> 00:33:23,480 all the homomorphism densities were equivalently subgraph 418 00:33:23,480 --> 00:33:24,660 densities, they converge. 419 00:33:27,250 --> 00:33:28,960 So this is the definition. 420 00:33:28,960 --> 00:33:30,380 It's not required. 421 00:33:30,380 --> 00:33:33,910 So this is basically rigorous as stated. 422 00:33:33,910 --> 00:33:38,890 Just as a remark, it's not required 423 00:33:38,890 --> 00:33:46,350 that the number of vertices goes to infinity, 424 00:33:46,350 --> 00:33:50,220 although you really should think that that is the case. 425 00:33:57,140 --> 00:33:59,000 So just to put it out there-- so I 426 00:33:59,000 --> 00:34:00,620 can have a sequence of constant graphs 427 00:34:00,620 --> 00:34:02,037 and they will still be convergent. 428 00:34:02,037 --> 00:34:03,283 And that's still OK. 429 00:34:03,283 --> 00:34:04,700 But you should think of the number 430 00:34:04,700 --> 00:34:05,980 of vertices going to infinity. 431 00:34:05,980 --> 00:34:06,874 Yeah? 432 00:34:06,874 --> 00:34:09,560 AUDIENCE: What is F in the definition? 433 00:34:09,560 --> 00:34:10,820 YUFEI ZHAO: F is H. Thank you. 434 00:34:14,670 --> 00:34:15,868 Any other questions? 435 00:34:32,380 --> 00:34:35,280 So there are some questions that we'd like to discuss. 436 00:34:35,280 --> 00:34:37,980 And this will occupy the next few lectures 437 00:34:37,980 --> 00:34:41,080 in terms of proving the following statements. 438 00:34:41,080 --> 00:34:44,699 One is do you always have graph limits? 439 00:34:44,699 --> 00:34:48,699 If you have a convergent sequence of graphs, 440 00:34:48,699 --> 00:34:53,129 do they always approach a limit? 441 00:34:53,129 --> 00:34:55,440 And just because something is convergent 442 00:34:55,440 --> 00:34:58,670 doesn't mean you can represent the limit necessarily. 443 00:34:58,670 --> 00:35:01,080 So it turns out the answer is yes. 444 00:35:01,080 --> 00:35:03,200 It turns out that-- and this makes 445 00:35:03,200 --> 00:35:05,400 it a good theory, a good, useful theory, 446 00:35:05,400 --> 00:35:07,870 and an easy theory to use, that there is always a limit 447 00:35:07,870 --> 00:35:11,530 object whenever you have convergence. 448 00:35:11,530 --> 00:35:14,580 And the other question is while we 449 00:35:14,580 --> 00:35:18,880 have described intuitively one notion of convergence 450 00:35:18,880 --> 00:35:23,440 and also defined more rigorously another definition 451 00:35:23,440 --> 00:35:27,370 of convergence, are these two notions compatible? 452 00:35:27,370 --> 00:35:29,410 And what does this even mean, this idea 453 00:35:29,410 --> 00:35:32,720 of image becoming closer and closer to a final image? 454 00:35:32,720 --> 00:35:34,303 What does that even mean? 455 00:35:34,303 --> 00:35:35,720 So these are some of the questions 456 00:35:35,720 --> 00:35:37,290 that I would like to address. 457 00:35:37,290 --> 00:35:42,440 So in the next few things that I would like to discuss, 458 00:35:42,440 --> 00:35:44,870 first, I want to give you a definition 459 00:35:44,870 --> 00:35:50,390 of a distance between two graphons or two graphs. 460 00:35:50,390 --> 00:35:54,980 If I give you two graphs, how similar or dissimilar 461 00:35:54,980 --> 00:35:56,860 are they-- 462 00:35:56,860 --> 00:35:58,770 so that we have this metric. 463 00:35:58,770 --> 00:36:01,460 And then we can talk about convergence in metric spaces. 464 00:36:04,040 --> 00:36:06,150 So let's take a quick break. 465 00:36:09,610 --> 00:36:11,700 So given this notion of convergence, 466 00:36:11,700 --> 00:36:14,620 I would like to define the notion of distance 467 00:36:14,620 --> 00:36:18,520 between graphs so that convergence corresponds 468 00:36:18,520 --> 00:36:21,220 to convergence in the metric space 469 00:36:21,220 --> 00:36:23,980 sense of distance going to 0. 470 00:36:23,980 --> 00:36:25,240 So how can we define distance? 471 00:36:36,742 --> 00:36:38,825 First, let me tell you that there's a trivial way. 472 00:36:38,825 --> 00:36:41,200 And so there's a way in which you look at that definition 473 00:36:41,200 --> 00:36:42,830 and produce a distance out. 474 00:36:42,830 --> 00:36:45,800 And here's what you can do. 475 00:36:45,800 --> 00:36:50,000 I can convert that definition to a metric 476 00:36:50,000 --> 00:37:00,310 by setting the distance between two 477 00:37:00,310 --> 00:37:10,150 graphs G and G prime to be the following quantity, obtained 478 00:37:10,150 --> 00:37:11,920 by-- 479 00:37:11,920 --> 00:37:12,920 what would I like to do? 480 00:37:12,920 --> 00:37:15,350 I would like to say the distance goes to 0 if and only 481 00:37:15,350 --> 00:37:21,010 if the homomorphism densities, they 482 00:37:21,010 --> 00:37:22,660 are all close to each other. 483 00:37:22,660 --> 00:37:30,290 And so I can sum up all the homomorphism densities 484 00:37:30,290 --> 00:37:37,220 and look at their differences between G and G prime. 485 00:37:37,220 --> 00:37:42,980 And I simply enumerate the list of all possible graphs. 486 00:37:55,450 --> 00:37:57,950 I want to be just slightly more careful with this definition 487 00:37:57,950 --> 00:38:00,390 here because I want something which-- 488 00:38:00,390 --> 00:38:03,380 so when I write this, this number 489 00:38:03,380 --> 00:38:06,770 might be infinite for all pairs G and G prime. 490 00:38:06,770 --> 00:38:12,400 So if I just add a scaling factor here, then-- 491 00:38:12,400 --> 00:38:15,140 and this is some distance. 492 00:38:15,140 --> 00:38:16,340 So this is some distance. 493 00:38:16,340 --> 00:38:18,890 And you see that it matches the definition up there. 494 00:38:18,890 --> 00:38:21,050 But it's completely useless. 495 00:38:21,050 --> 00:38:22,040 It might as well-- 496 00:38:22,040 --> 00:38:23,540 might as well not have said anything 497 00:38:23,540 --> 00:38:29,080 because it's tautologically the same as what happened up there. 