1 00:00:16,533 --> 00:00:17,450 YUFEI ZHAO: All right. 2 00:00:17,450 --> 00:00:20,870 Last time we started talking about pseudorandom graphs, 3 00:00:20,870 --> 00:00:25,740 and we considered this theorem of Chung, Graham, and Wilson, 4 00:00:25,740 --> 00:00:30,470 which, for dense graphs, gave several equivalent notions 5 00:00:30,470 --> 00:00:34,800 of quasi-randomness that, at least the phase values, 6 00:00:34,800 --> 00:00:37,420 do not appear to be all that equivalent. 7 00:00:37,420 --> 00:00:41,870 But they are actually-- you can deduce one from the other. 8 00:00:41,870 --> 00:00:44,510 There was one condition at the very end which 9 00:00:44,510 --> 00:00:46,850 had to do with eigenvalues. 10 00:00:46,850 --> 00:00:51,125 And, basically, it said that if your second largest eigenvalue 11 00:00:51,125 --> 00:00:56,690 in absolute value is small, then the graph is pseudorandom. 12 00:00:56,690 --> 00:00:59,660 So that's something that I want to explore further today 13 00:00:59,660 --> 00:01:02,240 to better understand the relationship 14 00:01:02,240 --> 00:01:06,410 between eigenvalues of a graph and the pseudorandomness 15 00:01:06,410 --> 00:01:07,455 properties. 16 00:01:10,310 --> 00:01:12,860 For much of-- pretty much all of today, 17 00:01:12,860 --> 00:01:15,470 we're going to look at a special class of graphs 18 00:01:15,470 --> 00:01:17,420 known as n, d, lambda graphs. 19 00:01:22,080 --> 00:01:25,290 This just means we have n vertices, 20 00:01:25,290 --> 00:01:27,610 and we're only going to consider, mostly 21 00:01:27,610 --> 00:01:30,300 out of convenience, d regular graphs. 22 00:01:33,680 --> 00:01:36,110 So this will make our life somewhat simpler. 23 00:01:36,110 --> 00:01:38,210 And the lambda stands for that-- 24 00:01:38,210 --> 00:01:41,940 if you look at the adjacency matrix, 25 00:01:41,940 --> 00:01:45,860 and if you write down the eigenvalues 26 00:01:45,860 --> 00:01:56,380 of the adjacency matrix, then, well, 27 00:01:56,380 --> 00:01:57,630 what are these eigenvalues? 28 00:01:57,630 --> 00:02:03,030 The top one, because it's d regular, is equal to d. 29 00:02:03,030 --> 00:02:07,710 And lambda corresponds to the statement 30 00:02:07,710 --> 00:02:13,170 that all the other eigenvalues are, at most, 31 00:02:13,170 --> 00:02:16,760 lambda in absolute value. 32 00:02:16,760 --> 00:02:18,530 So the top one is equal to d. 33 00:02:18,530 --> 00:02:20,750 All the other ones in absolute value-- 34 00:02:20,750 --> 00:02:25,520 so it could be basically the maximum of these two-- 35 00:02:25,520 --> 00:02:28,890 is bounded above by lambda. 36 00:02:28,890 --> 00:02:32,970 And at the end of last time, we showed this expander 37 00:02:32,970 --> 00:02:48,205 mixing lemma, which, in this language, says that if G is n, 38 00:02:48,205 --> 00:02:58,540 d, lambda, then one has the following discrepancy type. 39 00:02:58,540 --> 00:03:01,770 So the randomness property, namely 40 00:03:01,770 --> 00:03:04,290 that if you look at two vertex sets 41 00:03:04,290 --> 00:03:07,170 and look at how many actual edges are between them compared 42 00:03:07,170 --> 00:03:09,330 to what you expect if this were a random graph 43 00:03:09,330 --> 00:03:14,050 of a similar density, then these two numbers are very similar, 44 00:03:14,050 --> 00:03:19,290 and the amount of error is controlled by your lambda. 45 00:03:19,290 --> 00:03:23,940 In particular, a smaller lambda gives you a more pseudorandom 46 00:03:23,940 --> 00:03:25,800 graph. 47 00:03:25,800 --> 00:03:27,360 So the second part of today's class, 48 00:03:27,360 --> 00:03:29,040 I want to explore the question of how 49 00:03:29,040 --> 00:03:31,590 small this lambda can be. 50 00:03:31,590 --> 00:03:36,170 So what's the optimal amount of pseudorandomness? 51 00:03:36,170 --> 00:03:39,150 But, first, I want to show you some examples. 52 00:03:39,150 --> 00:03:42,730 So, so far, we've been talking about pseudorandom graphs, 53 00:03:42,730 --> 00:03:45,580 and the only example, really, I've 54 00:03:45,580 --> 00:03:49,340 talked about is that a random graph is pseudorandom. 55 00:03:49,340 --> 00:03:50,100 Which is true. 56 00:03:50,100 --> 00:03:53,460 A random graph is pseudorandom with high probability, 57 00:03:53,460 --> 00:03:55,360 but some of the spirit of pseudorandomness 58 00:03:55,360 --> 00:03:58,390 is to come up with non-random examples, come up 59 00:03:58,390 --> 00:04:01,390 with deterministic constructions that give you 60 00:04:01,390 --> 00:04:03,920 pseudorandom properties. 61 00:04:03,920 --> 00:04:06,310 So I want to begin today with an example. 62 00:04:14,720 --> 00:04:17,690 A lot of examples, especially for pseudorandomness, 63 00:04:17,690 --> 00:04:20,779 come from this class of graphs called Cayley graphs, 64 00:04:20,779 --> 00:04:24,710 which are built from a group. 65 00:04:24,710 --> 00:04:27,440 So we're going to reserve the letter G for graphs, so I'm 66 00:04:27,440 --> 00:04:30,300 going to use gamma for a group. 67 00:04:30,300 --> 00:04:37,970 And I have a subset S of gamma, and S is symmetric, 68 00:04:37,970 --> 00:04:40,980 in that if you invert the elements of S, 69 00:04:40,980 --> 00:04:42,770 they remain in S. 70 00:04:42,770 --> 00:04:53,900 Then we define the Cayley graph given by this group 71 00:04:53,900 --> 00:05:01,400 and the set S to be the following graph, 72 00:05:01,400 --> 00:05:07,190 where V, the set of vertices, is just the set of group elements. 73 00:05:07,190 --> 00:05:16,240 And the edges are obtained by taking a group element 74 00:05:16,240 --> 00:05:20,340 and multiplying it by S to go to its neighbor. 75 00:05:26,480 --> 00:05:28,030 So this is a Cayley graph. 76 00:05:28,030 --> 00:05:30,670 And Cayley graphs are-- 77 00:05:30,670 --> 00:05:33,162 start with any group, start with any subset of the group, 78 00:05:33,162 --> 00:05:34,120 you get a Cayley graph. 79 00:05:34,120 --> 00:05:37,260 And this is a very important construction of graphs. 80 00:05:37,260 --> 00:05:39,630 They have lots of nice properties 81 00:05:39,630 --> 00:05:44,230 And, in particular, an example of a Cayley graph 82 00:05:44,230 --> 00:05:45,700 is a Paley graph. 83 00:05:49,830 --> 00:05:51,750 They're not related. 84 00:05:51,750 --> 00:05:54,650 So a Paley graph is a special case 85 00:05:54,650 --> 00:06:00,260 of a Cayley graph obtained by considering 86 00:06:00,260 --> 00:06:08,570 the group, the cyclic group mod p, where p is prime 1 mod 4. 87 00:06:12,110 --> 00:06:17,190 And I'm looking at S being the set of quadratic residues, mod 88 00:06:17,190 --> 00:06:17,690 p. 89 00:06:24,810 --> 00:06:28,580 It's actually nonzero quadratic residues. 90 00:06:28,580 --> 00:06:33,970 So elements of mod p that could be a square. 91 00:06:33,970 --> 00:06:38,130 So we will show in a second that this Paley graph has 92 00:06:38,130 --> 00:06:40,830 nice pseudorandom properties by showing 93 00:06:40,830 --> 00:06:45,030 that it is an n, d, lambda graph with lambda fairly small 94 00:06:45,030 --> 00:06:47,280 compared to the degree. 95 00:06:47,280 --> 00:06:49,800 Just a historical note-- 96 00:06:49,800 --> 00:06:55,240 so Raymond Paley-- so the Paley graph named after him-- 97 00:06:55,240 --> 00:06:59,380 he actually-- he was from the earlier part of 20th century. 98 00:06:59,380 --> 00:07:02,961 So from 1907 to 1932. 99 00:07:02,961 --> 00:07:05,210 So he died very young at the age of 26, 100 00:07:05,210 --> 00:07:07,220 and he actually died in an avalanche 101 00:07:07,220 --> 00:07:09,510 when he was skiing by himself in Banff. 102 00:07:09,510 --> 00:07:13,150 So Banff is a national park in Alberta in Canada. 103 00:07:13,150 --> 00:07:17,400 And when I was in Banff earlier this year for a math 104 00:07:17,400 --> 00:07:20,580 conference-- so there's also a math conference center there-- 105 00:07:20,580 --> 00:07:22,740 so I had a chance to go visit the-- 106 00:07:25,770 --> 00:07:27,000 Raymond Paley's tomb. 107 00:07:27,000 --> 00:07:33,596 So there's a graveyard there where you can find his tomb. 108 00:07:33,596 --> 00:07:37,920 And it's very sad that, in his short mathematical timespan, 109 00:07:37,920 --> 00:07:41,730 actually he managed to do a lot of amazing mathematical-- 110 00:07:41,730 --> 00:07:44,100 find a lot of amazing mathematical discoveries. 111 00:07:44,100 --> 00:07:46,890 And there are many important concepts named after him. 112 00:07:46,890 --> 00:07:50,760 So things like Paley-Wiener theorem, Paley-Zygmund, 113 00:07:50,760 --> 00:07:54,270 Littlewood-Paley, all this important ideas and analysis 114 00:07:54,270 --> 00:07:55,380 named after Paley. 115 00:07:55,380 --> 00:08:00,660 And Paley graph is also one of his contributions. 116 00:08:00,660 --> 00:08:06,390 So what we'll claim is that this Paley graph has the desired 117 00:08:06,390 --> 00:08:10,890 pseudorandom properties, in that if you look at its eigenvalues, 118 00:08:10,890 --> 00:08:21,120 then the top eigenvalues, except-- 119 00:08:21,120 --> 00:08:22,810 so except for the top eigenvalue, 120 00:08:22,810 --> 00:08:28,350 all the other eigenvalue are quite small. 121 00:08:28,350 --> 00:08:30,870 So keep in mind that the size of S 122 00:08:30,870 --> 00:08:33,020 is basically half of the group. 123 00:08:33,020 --> 00:08:34,770 So p minus 1 over 2. 124 00:08:34,770 --> 00:08:38,970 So for especially larger values of p, 125 00:08:38,970 --> 00:08:41,760 p's eigenvalues are quite small compared to the degree. 126 00:08:46,480 --> 00:08:48,780 So the main way to show that Cayley graphs 127 00:08:48,780 --> 00:08:54,090 like that have small eigenvalues is to just compute 128 00:08:54,090 --> 00:08:55,833 what the eigenvalues are. 129 00:08:55,833 --> 00:08:58,500 And this is actually not so hard to do for Cayley graphs, so let 130 00:08:58,500 --> 00:09:02,200 me do this explicitly. 131 00:09:02,200 --> 00:09:04,850 So I will tell you very explicitly a set 132 00:09:04,850 --> 00:09:08,220 of eigenvectors. 