1 00:00:00,992 --> 00:00:04,410 It's actually delightful to be here 2 00:00:04,410 --> 00:00:07,126 So thanks to for inviting me. 3 00:00:07,126 --> 00:00:10,822 I am a mathematician. And you may not 4 00:00:10,822 --> 00:00:12,860 associate mathematicians with people that 5 00:00:12,860 --> 00:00:13,953 love art. 6 00:00:13,953 --> 00:00:15,787 But I happen to think that mathematics is 7 00:00:15,787 --> 00:00:17,164 the most beautiful thing on earth. 8 00:00:17,164 --> 00:00:19,247 And I want to convince you today that 9 00:00:19,247 --> 00:00:20,745 that's the case. 10 00:00:20,745 --> 00:00:21,797 Now we're going to use a few tricks. 11 00:00:21,797 --> 00:00:23,624 So we're not just going to stare at 12 00:00:23,624 --> 00:00:26,422 equations. I get really a wonderful 13 00:00:26,422 --> 00:00:29,441 feeling inside of me when I stare at a 14 00:00:29,441 --> 00:00:32,050 system of equations. But I'm not expecting 15 00:00:32,050 --> 00:00:33,448 you to have that same sort of feeling. 16 00:00:33,448 --> 00:00:35,497 So what I'd like to do is provide some 17 00:00:35,497 --> 00:00:37,265 color and some illustrations of the 18 00:00:37,265 --> 00:00:42,545 systems of equations. 19 00:00:42,545 --> 00:00:45,017 Now, matrices what are matrices, you know? 20 00:00:45,017 --> 00:00:47,873 Why, as a mathematician, do I love them so 21 00:00:47,873 --> 00:00:49,821 much? Well, let me just give you a little 22 00:00:49,821 --> 00:00:52,357 bit of a feel of the sort of problems I 23 00:00:52,357 --> 00:00:53,855 work on. I'm a computational mathematician 24 00:00:53,855 --> 00:00:58,075 And I really like things to do with fluid 25 00:00:58,075 --> 00:01:00,642 flow. So you give me a fluid, I like to 26 00:01:00,642 --> 00:01:03,646 simulate it. So I've done work on 27 00:01:03,646 --> 00:01:04,751 coastal ocean flows 28 00:01:04,751 --> 00:01:06,485 I've done work on oil and gas flows 29 00:01:06,485 --> 00:01:10,796 I've designed sails for the American cup 30 00:01:10,796 --> 00:01:13,126 For those of you who know about America's 31 00:01:13,126 --> 00:01:14,982 cup Right now. Not for this particular 32 00:01:14,982 --> 00:01:17,429 race but in 2000 and 2003 33 00:01:17,429 --> 00:01:19,472 but in all of these things are eminent 34 00:01:19,472 --> 00:01:21,776 flow I'd like to simulate it. 35 00:01:21,776 --> 00:01:23,386 and to simulate this 36 00:01:23,386 --> 00:01:26,641 we set up a really large system of 37 00:01:26,641 --> 00:01:29,039 equations relating pressures at all 38 00:01:29,039 --> 00:01:31,167 sort of different points in the flow 39 00:01:31,167 --> 00:01:32,720 fields and velocities 40 00:01:32,720 --> 00:01:35,059 and we need to solve it 41 00:01:35,059 --> 00:01:35,933 and ultimately the system of equations 42 00:01:35,933 --> 00:01:37,701 leads to something called a matrix. 43 00:01:37,701 --> 00:01:39,039 I'll show you an example. 44 00:01:39,039 --> 00:01:41,582 But also, because I like matrices so much 45 00:01:41,582 --> 00:01:42,975 In these matrix equations 46 00:01:42,975 --> 00:01:44,457 I can use that same 47 00:01:44,457 --> 00:01:47,061 knowledge to help with example 48 00:01:47,061 --> 00:01:49,221 of the design of a search engine. 49 00:01:49,221 --> 00:01:51,343 Or a recommender system. 50 00:01:51,343 --> 00:01:53,234 And that may sound really funny to you, 51 00:01:53,234 --> 00:01:54,141 but it's exactly the same math 52 00:01:54,141 --> 00:01:56,036 behind it. 53 00:01:56,036 --> 00:01:58,263 And the math behind it, 54 00:01:58,263 --> 00:01:59,123 is this matrix stuff. 55 00:01:59,123 --> 00:02:01,258 Now for some of you, 56 00:02:01,258 --> 00:02:02,804 matrices may actually have to do 57 00:02:02,804 --> 00:02:04,481 with something like this. 58 00:02:04,481 --> 00:02:06,864 Do you recognize that made with 59 00:02:06,864 --> 00:02:07,816 the movie the matrix 60 00:02:07,816 --> 00:02:10,234 which I happen to love. 61 00:02:10,234 --> 00:02:12,342 But has nothing to do with my field. 