498 00:38:29,080 --> 00:38:31,490 And if I give you two graphs, it doesn't really 499 00:38:31,490 --> 00:38:32,930 tell you all that much information 500 00:38:32,930 --> 00:38:35,360 except to encapsulate that definition 501 00:38:35,360 --> 00:38:37,590 into a single number. 502 00:38:37,590 --> 00:38:38,090 Great. 503 00:38:38,090 --> 00:38:40,730 So I'm just-- the point of this is just to tell you 504 00:38:40,730 --> 00:38:45,470 that there is always a trivial way to define distance. 505 00:38:45,470 --> 00:38:49,160 But we want some more interesting ways. 506 00:38:49,160 --> 00:38:51,920 So what can we do? 507 00:38:51,920 --> 00:38:55,310 So here is an attempt, which is that of an edit distance. 508 00:38:58,690 --> 00:39:01,460 So we have seen this before when we discussed removal lemmas. 509 00:39:01,460 --> 00:39:04,250 The edit distance is the number of edges 510 00:39:04,250 --> 00:39:09,780 you need to change to go from one graph to the other graph. 511 00:39:09,780 --> 00:39:12,060 And this seems like a pretty reasonable thing to do. 512 00:39:12,060 --> 00:39:16,240 And it is an important quantity for many applications, 513 00:39:16,240 --> 00:39:19,970 but turns out not the right one for all application. 514 00:39:19,970 --> 00:39:21,840 And here is the reason. 515 00:39:21,840 --> 00:39:25,960 So this is why the edit distance is-- 516 00:39:25,960 --> 00:39:30,180 by edit distance, I mean 1 over the number 517 00:39:30,180 --> 00:39:38,470 of vertex squared times the number of edge changes needed. 518 00:39:43,148 --> 00:39:45,440 So there's normalization so that the distance is always 519 00:39:45,440 --> 00:39:47,310 between 0 and 1. 520 00:39:47,310 --> 00:39:48,890 But this is not a very good notion 521 00:39:48,890 --> 00:39:51,420 for the following reason. 522 00:39:51,420 --> 00:39:58,970 If I take two copies of the Erdos-Reyni random graph G, n, 523 00:39:58,970 --> 00:40:04,180 1/2, what do you think is the edit distance between two 524 00:40:04,180 --> 00:40:05,320 such random graphs? 525 00:40:10,440 --> 00:40:11,080 How many edges? 526 00:40:11,080 --> 00:40:11,580 Yeah? 527 00:40:11,580 --> 00:40:14,045 AUDIENCE: Isn't it roughly one-half of the number of edges 528 00:40:14,045 --> 00:40:15,712 because there's like a one-half probably 529 00:40:15,712 --> 00:40:21,440 that won't be there or not be there [INAUDIBLE]?? 530 00:40:21,440 --> 00:40:23,607 YUFEI ZHAO: So yeah, so let me try to rephrase 531 00:40:23,607 --> 00:40:24,440 what you are saying. 532 00:40:24,440 --> 00:40:29,680 So suppose I have this G and G prime 533 00:40:29,680 --> 00:40:37,850 both sitting on top of the vertex set n. 534 00:40:37,850 --> 00:40:40,790 So if I'm not allowed to rearrange the vertices, 535 00:40:40,790 --> 00:40:45,860 how many edge changes do I need to go from one to the other? 536 00:40:45,860 --> 00:40:47,700 I need about 1/2. 537 00:41:01,890 --> 00:41:05,910 So one-half the time, I'm going to have a wrong edge there. 538 00:41:05,910 --> 00:41:09,020 Now you can make this number just slightly smaller 539 00:41:09,020 --> 00:41:11,330 by permuting the vertices. 540 00:41:11,330 --> 00:41:13,490 But actually you will not improve that much. 541 00:41:13,490 --> 00:41:17,060 It is still going to be roughly that edit distance, which 542 00:41:17,060 --> 00:41:18,560 is quite large. 543 00:41:18,560 --> 00:41:22,520 This is almost as large as you can possibly get 544 00:41:22,520 --> 00:41:25,960 between two arbitrary graphs. 545 00:41:25,960 --> 00:41:30,510 So if we want to say that random graphs, 546 00:41:30,510 --> 00:41:33,660 they approach a limit, a single limit, 547 00:41:33,660 --> 00:41:36,630 then this is not a very good notion because they are 548 00:41:36,630 --> 00:41:38,240 quite far apart for every n. 549 00:41:41,120 --> 00:41:45,110 So this is the reason why the more obvious suggestion 550 00:41:45,110 --> 00:41:49,590 of an edit distance might not be such a great idea. 551 00:41:49,590 --> 00:41:51,700 So what should we use instead? 552 00:41:51,700 --> 00:41:54,650 So we should take inspiration from what we 553 00:41:54,650 --> 00:41:56,595 discussed in quasirandomness. 554 00:41:56,595 --> 00:41:58,416 You have a question. 555 00:41:58,416 --> 00:42:01,017 AUDIENCE: Is the edit distance only for two graphs 556 00:42:01,017 --> 00:42:02,665 of the same vertex set? 557 00:42:02,665 --> 00:42:05,040 YUFEI ZHAO: So the question is, is the edit distance only 558 00:42:05,040 --> 00:42:07,480 for two graphs with the same vertex set? 559 00:42:07,480 --> 00:42:08,280 Let's say yes. 560 00:42:08,280 --> 00:42:10,500 So we'll see later on, you can also 561 00:42:10,500 --> 00:42:15,580 compare graphs with different number of vertices. 562 00:42:15,580 --> 00:42:18,540 So hold onto that thought. 563 00:42:18,540 --> 00:42:21,480 So I would like to come up with a notion of distance 564 00:42:21,480 --> 00:42:24,660 between graphs that is inspired by our discussion 565 00:42:24,660 --> 00:42:28,120 of quasirandomness earlier. 566 00:42:28,120 --> 00:42:34,940 So think about the discussion of quasirandomness or quasirandom 567 00:42:34,940 --> 00:42:35,440 graphs. 568 00:42:42,920 --> 00:42:56,660 In what sense can G be close to a constant, let's say p? 569 00:42:59,300 --> 00:43:03,050 And so this was the Chung-Graham-Wilson theorem 570 00:43:03,050 --> 00:43:05,460 that we proved a few lectures ago. 571 00:43:05,460 --> 00:43:07,790 So in what sense can G be close to p? 572 00:43:07,790 --> 00:43:10,874 And one of those definitions was discrepancy. 