133 00:09:08,220 --> 00:09:10,690 And they are-- the first eigenvector 134 00:09:10,690 --> 00:09:13,540 is just the all 1's vector. 135 00:09:13,540 --> 00:09:15,850 The second eigenvector is the vector 136 00:09:15,850 --> 00:09:19,500 coming from 1, omega, omega squared, 137 00:09:19,500 --> 00:09:22,300 so omega to the p minus 1, where omega 138 00:09:22,300 --> 00:09:24,905 is a parameter of p-th root of unity. 139 00:09:29,760 --> 00:09:35,305 The next one is 1, omega square, omega fourth, 140 00:09:35,305 --> 00:09:38,090 all the way to omega p-- 141 00:09:38,090 --> 00:09:40,890 omega to the 2 times p minus 1. 142 00:09:40,890 --> 00:09:42,850 And so on. 143 00:09:42,850 --> 00:09:45,870 So I want to have-- 144 00:09:45,870 --> 00:09:46,440 yes, so OK. 145 00:09:46,440 --> 00:09:50,510 So I make this list, and I have p of them. 146 00:09:59,510 --> 00:10:01,230 So these are my eigenvectors. 147 00:10:01,230 --> 00:10:04,660 And let me check that they are actually eigenvectors. 148 00:10:04,660 --> 00:10:07,150 And then we can also compute their eigenvalues. 149 00:10:07,150 --> 00:10:12,580 So the top eigenvector corresponds to d. 150 00:10:12,580 --> 00:10:14,360 So the all 1's in a d regular graph 151 00:10:14,360 --> 00:10:17,180 is always an eigenvector with eigenvalue d. 152 00:10:17,180 --> 00:10:19,617 And the other ones, we'll just do this computation. 153 00:10:22,860 --> 00:10:25,570 So instead of getting confused with indices, 154 00:10:25,570 --> 00:10:28,000 let me just compute, as an example, 155 00:10:28,000 --> 00:10:37,712 the j-th coordinate of the adjacency matrix times V2. 156 00:10:37,712 --> 00:10:42,360 So the j-th coordinate, so what it comes to, 157 00:10:42,360 --> 00:10:45,630 is the following sum. 158 00:10:45,630 --> 00:10:52,110 If I run over S, then omega raised to j plus s. 159 00:10:52,110 --> 00:10:56,090 So S is symmetric, so I don't have to worry so much about 160 00:10:56,090 --> 00:10:57,080 plus or minus. 161 00:10:57,080 --> 00:10:58,878 So I say j plus s. 162 00:10:58,878 --> 00:11:00,920 So if you think about what this Cayley graph, how 163 00:11:00,920 --> 00:11:05,000 it is defined, if you hit this vector with that matrix, 164 00:11:05,000 --> 00:11:09,370 the j-th coordinate is that sum there. 165 00:11:09,370 --> 00:11:15,690 But I can rewrite the sum by taking out this common factor 166 00:11:15,690 --> 00:11:16,700 omega to j. 167 00:11:22,640 --> 00:11:28,890 And you see that this is the j-th coordinate of V2. 168 00:11:33,790 --> 00:11:34,970 And this is true for all j. 169 00:11:34,970 --> 00:11:38,030 So this number here is lambda 2. 170 00:11:44,640 --> 00:11:55,410 And, more generally, lambda k is the following sum, 171 00:11:55,410 --> 00:11:59,340 for k from 0-- 172 00:12:04,010 --> 00:12:08,768 so from k being 1 through p. 173 00:12:08,768 --> 00:12:11,580 So when you plug in k equals to 1, you just get d. 174 00:12:11,580 --> 00:12:17,450 And the others are sums of these exponential sums. 175 00:12:17,450 --> 00:12:19,610 Now, this is a pretty straightforward computation. 176 00:12:19,610 --> 00:12:21,193 And, in fact, we're not using anything 177 00:12:21,193 --> 00:12:22,440 about quadratic residues. 178 00:12:22,440 --> 00:12:29,076 This is a generic fact about Cayley graphs of z mod p. 179 00:12:29,076 --> 00:12:35,010 So this is true for all Cayley graphs S, not necessarily 180 00:12:35,010 --> 00:12:38,000 for quadratic residues. 181 00:12:38,000 --> 00:12:40,350 And the basic reason is that, here, you 182 00:12:40,350 --> 00:12:45,990 have this set of eigenvectors, and they do not depend on S. 183 00:12:45,990 --> 00:12:48,240 So you might know this concept from other places, 184 00:12:48,240 --> 00:12:50,880 such as circular matrices and whatnot, 185 00:12:50,880 --> 00:12:55,430 but this is true in this simple computation. 186 00:12:55,430 --> 00:13:02,110 So now we have the values of lambda explicitly. 187 00:13:02,110 --> 00:13:04,870 I can now compute their sizes. 188 00:13:04,870 --> 00:13:07,780 I want to know how big this lambda is. 189 00:13:07,780 --> 00:13:09,830 Well, the first one, when k equals to 1, 190 00:13:09,830 --> 00:13:15,720 it's exactly d, the degree, which is p minus 1 over 2. 191 00:13:15,720 --> 00:13:18,420 But what about the other ones? 192 00:13:18,420 --> 00:13:22,350 So, for the other ones, we can do a computation as follows. 193 00:13:22,350 --> 00:13:26,570 So note that I can rewrite lambda k 194 00:13:26,570 --> 00:13:30,320 by noting that if I take twice it 195 00:13:30,320 --> 00:13:34,550 and plus 1, then I obtain the following sum. 196 00:13:41,500 --> 00:13:45,550 Because here I am using the S as a set of quadratic residues. 197 00:13:45,550 --> 00:13:48,750 So if I consider this sum here, every quadratic residue 198 00:13:48,750 --> 00:13:52,050 gets counted twice, except for 0, which gets counted once. 199 00:13:55,110 --> 00:13:59,140 And now I would like to evaluate the size of this sum, 200 00:13:59,140 --> 00:14:00,850 this exponential sum. 201 00:14:00,850 --> 00:14:03,610 And this is something that's known as a Gauss sum. 202 00:14:07,940 --> 00:14:09,470 So, basically, a Gauss sum is what 203 00:14:09,470 --> 00:14:11,525 happens when you have something that's 204 00:14:11,525 --> 00:14:15,950 like a quadratic, an exponential sum with a quadratic dependence 205 00:14:15,950 --> 00:14:17,930 in the exponent. 206 00:14:17,930 --> 00:14:22,340 And the trick here is to consider the square of the sum. 207 00:14:26,540 --> 00:14:29,080 So the magnitude squared. 208 00:14:29,080 --> 00:14:32,480 Now if I expand the square-- 209 00:14:32,480 --> 00:14:36,930 so squaring is a common feature of many of the things 210 00:14:36,930 --> 00:14:38,370 we do in this course. 211 00:14:38,370 --> 00:14:40,900 It really simplifies your life. 212 00:14:40,900 --> 00:14:43,140 You do the square, you expand the sum. 213 00:14:43,140 --> 00:14:54,258 You can re-parameterize one of the summands like that. 214 00:14:54,258 --> 00:14:57,850 So do two steps at once. 215 00:14:57,850 --> 00:15:01,390 I'm re-parameterizing and I'm expanding. 216 00:15:01,390 --> 00:15:10,630 But now you see, if I expand the exponent, we find-- 217 00:15:16,520 --> 00:15:18,350 so that's just algebra. 218 00:15:18,350 --> 00:15:25,960 And now you notice that this sum here, the sum over a 219 00:15:25,960 --> 00:15:29,680 is equal to-- 220 00:15:29,680 --> 00:15:36,850 when b is nonzero, I claim that this sum is 0. 221 00:15:36,850 --> 00:15:41,030 And when b is nonzero, then I'm summing over some permutations 222 00:15:41,030 --> 00:15:43,910 of the roots of unity. 223 00:15:43,910 --> 00:15:49,150 So here I'm assuming that k is bigger than-- 224 00:15:49,150 --> 00:15:52,080 let's say here k is not 0. 225 00:15:52,080 --> 00:15:54,445 So I'm re-parameterizing k a little bit. 226 00:15:54,445 --> 00:15:56,020 So k is not 0. 227 00:15:56,020 --> 00:15:59,740 Then when b is not 0, the sum over a is 0. 228 00:15:59,740 --> 00:16:01,720 And otherwise it equals to p. 229 00:16:01,720 --> 00:16:05,880 So the sum over here equals to p. 230 00:16:09,090 --> 00:16:13,671 And, therefore, lambda k, lambda sub k-- 231 00:16:17,520 --> 00:16:20,270 how about if I-- 232 00:16:20,270 --> 00:16:22,980 so what should I change that to? 233 00:16:22,980 --> 00:16:30,480 So if I-- k is 0, then I want this to be lambda sub k plus 1. 234 00:16:30,480 --> 00:16:38,350 Then lambda sub k plus 1 is equal to plus/minus 235 00:16:38,350 --> 00:16:45,470 p plus 1 over 2, for all lambda not equal to 0. 236 00:16:51,800 --> 00:16:54,230 So, really, except for the top eigenvalue, 237 00:16:54,230 --> 00:16:56,570 which is just the degree, all the other ones 238 00:16:56,570 --> 00:17:01,140 are one of these two values, and they're all quite small. 239 00:17:01,140 --> 00:17:03,230 So this is an explicit computation showing you 240 00:17:03,230 --> 00:17:06,829 that this Paley graph is indeed a pseudorandom graph. 241 00:17:06,829 --> 00:17:08,599 It's an example of a quasi-random graph. 242 00:17:08,599 --> 00:17:10,198 Yes. 243 00:17:10,198 --> 00:17:12,875 AUDIENCE: Do we know what the sign is? 244 00:17:12,875 --> 00:17:15,250 YUFEI ZHAO: The question is, do we know what the sign is? 245 00:17:15,250 --> 00:17:17,990 So we actually-- so here I am not 246 00:17:17,990 --> 00:17:21,398 telling you what the sign is, but you can look up. 247 00:17:21,398 --> 00:17:23,190 Actually, people have computed exactly what 248 00:17:23,190 --> 00:17:24,750 the sign should be. 249 00:17:24,750 --> 00:17:28,000 And this is something that you can find in a number theory 250 00:17:28,000 --> 00:17:29,530 textbook, like Aaron and Rosen. 251 00:17:33,218 --> 00:17:35,810 Any more questions? 252 00:17:35,810 --> 00:17:38,270 There is a concept here I just want 253 00:17:38,270 --> 00:17:43,110 to bring out, that you might recognize sums like this. 254 00:17:43,110 --> 00:17:44,150 So this kind of sum. 255 00:17:46,830 --> 00:17:50,930 That's a Fourier coefficient. 256 00:17:50,930 --> 00:17:53,320 So if you have some Fourier transform, I mean, 257 00:17:53,320 --> 00:17:56,890 this is exactly what Fourier transforms look like. 258 00:17:56,890 --> 00:18:02,150 And it is indeed the case that, in general, 259 00:18:02,150 --> 00:18:13,560 if you have an Abelian group, then the eigenvalues 260 00:18:13,560 --> 00:18:22,550 and the spectral information of the corresponding Cayley graph 261 00:18:22,550 --> 00:18:25,537 corresponds to Fourier coefficients. 262 00:18:29,020 --> 00:18:32,080 And this is the connection that we'll see also 263 00:18:32,080 --> 00:18:35,710 later on in the course when we consider additive combinatorics 264 00:18:35,710 --> 00:18:38,470 and giving a Fourier analytic proof of Roth's theorem. 265 00:18:38,470 --> 00:18:42,820 And there Fourier analysis will play a central role. 266 00:18:42,820 --> 00:18:45,800 But this is actually-- this analogy, as I've written it, 267 00:18:45,800 --> 00:18:47,360 is only for Abelian groups. 