62 00:02:12,342 --> 00:02:14,388 But if you type in matrices 63 00:02:14,388 --> 00:02:15,966 or you type in matrices for 64 00:02:15,966 --> 00:02:17,456 engineering applications 65 00:02:17,456 --> 00:02:19,396 in Google, you get millions and millions 66 00:02:19,396 --> 00:02:20,021 and millions of hits. 67 00:02:20,021 --> 00:02:21,814 And you can see lots of different 68 00:02:21,814 --> 00:02:23,631 applications, which some how use this 69 00:02:23,631 --> 00:02:25,521 This is just a snap shot 70 00:02:25,521 --> 00:02:26,923 of the images 71 00:02:26,923 --> 00:02:28,438 the first place of images you see 72 00:02:28,438 --> 00:02:29,991 when you type in matrices. 73 00:02:29,991 --> 00:02:32,941 74 00:02:32,941 --> 00:02:34,945 And when you go look at this, 75 00:02:34,945 --> 00:02:35,323 you see matrices come up in 76 00:02:35,323 --> 00:02:36,070 flow mechanics 77 00:02:36,070 --> 00:02:39,314 but also in structural mechanics 78 00:02:39,314 --> 00:02:41,981 you see it come up in social networks 79 00:02:41,981 --> 00:02:42,487 you see it come up in neurology, 80 00:02:42,487 --> 00:02:44,110 biology, so many different application areas 81 00:02:45,890 --> 00:02:46,368 use this matrices. 82 00:02:46,368 --> 00:02:47,109 And so, officialization of them is nice 83 00:02:47,109 --> 00:02:52,801 also because sometimes, when we stare 84 00:02:52,801 --> 00:02:55,691 at a big matrix 85 00:02:55,691 --> 00:02:57,716 which is just a table with numbers, 86 00:02:57,716 --> 00:02:59,946 it's very hard to discern patterns 87 00:02:59,946 --> 00:03:02,124 But sometimes when you start visualizing 88 00:03:02,124 --> 00:03:02,837 them, 89 00:03:02,837 --> 00:03:04,919 you can get a deeper understanding, 90 00:03:04,919 --> 00:03:07,084 also of the underlying math 91 00:03:07,084 --> 00:03:08,119 and the underlying physics. 92 00:03:08,119 --> 00:03:12,681 So, let's start with a little bit of 93 00:03:12,681 --> 00:03:13,664 algebra. 94 00:03:13,664 --> 00:03:17,630 So say, I have four unknowns, that I want 95 00:03:17,630 --> 00:03:19,670 to compute. So this brings 96 00:03:19,670 --> 00:03:21,071 you back to, maybe, high school algebra 97 00:03:21,071 --> 00:03:23,990 If I have four unknowns I want to 98 00:03:23,990 --> 00:03:24,962 compute, 99 00:03:24,962 --> 00:03:25,949 in this case they're called 100 00:03:25,949 --> 00:03:27,378 W, X, Y and Z 101 00:03:27,378 --> 00:03:29,462 We always use these sort of variables 102 00:03:29,462 --> 00:03:30,553 because mathematicians aren't 103 00:03:30,553 --> 00:03:32,325 that creative 104 00:03:32,325 --> 00:03:32,985 alright, so we use things like 105 00:03:32,985 --> 00:03:33,235 X1, X2, X3 106 00:03:33,235 --> 00:03:35,338 and so on, right? 107 00:03:35,338 --> 00:03:37,640 And so if I want to solve for 108 00:03:37,640 --> 00:03:38,588 four unknowns, 109 00:03:38,588 --> 00:03:40,263 I need four constraints. 110 00:03:40,263 --> 00:03:41,573 Four equations 111 00:03:41,573 --> 00:03:43,156 that tell me how they relate to 112 00:03:43,156 --> 00:03:43,843 each other 113 00:03:43,843 --> 00:03:45,369 and here I just made up 114 00:03:45,369 --> 00:03:46,804 four of those 115 00:03:46,804 --> 00:03:47,545 K? 116 00:03:47,545 --> 00:03:49,211 Now one of the things that you see, 117 00:03:49,211 --> 00:03:50,526 is that there is 118 00:03:50,526 --> 00:03:52,374 some empty spots here. 119 00:03:52,374 --> 00:03:54,576 And you also see that there is 120 00:03:54,576 --> 00:03:56,923 this pattern; W, X, Y Z 121 00:03:56,923 --> 00:03:58,324 They are always written in the 122 00:03:58,324 --> 00:03:59,121 same pattern. 123 00:03:59,121 --> 00:04:00,970 Sometime there is a one in front 124 00:04:00,970 --> 00:04:01,847 of it. 125 00:04:01,847 --> 00:04:03,152 1 times W, 1 times X 126 00:04:03,152 --> 00:04:04,425 and sometimes there is nothing. 127 00:04:04,425 --> 00:04:06,121 Which really means there was a zero 128 00:04:06,121 --> 00:04:07,627 in front of it, right? 