573 00:43:16,570 --> 00:43:25,840 And discrepancy says that if the following quantity 574 00:43:25,840 --> 00:43:31,540 is small for all subsets x and y, which 575 00:43:31,540 --> 00:43:34,370 are subsets of vertices of G-- 576 00:43:34,370 --> 00:43:37,390 so you remember, all of you remember, this part, 577 00:43:37,390 --> 00:43:41,550 the discrepancy hypothesis for quasirandomness. 578 00:43:41,550 --> 00:43:43,240 And this is a kind of definition that we 579 00:43:43,240 --> 00:43:45,590 would like to describe when two graphs are 580 00:43:45,590 --> 00:43:50,750 similar to each other, when they are close in this discrepancy 581 00:43:50,750 --> 00:43:52,270 sense. 582 00:43:52,270 --> 00:43:56,410 So now, instead of a graph and a number, 583 00:43:56,410 --> 00:43:59,280 what if now I have two graphs? 584 00:43:59,280 --> 00:44:05,820 I'll give you two graphs of G and G prime. 585 00:44:05,820 --> 00:44:11,750 And what I would like to say is that, if for now, 586 00:44:11,750 --> 00:44:21,490 so if they have the same vertex set, 587 00:44:21,490 --> 00:44:32,460 I want to say that there are close if I have 588 00:44:32,460 --> 00:44:37,170 that the number of edges between x and y in G 589 00:44:37,170 --> 00:44:42,390 is very close to the number of edges 590 00:44:42,390 --> 00:44:46,860 between x and y in G prime. 591 00:44:46,860 --> 00:44:55,245 And I normalize by the number of vertices squared, 592 00:44:55,245 --> 00:44:58,040 so n this number of vertices. 593 00:44:58,040 --> 00:45:01,800 And I would like to find out the worst possible scenario, so 594 00:45:01,800 --> 00:45:07,130 overall, x and y subsets of the vertex set. 595 00:45:07,130 --> 00:45:12,408 If this quantity is small, then I 596 00:45:12,408 --> 00:45:14,950 would like to say that G and G prime are close to each other. 597 00:45:18,240 --> 00:45:20,810 So this is inspired by this discrepancy notion. 598 00:45:24,850 --> 00:45:28,142 Can you see anything wrong with this definition here? 599 00:45:28,142 --> 00:45:28,642 Yeah? 600 00:45:28,642 --> 00:45:30,620 AUDIENCE: [INAUDIBLE] 601 00:45:30,620 --> 00:45:33,230 YUFEI ZHAO: So permutations are vertices. 602 00:45:33,230 --> 00:45:37,140 So just like in the checkerboard example we saw earlier, 603 00:45:37,140 --> 00:45:38,460 you have two graphs. 604 00:45:38,460 --> 00:45:40,860 And if they are indeed labeled graphs 605 00:45:40,860 --> 00:45:43,920 in the same labeled vertex set, then this 606 00:45:43,920 --> 00:45:46,470 is the definition more or less what we used. 607 00:45:46,470 --> 00:45:48,600 I will define it more precisely in a second. 608 00:45:48,600 --> 00:45:51,690 But if they are unlabeled vertices, 609 00:45:51,690 --> 00:46:01,480 we need to possibly optimize permutations 610 00:46:01,480 --> 00:46:12,470 over rearrangements of vertices, which actually turns out 611 00:46:12,470 --> 00:46:13,630 to be quite subtle. 612 00:46:13,630 --> 00:46:16,160 So I'm going to give precise definitions in a second. 613 00:46:16,160 --> 00:46:19,133 But this one here, so think about permuting vertices. 614 00:46:19,133 --> 00:46:21,050 But it's actually a bit more subtle than that. 615 00:46:26,680 --> 00:46:30,750 So here are some actual definitions. 616 00:46:30,750 --> 00:46:34,630 I'm going to define this quantity called a cut norm. 617 00:46:39,490 --> 00:46:43,360 So this chapter is all going to be somewhat functional analytic 618 00:46:43,360 --> 00:46:44,570 in nature. 619 00:46:44,570 --> 00:46:47,680 So get used to the analytic language. 620 00:46:47,680 --> 00:46:57,560 So the cut norm of W is defined to be the following quantity 621 00:46:57,560 --> 00:47:02,510 denoted by this norm with a box in the subscript, which 622 00:47:02,510 --> 00:47:04,860 is defined to be-- 623 00:47:04,860 --> 00:47:11,910 if I look at this W, and I integrate it over a box, 624 00:47:11,910 --> 00:47:15,420 and I would like to maximize this quantity here 625 00:47:15,420 --> 00:47:19,590 over choices of boxes S and T, they 626 00:47:19,590 --> 00:47:22,545 are subsets of the interval measurable subsets. 627 00:47:27,390 --> 00:47:31,950 So choose your-- so over all possible choices 628 00:47:31,950 --> 00:47:36,630 of measurable subsets S and T, if I 629 00:47:36,630 --> 00:47:39,630 integrate W over S cross T, what is 630 00:47:39,630 --> 00:47:41,760 the furthest I can get from 0? 631 00:47:47,117 --> 00:47:48,700 So this is the definition of cut norm. 632 00:47:48,700 --> 00:47:51,550 And you can already see that it has some relations to what 633 00:47:51,550 --> 00:47:53,100 were discussed up there. 634 00:47:53,100 --> 00:47:54,920 But while we're talking about norms, 635 00:47:54,920 --> 00:47:56,690 let me just mention a few other norms that 636 00:47:56,690 --> 00:48:00,950 might come up later on when we discuss graph limits. 637 00:48:00,950 --> 00:48:03,920 So there will be a lot of norms throughout. 638 00:48:03,920 --> 00:48:09,440 So in particular, the lp norm is going to play a frequent role. 639 00:48:09,440 --> 00:48:15,170 So lp norm is defined by looking at the peak norm 640 00:48:15,170 --> 00:48:17,690 of the absolute value, integrated and then 641 00:48:17,690 --> 00:48:20,960 raised to 1 over p. 642 00:48:20,960 --> 00:48:26,500 And so the infinity norm-- 643 00:48:26,500 --> 00:48:30,610 so this is almost, but not quite the same as the sup-- 644 00:48:34,940 --> 00:48:37,520 so almost the same as the supremum, 645 00:48:37,520 --> 00:48:44,210 but not quite because I need to ignore subsets of measure 0. 646 00:48:54,940 --> 00:48:57,760 So I can write down a formal definition in a second. 647 00:48:57,760 --> 00:48:59,470 But I need to-- 648 00:48:59,470 --> 00:49:01,447 if I change W on the subset of the measure 0, 649 00:49:01,447 --> 00:49:03,030 I shouldn't change any of these norms. 