268 00:18:47,360 --> 00:18:50,480 If you try to do the same for non-Abelian groups, 269 00:18:50,480 --> 00:18:55,530 you will get something somewhat different. 270 00:18:55,530 --> 00:18:58,790 So for non-Abelian groups, you do not 271 00:18:58,790 --> 00:19:01,790 have this nice notion of Fourier analysis, 272 00:19:01,790 --> 00:19:05,450 at least in the versions that generalizes what's 273 00:19:05,450 --> 00:19:07,450 above in a straightforward way. 274 00:19:07,450 --> 00:19:10,680 But, instead, you have something else, which many of you 275 00:19:10,680 --> 00:19:12,530 have seen before but under a different name. 276 00:19:12,530 --> 00:19:19,500 And that's representation theory, which, in some sense, 277 00:19:19,500 --> 00:19:21,540 is Fourier analysis, except, instead 278 00:19:21,540 --> 00:19:24,900 of one-dimensional objects and complex numbers, 279 00:19:24,900 --> 00:19:26,660 we're looking at higher-dimensional 280 00:19:26,660 --> 00:19:28,250 representations. 281 00:19:28,250 --> 00:19:31,120 So I just want to point out this connection, 282 00:19:31,120 --> 00:19:35,940 and we'll see more of it later on. 283 00:19:35,940 --> 00:19:36,980 Any questions? 284 00:19:41,780 --> 00:19:44,390 So let's talk more about Cayley graphs. 285 00:19:44,390 --> 00:19:46,970 So, last time, we mentioned these notions 286 00:19:46,970 --> 00:19:49,820 of quasi-randomness. 287 00:19:49,820 --> 00:19:51,980 And I said at the end of the class 288 00:19:51,980 --> 00:19:55,280 that many of these equivalences between quasi-random graphs, 289 00:19:55,280 --> 00:19:59,090 they fail for sparse graphs. 290 00:19:59,090 --> 00:20:02,220 If your density, if your x density is a constant, 291 00:20:02,220 --> 00:20:06,100 then the equivalences no longer hold. 292 00:20:06,100 --> 00:20:09,560 But what about for Cayley graphs? 293 00:20:09,560 --> 00:20:13,790 And, in particular, I would like to consider 294 00:20:13,790 --> 00:20:16,880 two specific notions that we discussed last time 295 00:20:16,880 --> 00:20:20,750 and try to understand how they relate to each other for Cayley 296 00:20:20,750 --> 00:20:21,250 graphs. 297 00:20:30,052 --> 00:20:33,848 So for dense Cayley graphs, it's a special case 298 00:20:33,848 --> 00:20:34,890 of what we did yesterday. 299 00:20:34,890 --> 00:20:37,430 So I'm really interested in sparser Cayley graphs, 300 00:20:37,430 --> 00:20:39,306 even down the degree. 301 00:20:39,306 --> 00:20:40,687 So even down the degree. 302 00:20:40,687 --> 00:20:42,270 So that's much sparser than the regime 303 00:20:42,270 --> 00:20:44,010 we were looking at last time. 304 00:20:44,010 --> 00:20:47,820 And the main result I want to tell you 305 00:20:47,820 --> 00:20:52,440 is that the DISC condition is, in a very strong sense, 306 00:20:52,440 --> 00:20:56,610 actually equivalent to the eigenvalue condition 307 00:20:56,610 --> 00:21:10,460 for all Cayley graphs, including non-Abelian Cayley graphs. 308 00:21:13,687 --> 00:21:15,520 So before telling you what the statement is, 309 00:21:15,520 --> 00:21:18,370 I first want to give an example showing you 310 00:21:18,370 --> 00:21:21,310 that this equivalence is definitely not true 311 00:21:21,310 --> 00:21:24,850 if you remove the assumption of Cayley graphs. 312 00:21:24,850 --> 00:21:27,830 For example, if you-- 313 00:21:27,830 --> 00:21:32,334 so example that this is false for non-Cayley. 314 00:21:37,080 --> 00:21:40,950 Because if you take, let's say, a large-- 315 00:21:40,950 --> 00:21:42,710 so let's say d regular graph. 316 00:21:42,710 --> 00:21:46,260 So let's say a large random d regular graph. 317 00:21:50,164 --> 00:21:53,390 d here can be a constant or growing with n, 318 00:21:53,390 --> 00:21:55,670 but this is a pretty robust example. 319 00:21:55,670 --> 00:22:00,842 And then I add to it an extra destroying copy of k sub d 320 00:22:00,842 --> 00:22:04,940 plus 1 that's much smaller in terms of number of vertices. 321 00:22:07,500 --> 00:22:09,360 The big, large random graph, well, 322 00:22:09,360 --> 00:22:11,370 by virtue of being a random graph, 323 00:22:11,370 --> 00:22:12,930 has the discrepancy property. 324 00:22:18,610 --> 00:22:22,320 And because we're only adding in a very small number 325 00:22:22,320 --> 00:22:25,710 of vertices, it does not destroy the discrepancy property. 326 00:22:25,710 --> 00:22:27,570 The discrepancy property, if you're just 327 00:22:27,570 --> 00:22:29,430 adding a small number of vertices, 328 00:22:29,430 --> 00:22:30,840 it doesn't change much. 329 00:22:30,840 --> 00:22:33,784 So this whole thing has discrepancy. 330 00:22:36,390 --> 00:22:38,190 However, what about the eigenvalues? 331 00:22:41,030 --> 00:22:43,040 Claim that the top two eigenvalues 332 00:22:43,040 --> 00:22:45,530 are in fact both equal to d. 333 00:22:45,530 --> 00:22:49,310 And that's because you have two eigenvectors, one which 334 00:22:49,310 --> 00:22:52,580 is the all 1's vector on this graph, another 335 00:22:52,580 --> 00:22:55,120 which is the all 1's vector on that graph. 336 00:22:55,120 --> 00:22:58,100 These two DISC components each give you a top eigenvector 337 00:22:58,100 --> 00:23:00,920 of d, so you get d twice. 338 00:23:00,920 --> 00:23:03,920 And, in particular, the second eigenvalue is not small. 339 00:23:07,200 --> 00:23:11,720 So the implication from DISC to eigenvalue really 340 00:23:11,720 --> 00:23:16,380 fails for non-Cayley graph for general graphs. 341 00:23:19,110 --> 00:23:21,890 The implication, the other direction is actually OK. 342 00:23:26,040 --> 00:23:30,240 In fact, the eigenvalue implies DISC is actually the content 343 00:23:30,240 --> 00:23:33,020 of the expander mixing lemma. 344 00:23:33,020 --> 00:23:40,920 So this follows by expander mixing lemma. 345 00:23:40,920 --> 00:23:43,470 And that's because, if you look at the expander mixing 346 00:23:43,470 --> 00:23:47,060 lemma for a Cayley graph-- 347 00:23:47,060 --> 00:23:50,700 or for a-- not for Cayley graph-- for-- 348 00:23:50,700 --> 00:23:52,890 if you have the eigenvalue condition, 349 00:23:52,890 --> 00:23:58,320 then, automatically, you would find that these two guys here 350 00:23:58,320 --> 00:23:59,380 are at most n. 351 00:24:02,520 --> 00:24:10,570 So if lambda is quite small compared to the degree, 352 00:24:10,570 --> 00:24:15,952 then you still have the desired type of quasi-randomness. 353 00:24:15,952 --> 00:24:18,160 So I'll make the statements more precise in a second. 354 00:24:21,420 --> 00:24:25,060 So the question is, how can we certify, how can we 355 00:24:25,060 --> 00:24:29,920 show that, in fact, DISC, which is seemingly weaker property, 356 00:24:29,920 --> 00:24:33,733 implies a stronger property of eigenvalue for Cayley graphs. 357 00:24:33,733 --> 00:24:35,650 And what is a special about Cayley graphs that 358 00:24:35,650 --> 00:24:38,350 would allow to do this, that the statement is generally 359 00:24:38,350 --> 00:24:40,758 false for non-Cayley graphs? 360 00:24:47,330 --> 00:24:49,430 So let me define-- 361 00:24:49,430 --> 00:24:53,290 so let me first tell you the result. 362 00:24:53,290 --> 00:24:56,015 So this is the result due to David Conlon 363 00:24:56,015 --> 00:24:57,920 and myself two years ago. 364 00:25:00,570 --> 00:25:05,200 So many of you may not have been to many seminar talks, where 365 00:25:05,200 --> 00:25:08,160 there's this convention in mathematics talks 366 00:25:08,160 --> 00:25:12,090 where you don't write out your full name, only by the initial. 367 00:25:12,090 --> 00:25:13,740 Although some kind of false modesty. 368 00:25:13,740 --> 00:25:17,280 But, of course, we all love talking about our own results, 369 00:25:17,280 --> 00:25:19,140 but somehow we don't like to write 370 00:25:19,140 --> 00:25:21,420 our own name for some reason. 371 00:25:21,420 --> 00:25:23,850 So here's the theorem. 372 00:25:23,850 --> 00:25:29,700 So I start with a finite group, gamma. 373 00:25:29,700 --> 00:25:34,168 And let me consider a subset S of gamma that is symmetric. 374 00:25:37,040 --> 00:25:39,320 And consider G the Cayley graph. 375 00:25:44,220 --> 00:25:49,060 Let me right n as the number of vertices, and d the size of S. 376 00:25:49,060 --> 00:25:52,370 So this is a d regular graph. 377 00:25:52,370 --> 00:25:54,662 Let me define the following properties. 378 00:26:01,550 --> 00:26:05,680 The first property, I'll call DISC with epsilon. 379 00:26:05,680 --> 00:26:07,930 So I give you an explicit parameter. 380 00:26:12,290 --> 00:26:14,960 The number of edges between x and y 381 00:26:14,960 --> 00:26:19,070 differs from the number of edges that you would expect. 382 00:26:19,070 --> 00:26:21,980 So as in the expander mixing lemma. 383 00:26:21,980 --> 00:26:25,490 So the DISC property is that this quantity is small relative 384 00:26:25,490 --> 00:26:26,980 to the total number of edges. 385 00:26:30,670 --> 00:26:35,710 The second property, which we'll call the eigenvalue property, 386 00:26:35,710 --> 00:26:50,110 EIG, is that G is an n, d, lambda graph, with lambda, 387 00:26:50,110 --> 00:26:52,796 at most, epsilon d. 388 00:26:52,796 --> 00:26:56,338 So lambda is quite small as a function of d. 389 00:26:59,326 --> 00:27:03,250 The conclusion of the theorem is that, up 390 00:27:03,250 --> 00:27:07,560 to a small change of parameters, these two properties 391 00:27:07,560 --> 00:27:08,790 are equivalent. 392 00:27:08,790 --> 00:27:13,800 In particular, eigenvalue implies a-- 393 00:27:13,800 --> 00:27:19,350 epsilon implies DISC of epsilon. 394 00:27:19,350 --> 00:27:23,430 And DISC of epsilon-- and this is 395 00:27:23,430 --> 00:27:26,160 the-- the second one is the more interesting direction-- 396 00:27:26,160 --> 00:27:29,550 it implies EIG. 397 00:27:29,550 --> 00:27:31,180 Well, you lose a little bit, but, 398 00:27:31,180 --> 00:27:33,200 at most, a constant factor. 399 00:27:33,200 --> 00:27:34,530 EIG of 8 epsilon. 