129 00:04:07,627 --> 00:04:09,205 But as a mathematician, 130 00:04:09,205 --> 00:04:11,208 I look at these, I'm a very organized 131 00:04:11,208 --> 00:04:11,914 person 132 00:04:11,914 --> 00:04:13,501 So I write all of these things in 133 00:04:13,501 --> 00:04:15,357 this particular order. 134 00:04:15,357 --> 00:04:17,219 And I see there are four equations, and 135 00:04:17,219 --> 00:04:19,945 each equation has a bunch of 136 00:04:19,945 --> 00:04:21,487 coefficient corresponding to W. 137 00:04:21,487 --> 00:04:23,581 Bunch corresponding to X, 138 00:04:23,581 --> 00:04:25,120 to Y, and to Z. 139 00:04:25,120 --> 00:04:27,235 And all I really need to remember 140 00:04:27,235 --> 00:04:30,376 for the system, is these coefficients. 141 00:04:30,376 --> 00:04:32,280 Right? As soon as I fix the order, 142 00:04:32,280 --> 00:04:34,587 all I need to remember are these 143 00:04:34,587 --> 00:04:35,771 coefficients. 144 00:04:35,771 --> 00:04:37,630 So what I'm going to do, 145 00:04:37,630 --> 00:04:38,858 is I'm going to re-write this 146 00:04:38,858 --> 00:04:39,574 a little bit. 147 00:04:39,574 --> 00:04:40,543 And I'm going to create, 148 00:04:40,543 --> 00:04:43,116 this table here, which has these 149 00:04:43,116 --> 00:04:44,501 coefficients, now I left out 150 00:04:44,501 --> 00:04:46,500 the zeros. 151 00:04:46,500 --> 00:04:47,743 Now we are also very lazy, 152 00:04:47,743 --> 00:04:48,901 mathematicians. 153 00:04:48,901 --> 00:04:51,211 Zeros, we never write down. 154 00:04:51,211 --> 00:04:52,289 Also, because with these 155 00:04:52,289 --> 00:04:54,235 large systems of equations 156 00:04:54,235 --> 00:04:54,908 that we have to deal with, 157 00:04:54,908 --> 00:04:56,074 say for recommender system 158 00:04:56,074 --> 00:04:58,303 or page rank, or search algorithm 159 00:04:58,303 --> 00:04:59,524 there are many, many many zeros. 160 00:04:59,524 --> 00:05:00,282 So we leave them out. 161 00:05:00,282 --> 00:05:01,074 K? 162 00:05:01,074 --> 00:05:02,896 But, arguably, all that's really 163 00:05:02,896 --> 00:05:05,866 important to me, for this particular 164 00:05:05,866 --> 00:05:07,148 system, whatever that represents; 165 00:05:07,148 --> 00:05:10,077 maybe it's pressure of velocity 166 00:05:10,077 --> 00:05:11,701 relationships in fluid flow, 167 00:05:11,701 --> 00:05:13,559 maybe it's a friend networking 168 00:05:13,559 --> 00:05:14,593 algorithm. I don't really care. 169 00:05:14,593 --> 00:05:16,936 All that's really important 170 00:05:16,936 --> 00:05:19,281 is this table with the numbers 171 00:05:19,281 --> 00:05:20,667 this table, that we call a matrix 172 00:05:20,667 --> 00:05:24,002 K? So that's the matrix. 173 00:05:24,002 --> 00:05:25,910 Now 174 00:05:25,910 --> 00:05:28,435 sometimes we use very simple 175 00:05:28,435 --> 00:05:29,977 visualizations. 176 00:05:29,977 --> 00:05:31,747 Mine is called, Spy Plot. 177 00:05:31,747 --> 00:05:33,794 And all I do 178 00:05:33,794 --> 00:05:36,458 is, you know, I had all these ones 179 00:05:36,458 --> 00:05:38,783 and everywhere a one appears 180 00:05:38,783 --> 00:05:41,012 I simply put a little dot. 181 00:05:41,012 --> 00:05:41,742 K? 182 00:05:41,742 --> 00:05:43,582 So in this case I'm not so interested in 183 00:05:43,582 --> 00:05:47,134 what size these coefficients are, 184 00:05:47,134 --> 00:05:48,732 whether they're one or two or ten 185 00:05:48,732 --> 00:05:50,373 or minus two-hundred. 186 00:05:50,373 --> 00:05:50,923 I don't really care. 187 00:05:50,923 --> 00:05:52,714 I just want to know where there are 188 00:05:52,714 --> 00:05:54,858 non-zeros 189 00:05:54,858 --> 00:05:56,596 because these non-zeros 190 00:05:56,596 --> 00:05:59,471 give me exactly, how these things relate. 191 00:05:59,471 --> 00:06:02,944 If i have a non-zero right here, and a non 192 00:06:02,944 --> 00:06:07,033 zero there, I know that W and Y are an 193 00:06:07,033 --> 00:06:07,970 equation together. 194 00:06:07,970 --> 00:06:10,909 And so, the patterns, of these non zeros 195 00:06:10,909 --> 00:06:12,240 give me information about the system. 