650 00:49:03,030 --> 00:49:07,090 And so the one way to define this essential supremum-- 651 00:49:07,090 --> 00:49:09,630 it's called an essential sup-- 652 00:49:09,630 --> 00:49:14,360 is that it is the largest-- 653 00:49:14,360 --> 00:49:20,920 so it is the smallest lambda such that-- 654 00:49:20,920 --> 00:49:25,210 so the smallest number m such that the measure 655 00:49:25,210 --> 00:49:40,700 of the set taking value bigger than m this set has measure 0. 656 00:49:40,700 --> 00:49:46,140 So it's the threshold above which you-- 657 00:49:46,140 --> 00:49:47,670 this, it has measure 0. 658 00:49:50,890 --> 00:49:53,983 And the l2 norm will play a particularly special role. 659 00:49:53,983 --> 00:49:55,400 And for the l2 norm, you're really 660 00:49:55,400 --> 00:49:57,350 in the Hilbert space, in which case 661 00:49:57,350 --> 00:50:00,770 we are going to have inner products. 662 00:50:00,770 --> 00:50:04,380 And we denote inner products using the square-- 663 00:50:04,380 --> 00:50:06,430 using these brackets. 664 00:50:06,430 --> 00:50:07,590 So everything is real. 665 00:50:07,590 --> 00:50:10,231 I don't have to worry about complex conjugates. 666 00:50:16,080 --> 00:50:17,910 So comparing with the discussion up there, 667 00:50:17,910 --> 00:50:21,535 we see that a sequence of-- 668 00:50:21,535 --> 00:50:32,328 so sequence Gn of quasirandom graphs 669 00:50:32,328 --> 00:50:40,050 has a property that the associated graphons converge 670 00:50:40,050 --> 00:50:42,470 to p in the cut norm. 671 00:50:55,610 --> 00:50:58,580 For quasirandom graphs, there is no issue 672 00:50:58,580 --> 00:51:02,510 having to do with permutations because the target is 673 00:51:02,510 --> 00:51:04,670 invariant upon permutations. 674 00:51:04,670 --> 00:51:06,410 But if I give you two different graphs, 675 00:51:06,410 --> 00:51:09,110 then I need to think about their permutations. 676 00:51:09,110 --> 00:51:12,530 And to study permutations of vertices, 677 00:51:12,530 --> 00:51:15,290 the right way to do this is to consider 678 00:51:15,290 --> 00:51:19,640 measure-preserving transformations. 679 00:51:19,640 --> 00:51:27,870 So we say that phi from the interval to the interval 680 00:51:27,870 --> 00:51:36,790 is measure-preserving because first of all, 681 00:51:36,790 --> 00:51:38,040 it has to be a measurable map. 682 00:51:38,040 --> 00:51:40,500 And everything I'm going to talk about are measurable. 683 00:51:40,500 --> 00:51:43,290 So sometimes I will even omit mentioning it. 684 00:51:43,290 --> 00:51:53,480 So it is measure-preserving if, for all measurable subsets 685 00:51:53,480 --> 00:52:07,160 A of this interval, one has that the pullback of A 686 00:52:07,160 --> 00:52:09,200 has the same measure as A itself. 687 00:52:13,720 --> 00:52:15,210 Let me give you an example. 688 00:52:15,210 --> 00:52:17,710 So you have to be also slightly careful with this definition 689 00:52:17,710 --> 00:52:22,090 if you think about the pushforward that's false. 690 00:52:22,090 --> 00:52:23,650 It has to be the pullback. 691 00:52:23,650 --> 00:52:30,930 So for example, the map which sends-- 692 00:52:30,930 --> 00:52:39,860 so an easy example, the map which sends x to x plus 1/2-- 693 00:52:39,860 --> 00:52:43,720 so think about a circle as your space. 694 00:52:43,720 --> 00:52:47,900 And here I am just rotating the circle by one-half rotation. 695 00:52:47,900 --> 00:52:49,625 So it's obviously measure-preserving. 696 00:52:49,625 --> 00:52:53,350 I am not changing any measures. 697 00:52:53,350 --> 00:52:56,020 Slightly more interesting example, 698 00:52:56,020 --> 00:53:00,340 quite a bit more interesting example is setting x to 2x. 699 00:53:00,340 --> 00:53:02,350 This is also measure-preserving. 700 00:53:02,350 --> 00:53:04,940 And you might be puzzled for a second why 701 00:53:04,940 --> 00:53:07,680 it's measure-preserving because it sounds like it's dilating 702 00:53:07,680 --> 00:53:10,262 everything by a factor of 2. 703 00:53:10,262 --> 00:53:12,220 But if you look at the definition-- and so here 704 00:53:12,220 --> 00:53:15,660 is again mod 1. 705 00:53:15,660 --> 00:53:25,790 If you look at the definition, if you look at, 706 00:53:25,790 --> 00:53:32,300 let's say, a subset A, which is-- 707 00:53:35,020 --> 00:53:36,770 so what should I think? 708 00:53:40,018 --> 00:53:46,660 For example, so if that is my A, so what's the inverse of A? 709 00:53:51,460 --> 00:53:54,250 So it's this set. 710 00:53:59,420 --> 00:54:03,105 So the measure is preserved upon this pullback. 711 00:54:03,105 --> 00:54:05,360 And so if you pushforward, then you 712 00:54:05,360 --> 00:54:06,680 might dilate by a factor of 2. 713 00:54:06,680 --> 00:54:08,857 But when you pullback, the measure gets preserved. 714 00:54:15,400 --> 00:54:17,920 So these measure-preserving transformations 715 00:54:17,920 --> 00:54:23,260 are going to play role of permutations of vertices. 716 00:54:23,260 --> 00:54:25,900 So it turns out that these things are actually-- they 717 00:54:25,900 --> 00:54:28,420 are quite subtle technically. 718 00:54:28,420 --> 00:54:31,510 And I am going to, as much as I can, 719 00:54:31,510 --> 00:54:35,340 ignore some of the measure theoretic technicalities. 720 00:54:35,340 --> 00:54:37,690 But they are quite subtle. 721 00:54:37,690 --> 00:54:40,180 So for example, so now let me give you 722 00:54:40,180 --> 00:54:45,240 a definition for the distance between two graphons. 723 00:54:45,240 --> 00:54:51,290 I write, starting with a symmetric measurable function 724 00:54:51,290 --> 00:55:07,290 W, so I write W superscript phi to denote the function obtained 725 00:55:07,290 --> 00:55:07,980 as follows. 726 00:55:07,980 --> 00:55:10,770 So I think of this as relabeling the vertices of a graph. 727 00:55:14,480 --> 00:55:20,390 And now I define this distance. 