400 00:27:40,680 --> 00:27:43,435 Any questions about the statement so far? 401 00:27:43,435 --> 00:27:46,070 And so, as I mentioned, this is completely false 402 00:27:46,070 --> 00:27:52,490 if you consider non-Cayley graphs. 403 00:27:52,490 --> 00:27:56,000 And we also, using expander mixing lemma, 404 00:27:56,000 --> 00:28:00,130 using that implication up there, this direction follows. 405 00:28:11,707 --> 00:28:13,290 One of the main reasons I want to show 406 00:28:13,290 --> 00:28:15,060 you a proof of this theorem is that it 407 00:28:15,060 --> 00:28:17,870 uses this tool which I think is worth knowing. 408 00:28:17,870 --> 00:28:21,030 And this is an important inequality known 409 00:28:21,030 --> 00:28:22,898 as Grothendieck's inequality. 410 00:28:35,460 --> 00:28:37,530 So many of you probably know Grothendieck 411 00:28:37,530 --> 00:28:41,100 as this famous French mathematician who 412 00:28:41,100 --> 00:28:44,040 reinvented modern algebraic geometry 413 00:28:44,040 --> 00:28:45,678 and spent the rest of his life writing 414 00:28:45,678 --> 00:28:47,220 tomes and tomes of text that have yet 415 00:28:47,220 --> 00:28:48,387 to be translated to English. 416 00:28:51,110 --> 00:28:53,600 But he also did some important foundational work 417 00:28:53,600 --> 00:28:56,480 in functional analysis before he became 418 00:28:56,480 --> 00:28:59,690 an algebraic geometry nerd. 419 00:28:59,690 --> 00:29:03,540 And this is one of the important results in that area that he-- 420 00:29:06,710 --> 00:29:08,580 so Grothendieck's inequality tells us 421 00:29:08,580 --> 00:29:13,140 that there exists some absolute constant k 422 00:29:13,140 --> 00:29:22,855 such that for every matrix A-- 423 00:29:22,855 --> 00:29:24,600 so a real-valued matrix-- 424 00:29:31,100 --> 00:29:33,020 we have that the-- 425 00:29:40,040 --> 00:29:41,330 so we have that, if you-- 426 00:29:41,330 --> 00:29:42,810 so here's the idea. 427 00:29:42,810 --> 00:29:46,070 Let's consider the supremum-- 428 00:29:46,070 --> 00:29:50,410 so let's consider the following quantity. 429 00:29:50,410 --> 00:29:51,530 This is a bilinear form. 430 00:29:55,310 --> 00:29:57,970 So this is a bilinear form. 431 00:29:57,970 --> 00:29:59,950 This is basically a-- 432 00:30:03,740 --> 00:30:07,770 so bilinear form, if you hit it by a vector x and y 433 00:30:07,770 --> 00:30:09,450 from the two sides. 434 00:30:09,450 --> 00:30:13,080 And I'm interested in what is the maximum value 435 00:30:13,080 --> 00:30:16,110 of this bilinear form if you are allowed 436 00:30:16,110 --> 00:30:23,480 to take x and y to be plus/minus 1-valued real numbers? 437 00:30:30,720 --> 00:30:32,920 So this is an important quantity, 438 00:30:32,920 --> 00:30:34,870 and it gives you a matrix. 439 00:30:34,870 --> 00:30:38,230 And it's basically asking you, you get a sign of plus or minus 440 00:30:38,230 --> 00:30:40,240 to each row and column, and I want 441 00:30:40,240 --> 00:30:42,323 to maximize this number here. 442 00:30:42,323 --> 00:30:43,990 This is an important quantity that we'll 443 00:30:43,990 --> 00:30:47,680 see actually much more in the next chapter on graph limits. 444 00:30:47,680 --> 00:30:49,420 But, for now, just take my word. 445 00:30:49,420 --> 00:30:51,580 This is a very important quantity. 446 00:30:51,580 --> 00:30:53,080 And this is actually a quantity that 447 00:30:53,080 --> 00:30:54,820 is very difficult to evaluate. 448 00:30:54,820 --> 00:30:56,800 If I give you a very large matrix 449 00:30:56,800 --> 00:30:59,710 and ask you to compute this number here, 450 00:30:59,710 --> 00:31:01,810 there is no good algorithm for it. 451 00:31:01,810 --> 00:31:05,710 And it's believed that there is no good algorithm for it. 452 00:31:05,710 --> 00:31:09,370 On the other hand, there is a relaxation 453 00:31:09,370 --> 00:31:13,130 of this problem, which is the following. 454 00:31:13,130 --> 00:31:19,430 It's still a sum, but now, instead of considering 455 00:31:19,430 --> 00:31:25,190 the bilinear form there, let's consider the xi's and yi's. 456 00:31:25,190 --> 00:31:32,600 Not-- take them not form real numbers, but take vectors. 457 00:31:32,600 --> 00:31:39,680 So let's consider the sum where I'm 458 00:31:39,680 --> 00:31:47,330 taking a similar-looking sum, except that xi's and yi's come 459 00:31:47,330 --> 00:31:51,770 from a unit ball in some vector space 460 00:31:51,770 --> 00:31:57,920 with an inner product, where B is the unit 461 00:31:57,920 --> 00:32:07,395 ball in some Rm, where here the dimension is actually not 462 00:32:07,395 --> 00:32:07,895 so relevant. 463 00:32:07,895 --> 00:32:10,470 The dimension is arbitrary. 464 00:32:10,470 --> 00:32:15,620 If you like, you can make m n or 2n because you only 465 00:32:15,620 --> 00:32:19,050 have that many vectors. 466 00:32:19,050 --> 00:32:22,160 So this quantity here, just by very definition, 467 00:32:22,160 --> 00:32:25,010 is a relaxation of the right-hand of this quantity 468 00:32:25,010 --> 00:32:25,510 here. 469 00:32:25,510 --> 00:32:27,620 So it's at least this large. 470 00:32:27,620 --> 00:32:30,650 So, in particular, if you have whatever plus/minus, 471 00:32:30,650 --> 00:32:34,160 you can always look at the same quantity with m equal to 1, 472 00:32:34,160 --> 00:32:35,660 and you obtain this quantity here. 473 00:32:35,660 --> 00:32:38,310 But this quantity may be substantially larger. 474 00:32:38,310 --> 00:32:43,880 So the x and y's have more room to put themselves in 475 00:32:43,880 --> 00:32:45,220 to maximize the sum. 476 00:32:48,330 --> 00:32:52,650 And Grothendieck's inequality tells us that the left-hand 477 00:32:52,650 --> 00:32:57,210 side actually cannot be too much larger than the right-hand 478 00:32:57,210 --> 00:32:58,110 side. 479 00:32:58,110 --> 00:33:02,650 It exceeds it by, at most, a constant factor. 480 00:33:02,650 --> 00:33:04,870 So, in other words, the left-hand side, 481 00:33:04,870 --> 00:33:15,060 which is known as a semi-definite relaxation, 482 00:33:15,060 --> 00:33:17,910 you are not losing by more than a constant factor 483 00:33:17,910 --> 00:33:20,730 compared to the original problem. 484 00:33:20,730 --> 00:33:22,500 And this is important in computer science 485 00:33:22,500 --> 00:33:24,210 because the left-hand side turns out 486 00:33:24,210 --> 00:33:27,960 to be a Semidefinite Program, an SDP, which 487 00:33:27,960 --> 00:33:32,740 does have efficient algorithms to compute. 488 00:33:32,740 --> 00:33:35,610 So you can give a constant factor approximation 489 00:33:35,610 --> 00:33:38,430 to this difficult compute but important quantity 490 00:33:38,430 --> 00:33:41,030 by using semidefinite relaxation. 491 00:33:41,030 --> 00:33:43,020 And Grothendieck's inequality promises us 492 00:33:43,020 --> 00:33:44,540 that it is a good relaxation. 493 00:33:50,560 --> 00:33:53,050 You might ask, what is the value of k? 494 00:33:53,050 --> 00:33:55,790 So I said there exists some constant k. 495 00:33:55,790 --> 00:33:58,720 So this is actually a mystery. 496 00:33:58,720 --> 00:34:04,000 So the current proofs have been improved over time. 497 00:34:04,000 --> 00:34:06,340 And Grothendieck himself proved this theorem, 498 00:34:06,340 --> 00:34:08,889 but it constantly has been improved over time. 499 00:34:08,889 --> 00:34:11,469 And, currently, the best-known result 500 00:34:11,469 --> 00:34:20,604 is something along the lines of k roughly 1.78 works. 501 00:34:20,604 --> 00:34:25,719 But the optimal value, which is known as Grothendieck's 502 00:34:25,719 --> 00:34:33,310 constant, is unknown. 503 00:34:40,600 --> 00:34:42,317 So this is Grothendieck's constant. 504 00:34:42,317 --> 00:34:43,900 Actually, this, what I've written down 505 00:34:43,900 --> 00:34:47,949 is what's called the real Grothendieck's constant. 506 00:34:47,949 --> 00:34:49,780 Because you can also write a version 507 00:34:49,780 --> 00:34:52,810 for complex numbers and complex vectors, 508 00:34:52,810 --> 00:34:55,130 and that's the complex Grothendieck's constant. 509 00:34:55,130 --> 00:34:55,630 Yes. 510 00:34:55,630 --> 00:34:57,518 AUDIENCE: Is there a lower bound that's 511 00:34:57,518 --> 00:34:59,305 known [INAUDIBLE] greater than 1? 512 00:34:59,305 --> 00:35:00,790 YUFEI ZHAO: Is there a lower bound that is known? 513 00:35:00,790 --> 00:35:01,290 Yes. 514 00:35:01,290 --> 00:35:03,430 It's known that it's strictly bigger than 1. 515 00:35:03,430 --> 00:35:04,922 AUDIENCE: Do we know [INAUDIBLE]?? 516 00:35:04,922 --> 00:35:06,880 YUFEI ZHAO: So there are some specific numbers, 517 00:35:06,880 --> 00:35:07,840 but I forget what they are. 518 00:35:07,840 --> 00:35:08,632 You can look it up. 519 00:35:12,350 --> 00:35:13,500 Any more questions? 520 00:35:16,080 --> 00:35:18,790 So we'll leave Grothendieck's inequality. 521 00:35:18,790 --> 00:35:20,650 We'll use it as a black box. 522 00:35:20,650 --> 00:35:22,440 So if you wish to learn the proof, 523 00:35:22,440 --> 00:35:23,810 I encourage you to do so. 524 00:35:23,810 --> 00:35:26,700 There are some quite nice proofs out there. 525 00:35:26,700 --> 00:35:29,520 And we'll use it to prove this theorem here 526 00:35:29,520 --> 00:35:31,360 about quasi-random Cayley graphs. 527 00:35:44,890 --> 00:35:49,040 So let's suppose DISC holds. 528 00:35:55,880 --> 00:35:58,160 So what would we like to-- what do we like to show? 529 00:35:58,160 --> 00:36:04,910 We want to show that this eigenvalue condition holds. 530 00:36:04,910 --> 00:36:07,570 And we'll use the-- 531 00:36:07,570 --> 00:36:11,240 some min-max characterization of eigenvalues. 532 00:36:11,240 --> 00:36:13,400 But, first, some preliminaries. 533 00:36:13,400 --> 00:36:19,700 Suppose you have vectors x and y which have plus/minus 1 534 00:36:19,700 --> 00:36:21,065 coordinate values. 535 00:36:24,270 --> 00:36:34,810 Then, by letting-- so let's consider the following vectors, 536 00:36:34,810 --> 00:36:37,820 where I split up x and y according 537 00:36:37,820 --> 00:36:40,503 to where they're positive and where they're negative. 