196 00:06:12,240 --> 00:06:14,453 Does that make sense? 197 00:06:14,453 --> 00:06:16,862 And so the simplest way, 198 00:06:16,862 --> 00:06:18,846 because again you know the start 199 00:06:18,846 --> 00:06:20,428 is very simple, and not super creative. 200 00:06:20,428 --> 00:06:22,137 In the beginning, we just put a little 201 00:06:22,137 --> 00:06:24,923 dot. And therefore very large systems 202 00:06:24,923 --> 00:06:27,291 of equations we may get things like this. 203 00:06:27,291 --> 00:06:30,102 So this is called, "Spy plot." 204 00:06:30,102 --> 00:06:33,425 So this is just a matrix, with dots 205 00:06:33,425 --> 00:06:35,242 where ever there is a non zero 206 00:06:35,242 --> 00:06:37,531 and you see there are patterns, 207 00:06:37,531 --> 00:06:38,975 now that I can see. 208 00:06:38,975 --> 00:06:41,247 And actually these come from a network 209 00:06:41,247 --> 00:06:44,432 a matrix can also represent a network, 210 00:06:44,432 --> 00:06:45,905 which we'll see in a little bit. 211 00:06:45,905 --> 00:06:47,977 They're Stanford Networks. 212 00:06:47,977 --> 00:06:50,407 This is, I think, uh Standford as a whole 213 00:06:50,407 --> 00:06:52,940 And this was the Internet. 214 00:06:52,940 --> 00:06:55,510 And all the pages in Ero Astro 215 00:06:55,510 --> 00:06:58,265 that we crolled in 2004 to create those 216 00:06:58,265 --> 00:06:59,063 "Spy plots." 217 00:06:59,063 --> 00:07:01,800 And looking at this, I can see 218 00:07:01,800 --> 00:07:03,480 because I know a little bit about how 219 00:07:03,480 --> 00:07:04,186 these were created, 220 00:07:04,186 --> 00:07:07,013 That here are clusters of websites 221 00:07:07,013 --> 00:07:09,514 that are correlated very strongly 222 00:07:09,514 --> 00:07:11,586 so they keep on referring to each other 223 00:07:11,586 --> 00:07:14,263 and internally. And they're not very 224 00:07:14,263 --> 00:07:15,147 much connected to this 225 00:07:15,147 --> 00:07:17,222 so that's central administration. 226 00:07:17,222 --> 00:07:18,102 They only talk to each other. 227 00:07:18,102 --> 00:07:19,532 Not so much to others. 228 00:07:19,532 --> 00:07:21,197 And they can look at this, and they 229 00:07:21,197 --> 00:07:22,046 can discern 230 00:07:22,046 --> 00:07:23,082 organizational structures. 231 00:07:23,082 --> 00:07:24,868 And it's amazing how you can 232 00:07:24,868 --> 00:07:25,964 use these things. 233 00:07:25,964 --> 00:07:26,439 Right? 234 00:07:26,439 --> 00:07:28,502 And of course, some other equations 235 00:07:28,502 --> 00:07:30,394 lead to more interesting patterns. 236 00:07:30,394 --> 00:07:32,378 This is a "spy plot" 237 00:07:32,378 --> 00:07:33,535 Of a particular 238 00:07:33,535 --> 00:07:35,533 equation, or system of equation 239 00:07:35,533 --> 00:07:37,433 that looked at an oil resovoir modeling. 240 00:07:37,433 --> 00:07:40,317 And when look at these particular 241 00:07:40,317 --> 00:07:41,214 patterns, I can say something about 242 00:07:41,214 --> 00:07:45,426 the behavior, not that much 243 00:07:45,426 --> 00:07:47,281 But it's still, you know, resonably 244 00:07:47,281 --> 00:07:47,944 pretty. 245 00:07:47,944 --> 00:07:49,119 And then what I could do, 246 00:07:49,119 --> 00:07:52,383 if the actual non zeros change in size 247 00:07:52,383 --> 00:07:54,302 so some if they're bigger than others 248 00:07:54,302 --> 00:07:56,013 I could give them a color, 249 00:07:56,013 --> 00:07:56,766 depending on the size. 250 00:07:56,766 --> 00:07:58,406 Now we're getting very artsy, 251 00:07:58,406 --> 00:07:59,405 for a mathmatician. 252 00:07:59,405 --> 00:08:00,221 Which is pretty amazing. 253 00:08:00,221 --> 00:08:03,695 And I get things like this. 254 00:08:03,695 --> 00:08:06,603 Now there are many more interesting ways 255 00:08:06,603 --> 00:08:07,883 to do this though, and this is really 256 00:08:07,883 --> 00:08:09,946 what I'd like to show you today. 257 00:08:09,946 --> 00:08:11,809 Is to use graphs. 