728 00:55:20,390 --> 00:55:31,680 So this is going to be called the cut distance between two 729 00:55:31,680 --> 00:55:34,860 symmetric measurable functions, U and W, 730 00:55:34,860 --> 00:55:48,520 to be the infimum over all measure-preserving bijections. 731 00:56:02,310 --> 00:56:04,340 So this is the definition for the distance 732 00:56:04,340 --> 00:56:05,410 between two graphons. 733 00:56:09,648 --> 00:56:13,020 To take the optimal-- 734 00:56:13,020 --> 00:56:15,440 and my question does it-- 735 00:56:15,440 --> 00:56:16,930 I am looking at nth. 736 00:56:16,930 --> 00:56:18,700 So I haven't told you yet whether you 737 00:56:18,700 --> 00:56:19,710 can take a single one. 738 00:56:19,710 --> 00:56:21,335 And it turns out that's a subtle issue. 739 00:56:21,335 --> 00:56:23,050 And generally it doesn't exist. 740 00:56:23,050 --> 00:56:26,320 But I look over all measure-preserving bijections 741 00:56:26,320 --> 00:56:27,010 phi. 742 00:56:27,010 --> 00:56:31,450 And I look at the distance between W and Wv, 743 00:56:31,450 --> 00:56:35,061 optimized over the best possible measure-preserving bijection. 744 00:56:39,680 --> 00:56:41,571 So this nth is really an nth. 745 00:56:45,128 --> 00:56:46,170 It's not always obtained. 746 00:56:49,270 --> 00:56:54,210 And actually, this example here is a great example for-- 747 00:56:54,210 --> 00:56:58,360 you can create an example for why nth is not always obtained 748 00:56:58,360 --> 00:57:00,580 from the discussion over here. 749 00:57:00,580 --> 00:57:08,390 For example, if U is the function x times y, 750 00:57:08,390 --> 00:57:18,020 this is a graphon xy and W is Uv, 751 00:57:18,020 --> 00:57:28,793 where v is the map distance x to 2x, then in your mind, 752 00:57:28,793 --> 00:57:31,210 you should think of these two as really the same graphons. 753 00:57:31,210 --> 00:57:32,410 You are applying the measure-preserving 754 00:57:32,410 --> 00:57:33,035 transformation. 755 00:57:33,035 --> 00:57:35,010 It's like doing a permutation. 756 00:57:35,010 --> 00:57:39,550 But because phi is not bijective, 757 00:57:39,550 --> 00:57:43,980 you cannot putting phi here to get these two things to be 758 00:57:43,980 --> 00:57:44,480 the same. 759 00:57:48,812 --> 00:57:50,020 So there are some subtleties. 760 00:57:50,020 --> 00:57:52,330 So this is really an example just 761 00:57:52,330 --> 00:57:54,370 to highlight there's some subtleties here, 762 00:57:54,370 --> 00:57:58,420 which I am going to try to ignore as much as possible. 763 00:57:58,420 --> 00:58:02,067 But I will always give you correct definitions. 764 00:58:02,067 --> 00:58:02,650 Any questions? 765 00:58:02,650 --> 00:58:03,150 Yeah? 766 00:58:06,533 --> 00:58:07,033 Yeah? 767 00:58:07,033 --> 00:58:09,575 AUDIENCE: So can we expect the cut distance between these two 768 00:58:09,575 --> 00:58:10,733 sets to be 0 [INAUDIBLE]? 769 00:58:10,733 --> 00:58:12,150 YUFEI ZHAO: So the question, do we 770 00:58:12,150 --> 00:58:14,640 expect the cut distance between these two to be 0? 771 00:58:14,640 --> 00:58:16,280 And the answer is yes. 772 00:58:16,280 --> 00:58:19,510 So we do expect them to be 0. 773 00:58:19,510 --> 00:58:22,100 And they are 0. 774 00:58:22,100 --> 00:58:24,620 They are equal to 0. 775 00:58:24,620 --> 00:58:27,890 And let me just tell you one something that is new. 776 00:58:27,890 --> 00:58:31,190 And this is one of those statements 777 00:58:31,190 --> 00:58:34,250 that has a lot of measure theoretic technicalities. 778 00:58:34,250 --> 00:58:42,650 For all graphons U and W, it turns out that there exist 779 00:58:42,650 --> 00:58:43,940 measure-preserving maps-- 780 00:58:52,020 --> 00:58:57,460 so not necessarily bijections, but measure-preserving maps 781 00:58:57,460 --> 00:59:02,720 from 0, 1 interval to itself, such that the distance between 782 00:59:02,720 --> 00:59:09,530 U and W, the cut distance, is obtained by the cut norm 783 00:59:09,530 --> 00:59:10,730 difference between-- 784 00:59:10,730 --> 00:59:14,600 the difference between U phi and W psi. 785 00:59:27,314 --> 00:59:29,770 So don't worry about it. 786 00:59:40,840 --> 00:59:43,500 So far, we have defined this notion of a cut 787 00:59:43,500 --> 00:59:47,170 distance between two graphons. 788 00:59:47,170 --> 00:59:49,750 But now I'll give you two graphs. 789 00:59:49,750 --> 00:59:53,855 So what do you do for two graphs? 790 00:59:53,855 --> 00:59:55,396 Or I can-- yeah? 791 00:59:55,396 --> 00:59:57,438 AUDIENCE: You can take the graphon associated it. 792 00:59:57,438 --> 00:59:58,188 YUFEI ZHAO: Great. 793 00:59:58,188 --> 01:00:00,550 So take the graphon associated with these graphs 794 01:00:00,550 --> 01:00:03,020 and consider their cut distance. 795 01:00:03,020 --> 01:00:12,310 So for graphs G and G prime, and potentially even 796 01:00:12,310 --> 01:00:19,570 a different number of vertices, I 797 01:00:19,570 --> 01:00:25,540 can define the distance, the cut distance between these two 798 01:00:25,540 --> 01:00:34,650 graphs to be the distance between the associated 799 01:00:34,650 --> 01:00:35,150 graphons. 800 01:00:40,100 --> 01:00:47,338 And similarly, if I have a graph and a graphon, 801 01:00:47,338 --> 01:00:48,755 I can also compare their distance. 802 01:01:01,660 --> 01:01:03,340 So what does this actually mean? 803 01:01:03,340 --> 01:01:05,190 So if I give you two graphs, even 804 01:01:05,190 --> 01:01:08,240 with the same number of vertices, 805 01:01:08,240 --> 01:01:12,620 it's not quite the same thing as a permutation of vertices. 806 01:01:12,620 --> 01:01:14,930 It's a bit more subtle. 807 01:01:14,930 --> 01:01:17,520 Now why is it more subtle than just permuting the vertices? 808 01:01:20,650 --> 01:01:23,180 So here we are using measure-preserving 809 01:01:23,180 --> 01:01:27,020 transformations, which doesn't see your atomic vertices. 810 01:01:27,020 --> 01:01:30,120 So we might split up your vertices. 