538 00:36:46,070 --> 00:36:50,740 So, here, these are such that x plus is equal to-- 539 00:36:50,740 --> 00:36:55,396 so if I evaluate it on a coordinate g, then it's 1. 540 00:36:55,396 --> 00:37:04,861 So if x sub g is plus 1, and 0 otherwise. 541 00:37:04,861 --> 00:37:12,720 xg sub minus is 1 if x sub g is minus 1. 542 00:37:12,720 --> 00:37:14,150 0 otherwise. 543 00:37:14,150 --> 00:37:19,360 So x splits into x plus minus x minus, and y splits 544 00:37:19,360 --> 00:37:22,450 into y plus minus y minus. 545 00:37:25,710 --> 00:37:39,310 Let's consider a matrix A where the g comma h entry of A 546 00:37:39,310 --> 00:37:41,970 is the following quantity. 547 00:37:41,970 --> 00:37:49,090 I have the set S, and I look at whether g inverse h lies in S. 548 00:37:49,090 --> 00:37:50,960 And I can consider an indicator of that. 549 00:37:50,960 --> 00:37:52,880 So it's 1 or 0. 550 00:37:52,880 --> 00:38:01,010 And then subtract d over n so that this value has mean 0. 551 00:38:01,010 --> 00:38:03,950 So this is a matrix. 552 00:38:03,950 --> 00:38:10,720 And now if I consider the bilinear form, 553 00:38:10,720 --> 00:38:14,680 hit A from left and right with x and y, 554 00:38:14,680 --> 00:38:22,330 then the bilinear form splits according to the plus 555 00:38:22,330 --> 00:38:24,835 and minuses of the x's. 556 00:38:39,230 --> 00:38:41,330 And I claim that each one of these terms 557 00:38:41,330 --> 00:38:42,950 is controlled because of DISC. 558 00:38:45,674 --> 00:38:54,010 So, for example, the first term is, 559 00:38:54,010 --> 00:38:55,820 if you expand out what this guy is-- 560 00:38:55,820 --> 00:38:57,070 so here's an indicator vector. 561 00:38:57,070 --> 00:38:58,480 That's an indicator vector. 562 00:38:58,480 --> 00:39:00,400 And if you look at the definition, 563 00:39:00,400 --> 00:39:05,830 then this is precisely the number of edges between x plus 564 00:39:05,830 --> 00:39:10,960 and y plus minus d over n times the size 565 00:39:10,960 --> 00:39:14,410 of x plus times the size of y plus, 566 00:39:14,410 --> 00:39:21,140 where x plus is the set of group elements 567 00:39:21,140 --> 00:39:26,236 such that x sub g is 1, and so on. 568 00:39:34,668 --> 00:39:35,660 All right. 569 00:39:43,110 --> 00:39:47,830 So the punchline up there is that this quantity-- 570 00:39:47,830 --> 00:40:01,520 so this quantity is, at most, by discrepancy, epsilon dn. 571 00:40:05,020 --> 00:40:08,410 So this sum here, by triangle inequality, 572 00:40:08,410 --> 00:40:12,920 is, at most, 4 epsilon dn. 573 00:40:53,580 --> 00:40:56,548 All right. 574 00:40:56,548 --> 00:40:59,920 So, so far, we've reinterpreted the discrepancy property. 575 00:40:59,920 --> 00:41:02,460 And what we really want to show is that this graph 576 00:41:02,460 --> 00:41:05,255 satisfies eigenvalue condition. 577 00:41:05,255 --> 00:41:07,005 So what does that actually mean to satisfy 578 00:41:07,005 --> 00:41:08,510 the eigenvalue condition? 579 00:41:08,510 --> 00:41:11,440 So by the min-max characterization 580 00:41:11,440 --> 00:41:21,670 of eigenvalues, it follows that the maximum of these two 581 00:41:21,670 --> 00:41:23,440 eigenvalues, which is the quantity that we 582 00:41:23,440 --> 00:41:33,010 would like to control, is equal to the following. 583 00:41:33,010 --> 00:41:41,400 It is equal to the supremum of this bilinear form 584 00:41:41,400 --> 00:41:46,770 when x and y are unit-length vectors. 585 00:41:53,240 --> 00:41:56,900 And this is simply because A is the matrix-- 586 00:41:56,900 --> 00:41:59,390 it's not the adjacency matrix. 587 00:41:59,390 --> 00:42:01,570 A is not the adjacency matrix. 588 00:42:01,570 --> 00:42:04,790 A is the matrix obtained by essentially taking 589 00:42:04,790 --> 00:42:07,250 the adjacency matrix and subtracting 590 00:42:07,250 --> 00:42:09,500 that constant there. 591 00:42:09,500 --> 00:42:12,900 And subtracting that constant gets rid of the top eigenvalue. 592 00:42:12,900 --> 00:42:15,552 And what you remained is whatever that's left. 593 00:42:15,552 --> 00:42:17,510 And you want to show that whatever you remained 594 00:42:17,510 --> 00:42:20,550 has small spectral radius. 595 00:42:20,550 --> 00:42:22,910 So we would like to show that this quantity here 596 00:42:22,910 --> 00:42:23,600 is quite small. 597 00:42:27,880 --> 00:42:29,370 Well, let's do it. 598 00:42:29,370 --> 00:42:33,360 So give me a pair of vectors, x and y. 599 00:42:37,340 --> 00:42:44,290 And let's set the following quantities, 600 00:42:44,290 --> 00:42:49,090 where I take a twist on this x vector 601 00:42:49,090 --> 00:42:57,480 by rotating the coordinates, setting 602 00:42:57,480 --> 00:43:04,720 x super s sub g, the coordinate g, to be x sub sg. 603 00:43:04,720 --> 00:43:08,820 So x is a vector indexed by the group elements, 604 00:43:08,820 --> 00:43:14,520 and then rotating this indexing of the group elements by s. 605 00:43:14,520 --> 00:43:18,180 So that's what I mean by superscript s. 606 00:43:18,180 --> 00:43:24,904 And, likewise, y superscript s is defined similarly. 607 00:43:32,140 --> 00:43:36,760 So I claim that these twists, these rotations, 608 00:43:36,760 --> 00:43:40,820 do not change the norm of these vectors. 609 00:43:40,820 --> 00:43:43,080 And that should be pretty clear, because I'm simply 610 00:43:43,080 --> 00:43:47,540 relabeling the coordinates in a uniform way. 611 00:43:47,540 --> 00:43:50,130 And, likewise, same for y. 612 00:44:00,660 --> 00:44:09,970 So I would like to show this quantity up here is small. 613 00:44:09,970 --> 00:44:13,120 So let's consider two unit vectors. 614 00:44:25,450 --> 00:44:28,410 And consider this bilinear form. 615 00:44:28,410 --> 00:44:42,490 If I expand out this bilinear form, it looks like that. 616 00:44:42,490 --> 00:44:45,080 I'm just writing it out. 617 00:44:45,080 --> 00:44:52,390 But now let me just throw in an extra variable of summation. 618 00:44:52,390 --> 00:45:04,270 What we'll do is essentially look at the same sum, 619 00:45:04,270 --> 00:45:18,620 but now I add in an extra s, and put this s over here. 620 00:45:18,620 --> 00:45:21,450 So convince yourself that this is the same sum. 621 00:45:24,220 --> 00:45:27,890 So it's simply re-parameterizing the sum. 622 00:45:27,890 --> 00:45:28,890 So this is the same sum. 623 00:45:33,590 --> 00:45:36,080 But now, if you look at the definition of A, 624 00:45:36,080 --> 00:45:37,986 there's this cancellation. 625 00:45:41,720 --> 00:45:44,800 So the two s's cancel out. 626 00:45:44,800 --> 00:45:46,360 So let's rewrite the sum. 627 00:45:49,630 --> 00:45:54,541 1 over n, then g, h, s, all group elements. 628 00:45:54,541 --> 00:46:04,040 Then-- now, if I bring this summation of s, 629 00:46:04,040 --> 00:46:08,880 now I bring it inside, and then you see that what's inside is 630 00:46:08,880 --> 00:46:16,695 simply the inner product between the two vectors, x sub g-- 631 00:46:16,695 --> 00:46:17,695 between the two vectors. 632 00:46:24,490 --> 00:46:26,730 So this is-- so what's inside is simply 633 00:46:26,730 --> 00:46:36,600 the product, inner product, between these two. 634 00:46:36,600 --> 00:46:42,100 So I may need to redefine. 635 00:46:45,840 --> 00:46:46,340 Yes. 636 00:46:46,340 --> 00:46:47,510 So when you're looking at-- when you're 637 00:46:47,510 --> 00:46:48,930 talking about non-Abelian groups, 638 00:46:48,930 --> 00:46:50,630 it's always a question of which side 639 00:46:50,630 --> 00:46:53,160 should you multiply things by. 640 00:46:53,160 --> 00:46:57,220 And you guys are OK? 641 00:46:57,220 --> 00:46:59,770 Or I need to change this s to over here. 642 00:46:59,770 --> 00:47:03,650 But anyway, it should work. 643 00:47:03,650 --> 00:47:04,250 Yes, question. 644 00:47:04,250 --> 00:47:05,995 AUDIENCE: yh [INAUDIBLE]. 645 00:47:05,995 --> 00:47:06,620 YUFEI ZHAO: yh. 646 00:47:06,620 --> 00:47:07,606 Thank you. 647 00:47:17,480 --> 00:47:19,830 Yes, I think-- 648 00:47:19,830 --> 00:47:20,820 OK. 649 00:47:20,820 --> 00:47:21,320 Question. 650 00:47:21,320 --> 00:47:26,912 AUDIENCE: [INAUDIBLE] 651 00:47:26,912 --> 00:47:29,900 YUFEI ZHAO: Great. 652 00:47:29,900 --> 00:47:32,883 So maybe I need to switch the definition here, 653 00:47:32,883 --> 00:47:35,050 but, in any case, some version of this should be OK. 654 00:47:35,050 --> 00:47:35,550 Yes. 655 00:47:35,550 --> 00:47:37,220 So figure it out later in the notes. 656 00:47:39,800 --> 00:47:45,010 But now-- OK. 657 00:47:45,010 --> 00:47:48,190 So you have this-- 658 00:47:48,190 --> 00:47:49,770 we have this here. 659 00:47:49,770 --> 00:47:55,720 And if you look at this quantity here, 660 00:47:55,720 --> 00:47:57,850 it is the kind of quantity that comes up 661 00:47:57,850 --> 00:48:00,521 in Grothendieck's inequality. 662 00:48:03,565 --> 00:48:05,190 So this is basically the left-hand side 663 00:48:05,190 --> 00:48:08,550 of Grothendieck's inequality. 664 00:48:08,550 --> 00:48:12,230 What about the right-hand side of Grothendieck's inequality? 665 00:48:12,230 --> 00:48:14,790 Well, we already controlled that. 666 00:48:14,790 --> 00:48:16,340 We already controlled that because we 667 00:48:16,340 --> 00:48:20,340 said, whenever you have up there little x and little y-- 668 00:48:26,000 --> 00:48:31,900 so the conclusion of this board was that-- 669 00:48:31,900 --> 00:48:34,210 let me erase over here. 670 00:48:34,210 --> 00:48:41,160 So the conclusion of this board was that this bilinear form 671 00:48:41,160 --> 00:48:45,690 is bounded by, at most, 4 epsilon d, 672 00:48:45,690 --> 00:48:52,450 for all x and y being plus/minus 1 coordinate valued. 673 00:48:52,450 --> 00:48:57,700 So combining them by Grothendieck, 674 00:48:57,700 --> 00:49:03,830 we have an upper bound, which is the Grothendieck constant times 675 00:49:03,830 --> 00:49:05,510 4 epsilon-- 676 00:49:09,030 --> 00:49:11,870 so 4 epsilon dn. 677 00:49:15,570 --> 00:49:16,820 There's a-- sorry. 