258 00:08:11,809 --> 00:08:13,542 So we're going to go back 259 00:08:13,542 --> 00:08:16,106 I hope 260 00:08:16,106 --> 00:08:16,620 Yep. 261 00:08:16,620 --> 00:08:18,377 To this matrix. 262 00:08:18,377 --> 00:08:19,004 OK? 263 00:08:19,004 --> 00:08:25,126 Now I want you to focus on, ehm, this 264 00:08:25,126 --> 00:08:28,828 equation here, this equation had W 265 00:08:28,828 --> 00:08:30,933 Plus zero times X 266 00:08:30,933 --> 00:08:32,789 X doesn't play a role 267 00:08:32,789 --> 00:08:33,894 Plus Y times nothing, times Z 268 00:08:33,894 --> 00:08:36,365 Is equal to one. 269 00:08:36,365 --> 00:08:38,712 That's really where that came from 270 00:08:38,712 --> 00:08:41,114 And from this, I know that there is 271 00:08:41,114 --> 00:08:44,794 a connection between W and X. 272 00:08:44,794 --> 00:08:50,459 They're in the same equation. 273 00:08:50,459 --> 00:08:51,017 Right? 274 00:08:51,017 --> 00:08:51,869 W and Y, sorry. W and Y. 275 00:08:51,869 --> 00:08:54,101 I thought you said "Why," so I tried to 276 00:08:54,101 --> 00:08:57,251 explain it again. (Laughter). 277 00:08:57,251 --> 00:08:58,401 Because there is an equation connected. 278 00:08:58,401 --> 00:09:01,215 But W and Y. 279 00:09:01,215 --> 00:09:03,984 So all I'm going to remember now is 280 00:09:03,984 --> 00:09:05,294 that there is that connection. 281 00:09:05,294 --> 00:09:10,665 Now W is here in this location and Y 282 00:09:10,665 --> 00:09:13,594 is in this location. And this non zero 283 00:09:13,594 --> 00:09:16,442 I can see is really connecting 284 00:09:16,442 --> 00:09:17,254 those two together. 285 00:09:17,254 --> 00:09:21,904 So if I say I have a non zero in the 286 00:09:21,904 --> 00:09:23,782 first row and the third column, 287 00:09:23,782 --> 00:09:27,726 I know that W and Y are connected. 288 00:09:27,726 --> 00:09:29,665 If there is a non zero, say, in the 289 00:09:29,665 --> 00:09:31,155 second row and the fourth column 290 00:09:31,155 --> 00:09:34,436 I know that X and Z are connected. 291 00:09:34,436 --> 00:09:35,995 And that's what I am going to do. 292 00:09:35,995 --> 00:09:41,067 So I am just looking at these non zeros 293 00:09:41,067 --> 00:09:43,687 You know, these ones off of the diagonal 294 00:09:43,687 --> 00:09:46,874 The one that's saying W is connected 295 00:09:46,874 --> 00:09:47,554 To itself, but this one 296 00:09:47,554 --> 00:09:50,235 signifies a connection between W and Y 297 00:09:50,235 --> 00:09:52,285 This one between X and Y 298 00:09:52,285 --> 00:09:56,410 And so when I look at those non zeros 299 00:09:56,410 --> 00:09:58,520 I could also write it like this like a 300 00:09:58,520 --> 00:09:59,832 little graph. 301 00:09:59,832 --> 00:10:03,397 Now I see one and three are connected. 302 00:10:03,397 --> 00:10:05,149 The first element, and the third element, 303 00:10:05,149 --> 00:10:07,129 That is the W and the Y. 304 00:10:07,129 --> 00:10:10,744 Now I measure W, X, Y and Z. 305 00:10:10,744 --> 00:10:11,824 One, two, three, four. 306 00:10:11,824 --> 00:10:13,896 And so one and three are connected, 307 00:10:13,896 --> 00:10:15,673 there is another equation that connects 308 00:10:15,673 --> 00:10:17,854 Two with three, and there is an equation 309 00:10:17,854 --> 00:10:19,573 that connects two and four. 310 00:10:19,573 --> 00:10:20,941 And there is an equation that connects 311 00:10:20,941 --> 00:10:21,789 three and four. 312 00:10:21,789 --> 00:10:23,578 These are the connections that I have 313 00:10:23,578 --> 00:10:24,641 between these unknowns. 314 00:10:24,641 --> 00:10:27,097 Now I made it very abstract, right? 315 00:10:27,097 --> 00:10:29,210 Because I had this system of equations 316 00:10:29,210 --> 00:10:31,390 that told me exactly what these equations 317 00:10:31,390 --> 00:10:32,955 were. And I'm just removing that 318 00:10:32,955 --> 00:10:34,941 and say, I'm just interested now in 319 00:10:34,941 --> 00:10:36,593 connections. 320 00:10:36,593 --> 00:10:37,942 What is influenced by what? 321 00:10:37,942 --> 00:10:40,768 If W and Y are in an equation together, 322 00:10:40,768 --> 00:10:42,848 then the size of one influences 323 00:10:42,848 --> 00:10:43,742 the size of the other. 