811 01:01:30,120 --> 01:01:33,510 So you might take a vertex and chop it in half 812 01:01:33,510 --> 01:01:35,880 and send one half somewhere and another half 813 01:01:35,880 --> 01:01:38,080 somewhere else because these guys, 814 01:01:38,080 --> 01:01:41,060 they don't care about your vertices anymore. 815 01:01:41,060 --> 01:01:47,130 So it's not quite the same as permuting vertices. 816 01:01:52,450 --> 01:01:54,010 But it's some kind of-- 817 01:01:54,010 --> 01:01:59,530 so you allow some kind of splitting and rearrangement 818 01:01:59,530 --> 01:02:01,760 and overlays. 819 01:02:01,760 --> 01:02:04,510 So you can write out this distance 820 01:02:04,510 --> 01:02:08,230 in this format, find out the best way to split and overlay 821 01:02:08,230 --> 01:02:09,760 and to rearrange that way. 822 01:02:09,760 --> 01:02:13,250 But it's much cleaner to define it in terms of graphons. 823 01:02:13,250 --> 01:02:13,750 Yes? 824 01:02:13,750 --> 01:02:15,792 AUDIENCE: Is this why we take bijections up there 825 01:02:15,792 --> 01:02:16,972 [INAUDIBLE]? 826 01:02:16,972 --> 01:02:19,430 YUFEI ZHAO: The question is, is that why we take bijections 827 01:02:19,430 --> 01:02:19,930 up there? 828 01:02:19,930 --> 01:02:22,370 And no, so up there, if I wrote instead 829 01:02:22,370 --> 01:02:26,270 measure-preserving maps, it's still a correct definition 830 01:02:26,270 --> 01:02:28,740 and it's the same definition. 831 01:02:28,740 --> 01:02:30,930 And the fact that these two are equivalent 832 01:02:30,930 --> 01:02:35,660 goes to some measure theory, which I will not-- 833 01:02:35,660 --> 01:02:44,295 do not want to indulge yo Great. 834 01:02:44,295 --> 01:02:46,420 But the moral of the story is you take two graphons 835 01:02:46,420 --> 01:02:50,620 and rearrange the vertices in some way, in the best way, 836 01:02:50,620 --> 01:02:53,603 overlay them on top of each other and take the difference 837 01:02:53,603 --> 01:02:54,645 and look at the cut norm. 838 01:02:54,645 --> 01:02:55,810 And so that's the distance. 839 01:02:59,180 --> 01:03:04,570 So I want to finish by stating the main theorems 840 01:03:04,570 --> 01:03:07,890 that form graph limit theory. 841 01:03:07,890 --> 01:03:10,620 And these address the questions I mentioned right 842 01:03:10,620 --> 01:03:11,940 before the break. 843 01:03:11,940 --> 01:03:14,960 So do there exist limits? 844 01:03:14,960 --> 01:03:18,620 And do these two different notions of one 845 01:03:18,620 --> 01:03:20,900 having to do with distance and another 846 01:03:20,900 --> 01:03:24,410 having to do with homomorphism densities, 847 01:03:24,410 --> 01:03:26,400 how do they relate to each other? 848 01:03:26,400 --> 01:03:27,590 Are they consistent? 849 01:03:50,070 --> 01:03:54,420 So the first theorem, Theorem 1, has 850 01:03:54,420 --> 01:04:07,480 to do with the equivalence of the convergence, 851 01:04:07,480 --> 01:04:22,800 namely, that if you have a sequence of graphs or graphons, 852 01:04:22,800 --> 01:04:30,540 the sequence is convergent in the sense, up there, if 853 01:04:30,540 --> 01:04:38,380 and only if they are convergent in the sense 854 01:04:38,380 --> 01:04:40,440 of-- in this metric space. 855 01:04:40,440 --> 01:04:43,360 So remember what convergence means in the metric space 856 01:04:43,360 --> 01:04:46,310 is that of a Cauchy sequence-- 857 01:04:46,310 --> 01:04:55,600 so if and only if it is a Cauchy sequence with respect 858 01:04:55,600 --> 01:05:02,470 to this cut distance. 859 01:05:02,470 --> 01:05:05,360 So it's just-- maybe for many of you, 860 01:05:05,360 --> 01:05:07,140 it's been a while since you took 18-100. 861 01:05:07,140 --> 01:05:10,800 So let remind you a Cauchy sequence, in this case, 862 01:05:10,800 --> 01:05:20,310 it means that, if I look at the distance between two graphs, 863 01:05:20,310 --> 01:05:22,920 if I look far enough out, then I can 864 01:05:22,920 --> 01:05:28,500 contain the rest of the sequence in an arbitrarily small ball. 865 01:05:28,500 --> 01:05:33,150 So the sup positive m of this guy here, 866 01:05:33,150 --> 01:05:37,986 goes to 0 as n goes to infinity. 867 01:05:37,986 --> 01:05:42,173 But because we don't know yet whether the limit exists, 868 01:05:42,173 --> 01:05:44,340 so I can't talk about them getting closer and closer 869 01:05:44,340 --> 01:05:45,060 to a limit. 870 01:05:45,060 --> 01:05:47,110 But they mutually get closer to each other. 871 01:05:50,320 --> 01:05:54,350 So Theorem 1 tells us that these two notions, one having 872 01:05:54,350 --> 01:05:57,980 to do with homomorphism densities, is consistent 873 01:05:57,980 --> 01:06:02,360 and in fact equivalent to the appropriate notion 874 01:06:02,360 --> 01:06:03,450 in the metric space. 875 01:06:11,200 --> 01:06:16,380 So let's use a symbol. 876 01:06:16,380 --> 01:06:27,250 So we say that G sub n converges to W, or in the case 877 01:06:27,250 --> 01:06:29,020 of a sequence of graphons. 878 01:06:29,020 --> 01:06:30,520 So we can do that as well. 879 01:06:34,630 --> 01:06:43,440 So here we say that G sub n converges with W, 880 01:06:43,440 --> 01:06:48,960 if whenever you look at the F density in G sub n, 881 01:06:48,960 --> 01:06:56,610 this sequence converges to the corresponding f density in W 882 01:06:56,610 --> 01:07:01,360 for every f, and similarly, if you have 883 01:07:01,360 --> 01:07:04,480 a graphon instead of a graph. 884 01:07:04,480 --> 01:07:08,160 So that definition was just whether a sequence 885 01:07:08,160 --> 01:07:09,140 is convergent. 886 01:07:09,140 --> 01:07:16,190 Here it converges to this graphon W. 887 01:07:16,190 --> 01:07:21,380 And the question is, if you give me a convergent sequence, 888 01:07:21,380 --> 01:07:23,150 is there a limit? 889 01:07:23,150 --> 01:07:25,560 Does it converge to some limit? 