678 00:49:16,820 --> 00:49:19,250 There's an n missing here. 679 00:49:23,480 --> 00:49:26,720 And, therefore, because the Grothendieck constant 680 00:49:26,720 --> 00:49:30,846 is less than 2, we have a bound of 8 epsilon d. 681 00:49:34,580 --> 00:49:38,370 And this shows that this variational problem, which 682 00:49:38,370 --> 00:49:42,450 characterizes the largest eigenvalue in absolute value, 683 00:49:42,450 --> 00:49:47,090 is, at most, 8 epsilon d, thereby 684 00:49:47,090 --> 00:49:50,450 implying the eigenvalue property. 685 00:49:58,430 --> 00:50:00,680 So the main takeaway from this proof, two things. 686 00:50:00,680 --> 00:50:03,810 One is Grothendieck's inequality is a nice thing to know. 687 00:50:03,810 --> 00:50:06,170 So it's a semidefinite relaxation 688 00:50:06,170 --> 00:50:09,050 that changes the problem, which is initially somewhat 689 00:50:09,050 --> 00:50:11,750 intractable, to a semidefinite problem which 690 00:50:11,750 --> 00:50:13,730 is both, from a computer science point of view, 691 00:50:13,730 --> 00:50:15,590 algorithmically tractable, but also has 692 00:50:15,590 --> 00:50:17,680 nice mathematical properties. 693 00:50:17,680 --> 00:50:19,580 And for this application here, there's 694 00:50:19,580 --> 00:50:21,380 this nice trick in this proof where 695 00:50:21,380 --> 00:50:26,660 I'm symmetrizing the coordinates using the group symmetries. 696 00:50:26,660 --> 00:50:32,170 And that allows me to obtain this characterization showing 697 00:50:32,170 --> 00:50:35,680 that eigenvalue condition and this discrepancy condition 698 00:50:35,680 --> 00:50:38,900 are equivalent for Cayley graphs. 699 00:50:42,228 --> 00:50:43,270 Let's take a quick break. 700 00:50:46,465 --> 00:50:47,340 Any questions so far? 701 00:50:51,320 --> 00:50:54,220 So we've been talking about n, d, lambda graphs. 702 00:50:54,220 --> 00:50:55,250 So d regular graphs. 703 00:50:55,250 --> 00:50:57,170 And the next question I would like to address 704 00:50:57,170 --> 00:51:02,536 is, In an n, d, lambda graph, how small can lambda be? 705 00:51:02,536 --> 00:51:06,840 So smaller lambda corresponds to a more pseudorandom graph. 706 00:51:06,840 --> 00:51:09,830 So how small can this be? 707 00:51:09,830 --> 00:51:12,650 And the right kind of setting that I want you to think about 708 00:51:12,650 --> 00:51:15,770 is think of d as a constant. 709 00:51:15,770 --> 00:51:19,370 So think of d as a constant, and n getting large. 710 00:51:24,540 --> 00:51:29,830 So how small can lambda be. 711 00:51:32,860 --> 00:51:36,340 And it turns out there is a limit to how small it can be. 712 00:51:36,340 --> 00:51:46,520 And it is known as the Alon-Boppana bound, 713 00:51:46,520 --> 00:51:50,480 which tells you that if you have a fixed d-- 714 00:51:50,480 --> 00:52:00,440 and so G is an n-vertex graph with adjacency matrix 715 00:52:00,440 --> 00:52:08,620 eigenvalues lambda 1 through lambda n, 716 00:52:08,620 --> 00:52:11,940 sorted in non-increasing order. 717 00:52:11,940 --> 00:52:17,550 Then the second largest eigenvalue has to be at least, 718 00:52:17,550 --> 00:52:23,190 basically, 2 root d minus 1 minus a small error term, 719 00:52:23,190 --> 00:52:24,640 little on-- 720 00:52:24,640 --> 00:52:30,590 little o1, where the little o1 goes to 0 as n 721 00:52:30,590 --> 00:52:31,996 goes to infinity. 722 00:52:38,190 --> 00:52:46,700 So the Alon-Boppana bound tells you that the lambda cannot be 723 00:52:46,700 --> 00:52:49,120 below this quantity here. 724 00:52:49,120 --> 00:52:51,050 And I want to explain what is the significance 725 00:52:51,050 --> 00:52:54,080 of this quantity, and you will see it in the proof. 726 00:52:54,080 --> 00:52:56,080 And this quantity is the best possible. 727 00:52:56,080 --> 00:52:59,930 And it also says what do we know about the existence of graphs 728 00:52:59,930 --> 00:53:04,980 which have lambda 2 close to this number. 729 00:53:04,980 --> 00:53:07,580 So this is the optimal number you can put here. 730 00:53:07,580 --> 00:53:08,080 Question. 731 00:53:08,080 --> 00:53:09,747 AUDIENCE: Does it say anything about how 732 00:53:09,747 --> 00:53:11,813 negative lambda n can be? 733 00:53:11,813 --> 00:53:13,230 YUFEI ZHAO: Question-- does it say 734 00:53:13,230 --> 00:53:15,630 how negative lambda n can be? 735 00:53:15,630 --> 00:53:17,313 So I'll address that in a second, 736 00:53:17,313 --> 00:53:19,730 but, essentially, if you have a bipartite graph and lambda 737 00:53:19,730 --> 00:53:22,478 n equals to minus lambda 1. 738 00:53:22,478 --> 00:53:30,706 AUDIENCE: [INAUDIBLE] 739 00:53:30,706 --> 00:53:32,900 YUFEI ZHAO: More questions? 740 00:53:32,900 --> 00:53:36,780 So I want to show you a proof and, time permitting, 741 00:53:36,780 --> 00:53:39,320 a couple of proofs of Alon-Boppana bound. 742 00:53:39,320 --> 00:53:43,970 And they're all quite simple to execute, but the-- 743 00:53:43,970 --> 00:53:46,100 I think it's a good way to understand how 744 00:53:46,100 --> 00:53:47,560 these special techniques work. 745 00:53:56,490 --> 00:53:59,990 So, first, as with all of the proofs that we did concerning-- 746 00:53:59,990 --> 00:54:01,130 or most of them-- 747 00:54:01,130 --> 00:54:03,560 concerning eigenvalues, we're looking 748 00:54:03,560 --> 00:54:10,500 at the Courant-Fischer characterization 749 00:54:10,500 --> 00:54:14,340 of eigenvalues. 750 00:54:14,340 --> 00:54:25,980 It suffices to show, to exhibit some vector z-- 751 00:54:29,850 --> 00:54:33,000 so a nonzero vector-- 752 00:54:33,000 --> 00:54:39,010 such that z is orthogonal to the all 1's vector 753 00:54:39,010 --> 00:54:49,350 and this quotient is at least the claimed bound. 754 00:54:53,170 --> 00:54:54,920 So by the Courant-Fischer characterization 755 00:54:54,920 --> 00:54:57,650 of the second eigenvalue, if you vary 756 00:54:57,650 --> 00:55:00,020 over all such d that are orthogonal to the unit 757 00:55:00,020 --> 00:55:04,250 vector, then the maximum value this quantity attains 758 00:55:04,250 --> 00:55:06,450 is equal to lambda 2. 759 00:55:06,450 --> 00:55:09,080 So to show the lambda 2 is large, 760 00:55:09,080 --> 00:55:14,270 it suffices to exhibit such a z. 761 00:55:14,270 --> 00:55:18,550 So let me construct such a z for you. 762 00:55:18,550 --> 00:55:22,380 So let r be a positive integer. 763 00:55:22,380 --> 00:55:29,490 And let's pick an arbitrary vertex v. So v 764 00:55:29,490 --> 00:55:32,500 is a vertex in the graph. 765 00:55:32,500 --> 00:55:48,074 And let V sub i denote vertices at distance exactly i from V. 766 00:55:48,074 --> 00:55:55,670 From-- yes, from V. So, in particular, V0 is equal to V-- 767 00:55:55,670 --> 00:55:57,610 and I can just draw you a picture. 768 00:55:57,610 --> 00:56:05,110 So you have V0, and then the neighbors of V0, 769 00:56:05,110 --> 00:56:13,907 and each of them have more neighbors. 770 00:56:16,829 --> 00:56:18,780 Like that. 771 00:56:18,780 --> 00:56:26,270 So I'm calling V0 this stuff, big V0. 772 00:56:26,270 --> 00:56:31,470 And then big V1, V sub 2, and so on. 773 00:56:34,900 --> 00:56:39,130 So I'm going to define a vector, which I'll eventually 774 00:56:39,130 --> 00:56:41,560 make into z, by telling you what is 775 00:56:41,560 --> 00:56:48,440 the value of this vector on each of these vertices. 776 00:56:48,440 --> 00:56:53,920 I will do this by setting very explicitly-- 777 00:56:53,920 --> 00:57:03,660 so set x to be a vector with value 778 00:57:03,660 --> 00:57:14,820 x sub u to be wi, where wi is d minus 1 raised to power 779 00:57:14,820 --> 00:57:23,740 minus i over 2 whenever u lies in set big V sub i. 780 00:57:23,740 --> 00:57:27,580 So u is distance exactly i from V. I set it to this number. 781 00:57:27,580 --> 00:57:33,630 So notice that they decrease as you get further away from V. 782 00:57:33,630 --> 00:57:41,436 And I do this for all distances less than r. 783 00:57:44,760 --> 00:57:47,670 So this is my x vector. 784 00:57:47,670 --> 00:57:53,860 And I set all the other vect-- all the other corners 785 00:57:53,860 --> 00:58:02,050 to be 0 if the distance between u and V is at least r. 786 00:58:04,990 --> 00:58:06,690 So that gives you this vector. 787 00:58:06,690 --> 00:58:10,590 And I would like to compute that quotient over there 788 00:58:10,590 --> 00:58:12,450 for this vector. 789 00:58:12,450 --> 00:58:22,760 And I claim that this quotient here is at least 790 00:58:22,760 --> 00:58:24,060 the following quantity. 791 00:58:33,970 --> 00:58:36,577 But this is a computation, so let's just do it. 792 00:58:36,577 --> 00:58:37,410 So why is this true? 793 00:58:40,570 --> 00:58:45,210 Well, if you compute the norm of x-- 794 00:58:45,210 --> 00:58:47,460 so I'm just taking the sum of the squares 795 00:58:47,460 --> 00:58:49,530 of these coordinates. 796 00:58:49,530 --> 00:58:57,130 Well, that comes from adding up these values. 797 00:58:57,130 --> 00:59:01,370 So for each element in the i-th neighborhood. 798 00:59:01,370 --> 00:59:05,010 So I have wi squared. 799 00:59:05,010 --> 00:59:14,620 And if I look at that quantity up there, so what is this? 800 00:59:14,620 --> 00:59:17,120 A is the adjacency matrix. 801 00:59:17,120 --> 00:59:21,740 So over here, A is the adjacency matrix. 802 00:59:25,460 --> 00:59:35,170 So this quantity, I can write it as a sum over all vertices u. 803 00:59:35,170 --> 00:59:39,010 And I look at x sub u, and now I sum again 804 00:59:39,010 --> 00:59:50,600 over all neighbors of u, and consider x sub u prime. 805 00:59:50,600 --> 00:59:52,190 It's that sum there. 806 00:59:52,190 --> 00:59:57,570 But this sum, I have some control over, because it is-- 807 00:59:57,570 --> 01:00:00,000 so what's happening here? 808 01:00:00,000 --> 01:00:04,250 I claim it has at least the following quantity. 809 01:00:04,250 --> 01:00:05,660 Consider where u is. 810 01:00:05,660 --> 01:00:07,660 So u could be-- 811 01:00:07,660 --> 01:00:11,000 I mean, it's only nonzero if u lies 812 01:00:11,000 --> 01:00:13,031 in the r minus 1th neighborhood. 