324 00:10:43,742 --> 00:10:46,175 That's all I am interested in now. 325 00:10:46,175 --> 00:10:49,434 Now vice-versa, if I had a network like 326 00:10:49,434 --> 00:10:52,389 this, a connected graph, maybe 327 00:10:52,389 --> 00:10:55,234 friends. Friend one is connected to 328 00:10:55,234 --> 00:10:56,939 friend three and three is friends with 329 00:10:56,939 --> 00:10:58,329 four but four is not friends with one, 330 00:10:58,329 --> 00:11:01,641 then I could replace that network 331 00:11:01,641 --> 00:11:04,012 with this graph with a matrix. 332 00:11:04,012 --> 00:11:06,035 Right? I could go from one to the 333 00:11:06,035 --> 00:11:08,983 other. But now I'm going to take these 334 00:11:08,983 --> 00:11:10,452 matrices, maybe they come from 335 00:11:10,452 --> 00:11:12,723 fluid mechanics. And I have ten million 336 00:11:12,723 --> 00:11:17,013 columns this way, and I have 10 million 337 00:11:17,013 --> 00:11:19,414 rows this way. This is what we call small 338 00:11:19,414 --> 00:11:20,206 simulation. 339 00:11:20,206 --> 00:11:22,760 So I have a lot, and I'm going to create 340 00:11:22,760 --> 00:11:25,023 a graph out of that. Right? 341 00:11:25,023 --> 00:11:27,652 Now when I look at that graph I can see 342 00:11:27,652 --> 00:11:29,174 these connections, but of course 343 00:11:29,174 --> 00:11:30,906 you immediately say, well I control this 344 00:11:30,906 --> 00:11:32,230 in all sorts of different ways. 345 00:11:32,230 --> 00:11:35,832 You see the same graphs, just drawn 346 00:11:35,832 --> 00:11:38,026 a little differently. 347 00:11:38,026 --> 00:11:39,173 And then the question is, well 348 00:11:39,173 --> 00:11:41,143 which drawing do you prefer? 349 00:11:41,143 --> 00:11:42,654 Which makes it clearest, what the 350 00:11:42,654 --> 00:11:44,893 connections are. 351 00:11:44,893 --> 00:11:45,840 To you. 352 00:11:45,840 --> 00:11:47,277 Just by looking at it, what do you think? 353 00:11:47,277 --> 00:11:49,621 This one? 354 00:11:49,621 --> 00:11:50,836 (Audience) "That one" 355 00:11:50,836 --> 00:11:53,397 Yeah I like this one a lot too. 356 00:11:53,397 --> 00:11:55,521 So then, of course, this is just 357 00:11:55,521 --> 00:11:57,437 with four, right? I just had four 358 00:11:57,437 --> 00:12:00,154 One, two, three and four in connections 359 00:12:00,154 --> 00:12:02,272 Now suppose I have 10 million 360 00:12:02,272 --> 00:12:05,472 With maybe 50 million in connection total 361 00:12:05,472 --> 00:12:09,400 and, and I ask my student, make me 362 00:12:09,400 --> 00:12:11,773 a nice looking graph. 363 00:12:11,773 --> 00:12:14,006 So they can look at it and maybe 364 00:12:14,006 --> 00:12:15,822 discern a little bit of information about 365 00:12:15,822 --> 00:12:16,565 fluid flow. 366 00:12:16,565 --> 00:12:17,252 Right? 367 00:12:17,252 --> 00:12:19,147 So that would be very hard to do, by hand. 368 00:12:19,147 --> 00:12:22,240 So if I had something like this. 369 00:12:22,240 --> 00:12:24,240 And I say, now give me something 370 00:12:24,240 --> 00:12:26,555 that looks very good because now there 371 00:12:26,555 --> 00:12:28,369 are all these overlapping connections. 372 00:12:28,369 --> 00:12:30,744 So then, it's an interesting thing 373 00:12:30,744 --> 00:12:34,555 How do I pull a part this complicated 374 00:12:34,555 --> 00:12:36,902 looking graph and make something 375 00:12:36,902 --> 00:12:38,679 where structures and connections 376 00:12:38,679 --> 00:12:41,137 are much more easy to see. 377 00:12:41,137 --> 00:12:43,381 And that's what I want to show you, 378 00:12:43,381 --> 00:12:45,194 because we can do this. Now you may say 379 00:12:45,194 --> 00:12:46,837 this is a made up example. 380 00:12:46,837 --> 00:12:48,273 What has a messy network like this? 381 00:12:48,273 --> 00:12:49,001 Well let me just show you. 382 00:12:49,001 --> 00:12:49,864 Just one example. 