890 01:07:25,560 --> 01:07:27,430 And the answer is yes. 891 01:07:27,430 --> 01:07:34,070 And that's the second theorem, which 892 01:07:34,070 --> 01:07:37,840 tells us the existence of a limit, of the limit object. 893 01:07:41,990 --> 01:07:49,830 So the statement is that every convergent sequence 894 01:07:49,830 --> 01:08:04,650 of graph or graphons has a limit graphon. 895 01:08:18,346 --> 01:08:22,930 So now I want you to imagine this space of graphons. 896 01:08:22,930 --> 01:08:25,960 So we'll have this space containing all the graphons. 897 01:08:25,960 --> 01:08:30,149 And let me denote this space by this curly W0. 898 01:08:30,149 --> 01:08:32,279 So this, the 0 is-- don't worry about it. 899 01:08:32,279 --> 01:08:34,080 It's more just convention. 900 01:08:34,080 --> 01:08:37,600 But let me also put a tilde on top for the following reason. 901 01:08:37,600 --> 01:08:51,630 Let this be the space of graphons where we identify 902 01:08:51,630 --> 01:08:53,250 graphons with distance 0. 903 01:09:09,640 --> 01:09:17,290 So then the space combined with this metric is a metric space. 904 01:09:22,149 --> 01:09:23,380 It is the space of graphons. 905 01:09:26,060 --> 01:09:32,859 And so the third theorem is that it's 906 01:09:32,859 --> 01:09:35,529 the compactness of the space of graphons, 907 01:09:35,529 --> 01:09:37,345 namely, that this space is compact. 908 01:09:50,970 --> 01:09:53,819 Because we're in the metric space, 909 01:09:53,819 --> 01:09:59,340 compactness in the usual sense of every open cover has 910 01:09:59,340 --> 01:10:04,260 a finite subcover is equivalent to the slightly more intuitive 911 01:10:04,260 --> 01:10:06,550 notion of sequential compactness-- 912 01:10:06,550 --> 01:10:10,170 every sequence has a convergent subsequence. 913 01:10:10,170 --> 01:10:13,530 And then it's also, if you have a limit, 914 01:10:13,530 --> 01:10:15,710 so it converges to some limit. 915 01:10:18,330 --> 01:10:20,670 So how should you think of Theorem 3? 916 01:10:20,670 --> 01:10:24,720 So it's about compactness and some tautological notion. 917 01:10:24,720 --> 01:10:27,720 But intuitively, you should think of compactness 918 01:10:27,720 --> 01:10:29,170 as saying-- 919 01:10:29,170 --> 01:10:31,050 and the English word, the English meaning 920 01:10:31,050 --> 01:10:33,330 of the word compact is small. 921 01:10:33,330 --> 01:10:37,050 You should think of this space as being quite small, 922 01:10:37,050 --> 01:10:40,140 which is rather counterintuitive because we're 923 01:10:40,140 --> 01:10:43,970 looking at the space of graphons, certainly at least as 924 01:10:43,970 --> 01:10:48,150 large as the space of graphs, but really all functions 925 01:10:48,150 --> 01:10:49,875 from the square to the interval. 926 01:10:49,875 --> 01:10:53,010 This seems like a pretty large space. 927 01:10:53,010 --> 01:10:55,290 But this theorem here says that, in fact, that space 928 01:10:55,290 --> 01:10:57,950 is quite small. 929 01:10:57,950 --> 01:11:01,880 And where have we also seen that philosophy before? 930 01:11:06,120 --> 01:11:09,920 So in Szemeredi's Graph Regularity Lemma, 931 01:11:09,920 --> 01:11:12,350 the underlying philosophy there is 932 01:11:12,350 --> 01:11:15,560 that, even though the possibilities, the space 933 01:11:15,560 --> 01:11:19,400 of possibilities for graph is quite large, 934 01:11:19,400 --> 01:11:22,360 once you apply Szemeredi's Regularity Lemma, 935 01:11:22,360 --> 01:11:26,400 and once you are OK with some epsilon approximations, 936 01:11:26,400 --> 01:11:29,510 there is only a small description, 937 01:11:29,510 --> 01:11:32,280 this bounded description, of a graph. 938 01:11:32,280 --> 01:11:35,060 And you can work with that description. 939 01:11:35,060 --> 01:11:37,820 And these two philosophies, it's no coincidence 940 01:11:37,820 --> 01:11:41,270 that they are consistent with each other 941 01:11:41,270 --> 01:11:46,790 because we will use Szemeredi's Regularity Lemma to prove 942 01:11:46,790 --> 01:11:49,710 this compactness. 943 01:11:49,710 --> 01:11:51,900 In fact, we will use a slightly weaker version 944 01:11:51,900 --> 01:11:55,560 of Szemeredi's Regularity Lemma to prove compactness. 945 01:11:55,560 --> 01:11:58,890 And then you will see that, from the compactness, 946 01:11:58,890 --> 01:12:03,010 one can use properties of the compactness 947 01:12:03,010 --> 01:12:07,750 to boost to a stronger version of regularity. 948 01:12:07,750 --> 01:12:09,890 But the underlying philosophy here 949 01:12:09,890 --> 01:12:19,570 is that this compactness is in some sense a quantity. 950 01:12:19,570 --> 01:12:29,530 It's a qualitative reformulation, 951 01:12:29,530 --> 01:12:38,770 analytic reformulation of Szemeredi's Graph Regularity 952 01:12:38,770 --> 01:12:39,270 Lemma. 953 01:12:51,030 --> 01:12:53,780 OK, so-- 954 01:12:53,780 --> 01:12:55,560 So this topic, this graph limits, 955 01:12:55,560 --> 01:12:57,840 which we'll explore for the next few lecturers, 956 01:12:57,840 --> 01:13:01,620 including giving a proof of all three of these main theorems, 957 01:13:01,620 --> 01:13:05,250 nicely encapsulates the past couple of topics we have done. 958 01:13:05,250 --> 01:13:08,040 So on one hand, Szemeredi's Regularity Lemma, 959 01:13:08,040 --> 01:13:09,930 or some version of that, will be used 960 01:13:09,930 --> 01:13:14,520 in proving the existence of the limit and also the compactness. 961 01:13:14,520 --> 01:13:17,370 And also it's philosophically and in some sense 962 01:13:17,370 --> 01:13:19,830 related and very much equivalent in some sense 963 01:13:19,830 --> 01:13:22,830 and related to these notions. 