813 01:00:16,800 --> 01:00:21,480 So in that neighborhood, I have V sub i possible choices 814 01:00:21,480 --> 01:00:24,960 for the vertex u. 815 01:00:24,960 --> 01:00:31,990 For that choice, this x sub u is w sub i. 816 01:00:31,990 --> 01:00:35,230 But what about all its neighbors? 817 01:00:35,230 --> 01:00:38,280 So it could have neighbors, well, 818 01:00:38,280 --> 01:00:40,400 in the same set going left. 819 01:00:40,400 --> 01:00:44,500 But there's-- so there's one neighbor going left, 820 01:00:44,500 --> 01:00:47,380 and all the other neighbors are-- 821 01:00:47,380 --> 01:00:50,080 maybe it's in the same set, maybe it's in the next set. 822 01:00:50,080 --> 01:00:53,110 But, in any case, I have the following inequality. 823 01:00:53,110 --> 01:00:57,730 There's one neighbor in the same-- 824 01:00:57,730 --> 01:01:01,660 in the left, if you look at that picture just now. 825 01:01:01,660 --> 01:01:04,900 And then all the remaining neighbors 826 01:01:04,900 --> 01:01:12,640 have x sub u primes at least w sub i plus 1, 827 01:01:12,640 --> 01:01:14,860 because these weights are decreasing. 828 01:01:14,860 --> 01:01:17,840 So I can-- the worst case, so to speak, 829 01:01:17,840 --> 01:01:23,390 is if you-- all the neighbors point to the next set. 830 01:01:23,390 --> 01:01:24,760 So I had that inequality there. 831 01:01:28,600 --> 01:01:30,990 There's an issue. 832 01:01:30,990 --> 01:01:42,900 Because if you go to the very last set, 833 01:01:42,900 --> 01:01:45,000 if you go to the very last set and think 834 01:01:45,000 --> 01:01:49,410 about what happens, when i in in that very last set, 835 01:01:49,410 --> 01:01:55,260 I'm overcounting neighbors that no longer has weights. 836 01:01:55,260 --> 01:01:58,260 So I need to take them out. 837 01:01:58,260 --> 01:02:02,815 So I should subtract d minus 1 times-- 838 01:02:05,680 --> 01:02:09,590 and so this is the maximum possible weight 839 01:02:09,590 --> 01:02:12,010 sum I could have-- 840 01:02:12,010 --> 01:02:15,360 maximum possible overcount. 841 01:02:15,360 --> 01:02:18,430 So each product here has t minus 1 neighbors at most. 842 01:02:18,430 --> 01:02:19,330 All right. 843 01:02:19,330 --> 01:02:21,740 So this is-- should be pretty straightforward if you 844 01:02:21,740 --> 01:02:23,000 do the counting correctly. 845 01:02:23,000 --> 01:02:26,960 But now let's plug in what these weights are. 846 01:02:26,960 --> 01:02:30,620 And you'll find that this sum here, this quantity, 847 01:02:30,620 --> 01:02:32,780 is equal to-- so the key point here 848 01:02:32,780 --> 01:02:36,890 is that this thing simplifies very nicely if you consider 849 01:02:36,890 --> 01:02:39,500 what this is. 850 01:02:39,500 --> 01:02:42,680 So what ends up happening is that you 851 01:02:42,680 --> 01:02:45,250 get this extra factor of 2 root d minus 1. 852 01:02:45,250 --> 01:03:01,160 And then the sum minus 1/2 of V sub [INAUDIBLE].. 853 01:03:01,160 --> 01:03:05,320 It's pretty straightforward computation 854 01:03:05,320 --> 01:03:09,490 using the specific weights that we have. 855 01:03:09,490 --> 01:03:17,490 And one more thing is that notice that this-- 856 01:03:17,490 --> 01:03:22,250 so notice that the sizes of each neighborhood cannot expand 857 01:03:22,250 --> 01:03:30,050 by more than a factor of d minus 1, because, well, 858 01:03:30,050 --> 01:03:34,280 you only have d minus 1 outward edges going forward at each 859 01:03:34,280 --> 01:03:36,390 step. 860 01:03:36,390 --> 01:03:43,260 And, as a result, I can bound this guy. 861 01:03:43,260 --> 01:03:48,510 And so what you find is that this whole thing here 862 01:03:48,510 --> 01:03:53,826 is that least 2 times root d minus 1. 863 01:03:53,826 --> 01:03:56,520 The main term is the sum. 864 01:04:03,500 --> 01:04:08,750 And this here is less than each individual summand. 865 01:04:08,750 --> 01:04:12,440 So I can do 1 minus 1 over 2r. 866 01:04:19,150 --> 01:04:21,930 Putting these two together, you find the claim. 867 01:04:28,835 --> 01:04:30,400 All right. 868 01:04:30,400 --> 01:04:33,730 So I've exhibited this vector x, which 869 01:04:33,730 --> 01:04:36,160 has that quotient property. 870 01:04:36,160 --> 01:04:39,840 But that's not quite enough, because we need a vector-- 871 01:04:39,840 --> 01:04:42,100 so it's called z up here-- 872 01:04:42,100 --> 01:04:46,530 that is orthogonal to the all 1's vector. 873 01:04:46,530 --> 01:04:51,940 And that you can do, because if the number of vertices 874 01:04:51,940 --> 01:04:56,320 is quite a bit larger than-- 875 01:04:59,230 --> 01:05:11,080 compared to the degree, then I claim that there exists u and v 876 01:05:11,080 --> 01:05:12,640 vectors-- 877 01:05:12,640 --> 01:05:18,520 vertices that are at distance at least 2r. 878 01:05:22,480 --> 01:05:25,340 So if I let-- 879 01:05:30,150 --> 01:05:32,220 this is the size of this tree. 880 01:05:32,220 --> 01:05:35,520 So if you have-- 881 01:05:35,520 --> 01:05:37,950 everything is within distance r-- 882 01:05:37,950 --> 01:05:44,738 distance 2r from a vertex, then they all lie on this tree edge. 883 01:05:44,738 --> 01:05:46,780 If you count the number of vertices in that tree, 884 01:05:46,780 --> 01:05:49,840 it's what I have-- the sum I've written here. 885 01:05:49,840 --> 01:05:55,820 So if I consider these two vectors-- 886 01:05:55,820 --> 01:06:05,250 so be-- so x be the vector obtained above, which is, 887 01:06:05,250 --> 01:06:06,440 in some sense-- 888 01:06:06,440 --> 01:06:13,780 and I'm being somewhat informal here-- centered at v. 889 01:06:13,780 --> 01:06:19,280 And if I let y be the vector but I center it now at the vector-- 890 01:06:19,280 --> 01:06:27,020 at u, then I claim that, essentially, x and y 891 01:06:27,020 --> 01:06:32,830 are supported on disjoint vertex sets that have 892 01:06:32,830 --> 01:06:35,950 no edges even between them. 893 01:06:35,950 --> 01:06:40,020 So, in particular, this inner product-- 894 01:06:40,020 --> 01:06:43,870 this bilinear form-- not inner product but this bilinear 895 01:06:43,870 --> 01:06:45,100 form-- 896 01:06:45,100 --> 01:06:59,120 is equal to 0, since no edge between the supports of x 897 01:06:59,120 --> 01:07:00,056 and y. 898 01:07:04,550 --> 01:07:09,700 So now I have two vectors that do not interact, 899 01:07:09,700 --> 01:07:12,760 but both have this nice property above. 900 01:07:12,760 --> 01:07:14,998 And now I can take a linear combination. 901 01:07:18,590 --> 01:07:22,030 Let me choose a constant c-- 902 01:07:22,030 --> 01:07:24,860 so it's a real constant-- 903 01:07:24,860 --> 01:07:31,315 such that this z equal to x minus cy has-- 904 01:07:37,035 --> 01:07:39,110 and I can choose this constant. 905 01:07:39,110 --> 01:07:41,080 So x and y are both non-negative entries. 906 01:07:41,080 --> 01:07:43,840 They're both nonzero, and I can choose this constant c 907 01:07:43,840 --> 01:07:45,900 so that it is-- 908 01:07:45,900 --> 01:07:48,840 this z is orthogonal to the all 1's vector. 909 01:07:48,840 --> 01:07:53,570 And I now I have this extra property I want. 910 01:07:53,570 --> 01:07:55,830 But what about the inner products? 911 01:07:55,830 --> 01:08:01,140 Well, these two vectors, x and y, they do not interact at all. 912 01:08:01,140 --> 01:08:05,270 So their inner products split just fine, 913 01:08:05,270 --> 01:08:08,450 and the bilinear form splits just fine. 914 01:08:19,020 --> 01:08:30,810 So you have this inequality here, as desired. 915 01:08:30,810 --> 01:08:37,479 And r, notice that I can take r going 916 01:08:37,479 --> 01:08:43,740 to infinity as n going to infinity, because d is fixed. 917 01:08:43,740 --> 01:08:47,399 So if n goes to infinity, then r can go to infinity, roughly 918 01:08:47,399 --> 01:08:48,661 a logarithmic n. 919 01:08:52,990 --> 01:08:55,980 And that proves the Alon-Boppana bound. 920 01:08:55,980 --> 01:08:58,540 And just to recap, to prove this bound, 921 01:08:58,540 --> 01:09:01,990 we needed to exhibit by the Courant-Fischer some vector 922 01:09:01,990 --> 01:09:03,355 with a nice-- 923 01:09:03,355 --> 01:09:06,760 this quotient such that this quotient is large. 924 01:09:06,760 --> 01:09:11,350 And we exhibit this quotient by constructing the vector 925 01:09:11,350 --> 01:09:15,399 explicitly around the vertex and finding two such vertices that 926 01:09:15,399 --> 01:09:17,689 are far away from constructing these two vectors, 927 01:09:17,689 --> 01:09:19,660 taking the appropriate linear combination 928 01:09:19,660 --> 01:09:22,510 so that the final vector is orthogonal to the unit vector, 929 01:09:22,510 --> 01:09:26,270 to the all 1's vector, and then showing 930 01:09:26,270 --> 01:09:31,220 that the corresponding bilinear form has-- is large enough. 931 01:09:34,520 --> 01:09:35,396 Any questions? 932 01:09:38,140 --> 01:09:42,370 I want to show you a different proof which gives you 933 01:09:42,370 --> 01:09:47,229 a slightly worse result, but the proof is conceptually nice. 934 01:09:47,229 --> 01:09:51,569 So let me give you a second proof which 935 01:09:51,569 --> 01:09:54,864 is slightly weakening. 936 01:10:00,298 --> 01:10:03,290 And just that we'll show-- 937 01:10:03,290 --> 01:10:06,076 so we'll show that-- 938 01:10:06,076 --> 01:10:09,930 so the earlier proof showed that lambda 2 is quite large. 939 01:10:09,930 --> 01:10:17,100 But, next, we'll show that the max of lambda 2 and the lambda 940 01:10:17,100 --> 01:10:19,800 n is large. 941 01:10:24,500 --> 01:10:28,290 So not that the second largest eigenvalue is large, 942 01:10:28,290 --> 01:10:30,550 but the second largest eigenvalue in absolute value 943 01:10:30,550 --> 01:10:31,050 is large. 944 01:10:31,050 --> 01:10:34,290 So it's slightly weaker, but, for all intents and purposes, 945 01:10:34,290 --> 01:10:36,710 it's the same spirit. 