383 00:12:49,864 --> 00:12:51,366 Don't just thought off the web. 384 00:12:51,366 --> 00:12:54,535 I just looked at work by Allera Hall. 385 00:12:54,535 --> 00:12:56,778 And I looked at Saga's from Iceland, 386 00:12:56,778 --> 00:12:58,726 and he published this. 387 00:12:58,726 --> 00:13:00,651 So these are the connections between 388 00:13:00,651 --> 00:13:02,630 various sagas. 389 00:13:02,630 --> 00:13:04,330 Obviously this is very messy. 390 00:13:04,330 --> 00:13:05,701 And I think he should be using 391 00:13:05,701 --> 00:13:06,429 our software. 392 00:13:06,429 --> 00:13:08,980 (Audience Laughter) 393 00:13:08,980 --> 00:13:10,127 Maybe I should send it to him. 394 00:13:10,127 --> 00:13:11,527 So these things happen. 395 00:13:11,527 --> 00:13:13,090 So now the question is, 396 00:13:13,090 --> 00:13:16,171 if I have a bunch of nodes 397 00:13:16,171 --> 00:13:17,803 and I would just place the nodes, 398 00:13:17,803 --> 00:13:18,997 one, two, three, four, all the way 399 00:13:18,997 --> 00:13:20,273 to 10 million 400 00:13:20,273 --> 00:13:22,553 and I have connections between them, 401 00:13:22,553 --> 00:13:24,560 how do I figure out how to pull them 402 00:13:24,560 --> 00:13:27,476 apart and put them on a two 403 00:13:27,476 --> 00:13:28,566 dimensional piece of paper. 404 00:13:28,566 --> 00:13:29,873 So that I have a nice view. 405 00:13:29,873 --> 00:13:34,411 Okay? So how would you do it? 406 00:13:34,411 --> 00:13:37,120 (Audience "grab one and pull?") 407 00:13:37,120 --> 00:13:40,067 Somehow you need - no first of all you 408 00:13:40,067 --> 00:13:41,371 need two nodes that are really 409 00:13:41,371 --> 00:13:44,807 strongly connected to come close 410 00:13:44,807 --> 00:13:46,616 Right? So if node one and two are strongly 411 00:13:46,616 --> 00:13:47,840 connected, maybe because there was 412 00:13:47,840 --> 00:13:50,535 a big non-zero in this matrix then 413 00:13:50,535 --> 00:13:51,508 I want them to be close. 414 00:13:51,508 --> 00:13:55,503 Right? And if one is connected to two 415 00:13:55,503 --> 00:13:57,882 and two to four and four to 17 and 416 00:13:57,882 --> 00:14:00,680 17 to 300, I don't want 1 and 300 417 00:14:00,680 --> 00:14:02,218 to be too close because there is 418 00:14:02,218 --> 00:14:04,941 four degrees of separation. 419 00:14:04,941 --> 00:14:06,367 Right? So the question is, how do I do 420 00:14:06,367 --> 00:14:07,051 that? 421 00:14:07,051 --> 00:14:14,551 So we have these nodes, and 422 00:14:14,551 --> 00:14:16,613 we have these lines connecting them. 423 00:14:16,613 --> 00:14:17,737 K? Now what we're going to do is 424 00:14:17,737 --> 00:14:20,982 two things. Each of these lines will 425 00:14:20,982 --> 00:14:22,357 imagine it's a spring. 426 00:14:22,357 --> 00:14:26,998 So when we pull things apart, 427 00:14:26,998 --> 00:14:28,597 they pull back. 428 00:14:28,597 --> 00:14:30,979 Right? And the size of the spring - 429 00:14:30,979 --> 00:14:32,499 the strength of the spring, Guess what? 430 00:14:32,499 --> 00:14:35,022 That's determined by what? 431 00:14:35,022 --> 00:14:38,983 By the strength of that Non-zero. 432 00:14:38,983 --> 00:14:41,918 Right? OK, that's nice. But what would 433 00:14:41,918 --> 00:14:43,193 happen if I did this? 434 00:14:43,193 --> 00:14:44,677 If I had all of these nodes and I 435 00:14:44,677 --> 00:14:47,023 put springs on them, and let them go. 436 00:14:47,023 --> 00:14:51,097 What would happen if I didn't do anything 437 00:14:51,097 --> 00:14:51,944 else? 438 00:14:51,944 --> 00:14:53,753 Would they (shooo)? All get together? 439 00:14:53,753 --> 00:14:56,019 I don't want that either. 440 00:14:56,019 --> 00:14:56,735 I don't want them all to cluster, 441 00:14:56,735 --> 00:14:58,732 so they're not allowed to get too close. 442 00:14:58,732 --> 00:15:01,875 So how can I make things - I need some 443 00:15:01,875 --> 00:15:03,984 kind of repelling force. So when they get 444 00:15:03,984 --> 00:15:06,003 too close, they're not allowed to. 445 00:15:06,003 --> 00:15:06,933 So what do I do? 446 00:15:06,933 --> 00:15:09,856 I give every node an electric charge. 