964 01:13:22,830 --> 01:13:25,710 It is also related to quasirandomness-- 965 01:13:25,710 --> 01:13:28,320 in particular, quasirandom graphs 966 01:13:28,320 --> 01:13:31,680 that we did a few lectures ago, where in quasirandom graphs, 967 01:13:31,680 --> 01:13:35,030 we are really looking at the constant graphon 968 01:13:35,030 --> 01:13:36,890 in this language. 969 01:13:36,890 --> 01:13:39,340 And now we expand our horizons. 970 01:13:39,340 --> 01:13:43,570 And instead of just looking at the constant graphon, 971 01:13:43,570 --> 01:13:47,840 we can now consider arbitrary graphons. 972 01:13:47,840 --> 01:13:51,970 They are also this model for a very large graph. 973 01:13:54,530 --> 01:13:56,110 Any questions? 974 01:13:56,110 --> 01:13:56,610 Yeah? 975 01:13:56,610 --> 01:14:00,530 AUDIENCE: Can we prove the theorem analytically and then 976 01:14:00,530 --> 01:14:02,490 deduce the Regularity Lemma with it? 977 01:14:02,490 --> 01:14:03,950 YUFEI ZHAO: The question is, can we 978 01:14:03,950 --> 01:14:07,520 prove Theorem 3 analytically and deduce the Regularity Lemma? 979 01:14:07,520 --> 01:14:09,990 So you will see once you see the proof. 980 01:14:09,990 --> 01:14:11,600 It depends on what you mean. 981 01:14:11,600 --> 01:14:12,990 But roughly, the answer is yes. 982 01:14:12,990 --> 01:14:14,840 But there's a very important caveat. 983 01:14:14,840 --> 01:14:17,930 It's that, because we are using compactness, 984 01:14:17,930 --> 01:14:20,390 any argument involving compactness 985 01:14:20,390 --> 01:14:23,750 gives no quantitative bounds. 986 01:14:23,750 --> 01:14:28,940 So you will have a proof of the Szemeredi Regularity Lemma 987 01:14:28,940 --> 01:14:33,020 that tells you there is a bound for each epsilon. 988 01:14:33,020 --> 01:14:37,305 But it doesn't tell you what the bound is. 989 01:14:37,305 --> 01:14:38,778 Yeah? 990 01:14:38,778 --> 01:14:40,742 AUDIENCE: Doesn't Theorem 3 imply Theorem 1 991 01:14:40,742 --> 01:14:43,197 because of the [INAUDIBLE]? 992 01:14:46,150 --> 01:14:48,280 YUFEI ZHAO: Does Code Theorem 3 imply Theorem 1? 993 01:14:48,280 --> 01:14:50,800 And the answer is no because in Theorem 1, 994 01:14:50,800 --> 01:14:55,800 the notion of convergence is about homomorphism densities. 995 01:14:55,800 --> 01:14:59,000 So Theorem 1 is about these two different notions 996 01:14:59,000 --> 01:15:02,730 of convergence and that they are equivalent to each other. 997 01:15:02,730 --> 01:15:05,420 Theorem 3 is just about the metric. 998 01:15:05,420 --> 01:15:07,506 It's about the cut metric. 999 01:15:07,506 --> 01:15:10,340 And so Theorem 1 is-- the point of Theorem 1 is that you have 1000 01:15:10,340 --> 01:15:11,470 these two-- 1001 01:15:11,470 --> 01:15:13,670 you have these two notions of convergence, 1002 01:15:13,670 --> 01:15:16,340 one having to do with subgraph densities and the other 1003 01:15:16,340 --> 01:15:18,052 having to do with a cut distance. 1004 01:15:18,052 --> 01:15:19,760 And in fact, they are equivalent notions. 1005 01:15:23,140 --> 01:15:24,870 So all great questions-- 1006 01:15:24,870 --> 01:15:25,959 any others? 1007 01:15:25,959 --> 01:15:27,875 AUDIENCE: And for that F, is F a graphon 1008 01:15:27,875 --> 01:15:30,270 because the [INAUDIBLE]? 1009 01:15:30,270 --> 01:15:32,838 Is F a graphon or a graph? 1010 01:15:32,838 --> 01:15:35,130 YUFEI ZHAO: The question is, is F a graph or a graphon? 1011 01:15:35,130 --> 01:15:37,300 F is always a graph. 1012 01:15:37,300 --> 01:15:45,880 So in t F, W, I do not define this quantity for graphon F. 1013 01:15:45,880 --> 01:15:48,000 So this quantity here, I have only 1014 01:15:48,000 --> 01:15:50,580 allowed the second argument to be a graphon. 1015 01:15:50,580 --> 01:15:52,740 The first argument is not allowed to be a graphon. 1016 01:15:52,740 --> 01:15:53,657 It doesn't make sense. 1017 01:15:56,277 --> 01:15:56,777 Yeah? 1018 01:15:56,777 --> 01:16:00,810 AUDIENCE: Doesn't Theorem 1 and 2 together imply Theorem 3? 1019 01:16:00,810 --> 01:16:03,310 YUFEI ZHAO: The question is, doesn't Theorem 1 and Theorem 2 1020 01:16:03,310 --> 01:16:05,240 together imply Theorem 3? 1021 01:16:05,240 --> 01:16:07,780 So first of all, Theorem 1 is really-- 1022 01:16:07,780 --> 01:16:10,090 it's not about compactness. 1023 01:16:10,090 --> 01:16:11,590 So it's really about the equivalence 1024 01:16:11,590 --> 01:16:13,452 of two different notions of convergence. 1025 01:16:13,452 --> 01:16:15,160 It's like you have two different metrics. 1026 01:16:15,160 --> 01:16:16,618 I am showing that these two metrics 1027 01:16:16,618 --> 01:16:18,300 are equivalent to each other. 1028 01:16:18,300 --> 01:16:21,430 Theorem 2 and Theorem 3 are quite intimately related. 1029 01:16:21,430 --> 01:16:22,530 So Theorem 2 is about-- 1030 01:16:25,590 --> 01:16:27,788 Theorem 2, so they are quite related. 1031 01:16:27,788 --> 01:16:29,080 But they're not quite the same. 1032 01:16:29,080 --> 01:16:31,290 So let me just give you the real line analogy, 1033 01:16:31,290 --> 01:16:33,140 going back to what we said in the beginning. 1034 01:16:33,140 --> 01:16:37,410 So Theorem 2 is kind of like saying that the real numbers is 1035 01:16:37,410 --> 01:16:38,490 complete. 1036 01:16:38,490 --> 01:16:40,870 Every convergent sequence has a limit, 1037 01:16:40,870 --> 01:16:43,060 whereas Theorem 3 is more than that. 1038 01:16:43,060 --> 01:16:45,295 It's also bounded in some sense. 1039 01:16:45,295 --> 01:16:48,118 But here, there is no notion of bounded. 1040 01:16:48,118 --> 01:16:48,660 It's compact. 1041 01:16:51,570 --> 01:16:54,390 But the main-- you should think of these two 1042 01:16:54,390 --> 01:16:55,890 are very much related to each other. 1043 01:16:55,890 --> 01:16:59,347 But here it's-- but they are not equivalent. 1044 01:17:02,647 --> 01:17:03,230 Anything else? 1045 01:17:05,910 --> 01:17:06,410 Great. 1046 01:17:06,410 --> 01:17:08,410 So that's all for today.