946 01:10:36,710 --> 01:10:39,850 So I'll show this one here. 947 01:10:39,850 --> 01:10:42,040 And this is a nice illustration of what's 948 01:10:42,040 --> 01:10:48,940 called a trace method, sometimes also a moment method. 949 01:10:56,570 --> 01:10:59,090 Here's the idea. 950 01:10:59,090 --> 01:11:03,610 As we saw in the proof relating the quasi-randomness of C4 951 01:11:03,610 --> 01:11:06,380 and eigenvalues, well, C4's are-- 952 01:11:06,380 --> 01:11:09,320 eigenvalues are related to counting closed walks 953 01:11:09,320 --> 01:11:10,430 in a graph. 954 01:11:10,430 --> 01:11:14,540 And so we'll use that counting closed walks in a graph. 955 01:11:14,540 --> 01:11:21,110 And, specifically, the 2k-th moment of the spectrum 956 01:11:21,110 --> 01:11:28,610 is equal to the trace of the 2k-th power, which 957 01:11:28,610 --> 01:11:34,790 counts the number of closed walks of length exactly 2k. 958 01:11:49,680 --> 01:11:53,700 So to lower bound the left-hand side, 959 01:11:53,700 --> 01:11:56,880 we want to lower-bound the right-hand side. 960 01:11:56,880 --> 01:12:02,830 So let's consider closed walks starting at a fixed vertex. 961 01:12:02,830 --> 01:12:10,950 So the number of closed walks of length 962 01:12:10,950 --> 01:12:25,390 exactly 2k starting at a fixed vertex v. Here 963 01:12:25,390 --> 01:12:26,800 we're in a d regular graph. 964 01:12:26,800 --> 01:12:32,730 So here we are in a d regular graph. 965 01:12:32,730 --> 01:12:36,530 I claim, whatever this number is-- it maybe different 966 01:12:36,530 --> 01:12:42,860 for each d-- it is at least the same quantity if I 967 01:12:42,860 --> 01:12:46,070 do this walk in an infinite d regular tree. 968 01:12:53,280 --> 01:12:55,710 So infinite d regular tree is what? 969 01:12:58,530 --> 01:13:00,390 This is an infinite d regulator tree. 970 01:13:05,659 --> 01:13:09,300 We just start with the vertex, and go out d regular. 971 01:13:09,300 --> 01:13:12,880 So why is this true? 972 01:13:12,880 --> 01:13:14,665 So think about how you walk. 973 01:13:14,665 --> 01:13:15,820 So let me just explain. 974 01:13:15,820 --> 01:13:17,868 This is, I think, pretty easy once you 975 01:13:17,868 --> 01:13:18,910 see things the right way. 976 01:13:18,910 --> 01:13:22,360 So start with a vertex v. Think about how you walk. 977 01:13:22,360 --> 01:13:26,140 And whatever way you can walk, well, you 978 01:13:26,140 --> 01:13:31,740 can walk the same way on the infinite d regular tree. 979 01:13:31,740 --> 01:13:34,020 Well, I mean, sorry. 980 01:13:34,020 --> 01:13:37,560 Whatever walk you can do an infinite d regular tree, 981 01:13:37,560 --> 01:13:43,110 if you label the first vertex, the first edge, second edge, 982 01:13:43,110 --> 01:13:44,730 if you do a corresponding labeling 983 01:13:44,730 --> 01:13:46,350 on your original graph, you can do 984 01:13:46,350 --> 01:13:49,440 that walk on your original graph. 985 01:13:49,440 --> 01:13:52,020 Although the original graph may have some additional walks, 986 01:13:52,020 --> 01:13:54,110 namely things that involve cycles, 987 01:13:54,110 --> 01:13:56,575 that are not available on your tree. 988 01:13:56,575 --> 01:13:58,200 But, certainly, every walk, you can do. 989 01:13:58,200 --> 01:14:01,400 Every closed walk you can do on a tree, 990 01:14:01,400 --> 01:14:06,150 you can do the same walk on your graph. 991 01:14:06,150 --> 01:14:07,530 So you can make this more formal. 992 01:14:07,530 --> 01:14:09,447 So you can write down a bijection or injection 993 01:14:09,447 --> 01:14:12,192 to make this more formal, but it should be fairly convincing 994 01:14:12,192 --> 01:14:13,400 that this inequality is true. 995 01:14:16,940 --> 01:14:18,580 But this is just a number. 996 01:14:18,580 --> 01:14:21,740 So this is a number of 2k walks in a d regular tree starting 997 01:14:21,740 --> 01:14:22,410 on the vertex. 998 01:14:22,410 --> 01:14:23,993 And this number has been well studied, 999 01:14:23,993 --> 01:14:27,600 and we don't need to know the precise number. 1000 01:14:27,600 --> 01:14:29,430 We just need to know some good lower bound. 1001 01:14:29,430 --> 01:14:31,980 And here is one lower bound, which is that there's at least 1002 01:14:31,980 --> 01:14:34,080 a Catalan number, the k-th Catalan number, 1003 01:14:34,080 --> 01:14:43,230 times d minus 1 to the k, where C sub k is the k-th Catalan 1004 01:14:43,230 --> 01:14:45,867 number, which is equal to-- 1005 01:14:50,176 --> 01:14:53,860 so k 2k choose k divided by k plus 1. 1006 01:14:53,860 --> 01:14:55,750 So let me remind you what this is. 1007 01:14:55,750 --> 01:14:58,980 A wonder, has many combinatorial interpretations, 1008 01:14:58,980 --> 01:15:02,140 and it's a fun exercise to do bijections between them. 1009 01:15:02,140 --> 01:15:06,870 But, in particular, C3 is the equal to 5, 1010 01:15:06,870 --> 01:15:13,070 which counts the number of ups and down walks of length 1011 01:15:13,070 --> 01:15:17,423 6 that never dip below the horizontal line 1012 01:15:17,423 --> 01:15:18,090 where you start. 1013 01:15:23,780 --> 01:15:27,310 So, then, this corresponds to going away 1014 01:15:27,310 --> 01:15:32,200 from the root versus coming back to the root. 1015 01:15:32,200 --> 01:15:34,380 Soon you have at least that many ways. 1016 01:15:34,380 --> 01:15:37,210 And when you are moving away from the root, 1017 01:15:37,210 --> 01:15:42,196 you have d minus 1 choices on which branch to go to. 1018 01:15:42,196 --> 01:15:43,172 OK, good. 1019 01:15:49,030 --> 01:15:56,460 Given that, the right-hand side is at least, then, 1020 01:15:56,460 --> 01:16:02,700 n, the number of vertices, times the quantity above related 1021 01:16:02,700 --> 01:16:03,780 to Catalan numbers. 1022 01:16:11,310 --> 01:16:14,630 On the other hand, the left-hand side is at most-- 1023 01:16:14,630 --> 01:16:16,730 here we're using that 2k is an even number-- 1024 01:16:16,730 --> 01:16:23,220 is at most d to the 2k plus all the other eigenvalues that are 1025 01:16:23,220 --> 01:16:25,740 most lambda in absolute value. 1026 01:16:25,740 --> 01:16:28,920 So let me call this quantity lambda. 1027 01:16:32,570 --> 01:16:39,310 Rearranging this inequality, we find that lambda to the 2k 1028 01:16:39,310 --> 01:16:46,430 is at least this number here. 1029 01:16:56,390 --> 01:17:00,516 Just here, I'm changing n minus 1 to n. 1030 01:17:00,516 --> 01:17:01,930 So we have that. 1031 01:17:01,930 --> 01:17:04,180 And now what can we do? 1032 01:17:04,180 --> 01:17:10,900 We let n go to infinity and k go to infinity slowly enough. 1033 01:17:10,900 --> 01:17:16,690 So if k goes to infinity and n goes to-- 1034 01:17:16,690 --> 01:17:21,160 so k goes to infinity with n, but not too quickly. 1035 01:17:21,160 --> 01:17:25,260 But k is little of log n. 1036 01:17:25,260 --> 01:17:28,450 And we find that this quantity here 1037 01:17:28,450 --> 01:17:30,550 is essentially 2 to the k-- 1038 01:17:33,784 --> 01:17:34,730 2 to the 2k. 1039 01:17:37,250 --> 01:17:40,410 And this guy here is little o1. 1040 01:17:40,410 --> 01:17:45,850 So lambda is at least 2 root d minus 1 minus little o1. 1041 01:17:51,050 --> 01:17:53,570 That proves, essentially, the Alon-Boppana bound, 1042 01:17:53,570 --> 01:17:56,990 although a small weakening because we are-- 1043 01:17:56,990 --> 01:17:59,780 this big eigenvalue, you might find 1044 01:17:59,780 --> 01:18:03,230 might actually be very negative instead of very positive. 1045 01:18:03,230 --> 01:18:03,830 But that's OK. 1046 01:18:03,830 --> 01:18:09,200 For applications, this is not such a big deal. 1047 01:18:09,200 --> 01:18:10,790 These are two different proofs. 1048 01:18:10,790 --> 01:18:13,400 And now, we think about, are they really the same proof? 1049 01:18:13,400 --> 01:18:14,930 Are they different proofs? 1050 01:18:14,930 --> 01:18:16,800 Are they related to each other? 1051 01:18:16,800 --> 01:18:18,450 So it's worth thinking about. 1052 01:18:18,450 --> 01:18:20,750 They look very different, but how 1053 01:18:20,750 --> 01:18:22,500 are they related to each other? 1054 01:18:22,500 --> 01:18:24,510 And one final remark. 1055 01:18:24,510 --> 01:18:26,730 You already saw two different proofs as to-- 1056 01:18:26,730 --> 01:18:28,340 I mean, that shows you this number, 1057 01:18:28,340 --> 01:18:30,630 and you see where this number comes from. 1058 01:18:30,630 --> 01:18:32,750 And let me just offer one final remark on where 1059 01:18:32,750 --> 01:18:35,330 that number really comes from. 1060 01:18:35,330 --> 01:18:39,580 And it really comes from this infinite d regular tree. 1061 01:18:42,630 --> 01:18:45,300 So it turns out that 2 root d minus 1 1062 01:18:45,300 --> 01:18:55,320 exactly is the spectral radius of the infinite d regular tree. 1063 01:18:58,120 --> 01:19:00,400 And that is the reason, in some sense, 1064 01:19:00,400 --> 01:19:03,790 that this is the correct number occurring Alon-Boppana bound. 1065 01:19:07,380 --> 01:19:11,310 This is-- if you've seen things like algebraic topology 1066 01:19:11,310 --> 01:19:16,040 or topology, this is a universal cover for d regular graphs. 1067 01:19:16,040 --> 01:19:20,330 So I won't talk more about it, but just some general remarks, 1068 01:19:20,330 --> 01:19:22,310 and you already saw two different proofs. 1069 01:19:22,310 --> 01:19:24,540 So beginning of next time, I want to wrap this up 1070 01:19:24,540 --> 01:19:25,740 and to show you-- 1071 01:19:25,740 --> 01:19:28,590 to explain some-- what we know about 1072 01:19:28,590 --> 01:19:32,540 are there graphs for which this bound is tight? 1073 01:19:32,540 --> 01:19:36,240 And the answer is yes, and there are lots of major open problems 1074 01:19:36,240 --> 01:19:39,403 as well related to what happens there. 1075 01:19:39,403 --> 01:19:40,820 And then, after that, I would like 1076 01:19:40,820 --> 01:19:42,830 to start talking about graph limits. 1077 01:19:42,830 --> 01:19:45,760 So that's the next chapter of this course. 1078 01:19:45,760 --> 01:19:47,610 OK, good.