447 00:15:09,856 --> 00:15:15,208 So that they repel each other. 448 00:15:15,208 --> 00:15:17,382 K? So now I have a whole network of 449 00:15:17,382 --> 00:15:19,241 balls attached to springs, 450 00:15:19,241 --> 00:15:21,358 the springs have stiffness, the balls 451 00:15:21,358 --> 00:15:22,865 have an electric charge. 452 00:15:22,865 --> 00:15:24,533 And I let the whole thing drop on the 453 00:15:24,533 --> 00:15:25,559 floor. 454 00:15:25,559 --> 00:15:31,909 Because I want it two dimensional, right? 455 00:15:31,909 --> 00:15:35,260 And I let this thing organize itself. 456 00:15:35,260 --> 00:15:37,563 So it comes to an equilibrium shape. 457 00:15:37,563 --> 00:15:39,519 That's always minimizing some sort of 458 00:15:39,519 --> 00:15:40,265 energy. 459 00:15:40,265 --> 00:15:40,869 Right? 460 00:15:40,869 --> 00:15:43,000 It's beautiful, these systems do this 461 00:15:43,000 --> 00:15:44,193 automatically. 462 00:15:44,193 --> 00:15:46,029 And I just let them organize themselves. 463 00:15:46,029 --> 00:15:47,640 So it would look something like this. 464 00:15:47,640 --> 00:15:50,380 We start with this, 465 00:15:50,380 --> 00:15:51,874 we let it drop, 466 00:15:51,874 --> 00:15:53,783 and now it becomes this. 467 00:15:53,783 --> 00:15:56,782 So it's exactly that same configuration. 468 00:15:56,782 --> 00:15:59,997 Now here we've cheated a little, 469 00:15:59,997 --> 00:16:04,067 how big you make that electric charge 470 00:16:04,067 --> 00:16:07,103 and how big you make the strength of 471 00:16:07,103 --> 00:16:08,698 the springs. 472 00:16:08,698 --> 00:16:11,139 That determines what stuff you get out. 473 00:16:11,139 --> 00:16:13,710 So afterward, we need a lot of changing 474 00:16:13,710 --> 00:16:15,699 to make it look beautiful. 475 00:16:15,699 --> 00:16:16,994 K so this looks really easy. 476 00:16:16,994 --> 00:16:18,954 But some of the pictures I'm going to 477 00:16:18,954 --> 00:16:20,794 show you took a long, long time to create. 478 00:16:20,794 --> 00:16:22,843 Because there was a lot of 479 00:16:22,843 --> 00:16:23,305 twiggling. Put a little 480 00:16:23,305 --> 00:16:25,345 bit more spring strength here. 481 00:16:25,345 --> 00:16:26,227 And more repulsive charge there. 482 00:16:26,227 --> 00:16:29,685 But when you look at this it's beautiful 483 00:16:29,685 --> 00:16:32,676 structure. And it shows you very naturally 484 00:16:32,676 --> 00:16:36,192 these clusters and when you stare at 485 00:16:36,192 --> 00:16:36,898 these structures 486 00:16:36,898 --> 00:16:38,150 you can really get some information 487 00:16:38,150 --> 00:16:39,176 about the underlying system. 488 00:16:39,176 --> 00:16:41,389 No matter where this comes from. 489 00:16:41,389 --> 00:16:43,988 Now let me show you some 490 00:16:43,988 --> 00:16:45,993 really beautiful examples. 491 00:16:45,993 --> 00:16:50,438 In much larger systems than this. 492 00:16:50,438 --> 00:16:51,942 This is a financial portfolio 493 00:16:51,942 --> 00:16:53,323 optimization. 494 00:16:53,323 --> 00:16:56,276 So this is one of the matrices you 495 00:16:56,276 --> 00:16:58,707 would have come up in one of the 496 00:16:58,707 --> 00:16:59,994 simulations or computer programs you 497 00:16:59,994 --> 00:17:01,810 would have in financial portfolio 498 00:17:01,810 --> 00:17:02,753 optimizations. This looks much better 499 00:17:02,753 --> 00:17:06,285 than what you would imagine. From the 500 00:17:06,285 --> 00:17:08,010 2008 problems. 501 00:17:08,010 --> 00:17:08,812 Right? 502 00:17:08,812 --> 00:17:12,905 How did we know that this was behind it. 503 00:17:12,905 --> 00:17:16,533 This one, is another type of program 504 00:17:16,533 --> 00:17:18,880 that we often have in optimization. 505 00:17:18,880 --> 00:17:20,142 Called "prodredic(sp?)" programming. 506 00:17:20,142 --> 00:17:22,480 It's the matrix from one of those 507 00:17:22,480 --> 00:17:24,074 simulations, here is the close up. 508 00:17:24,074 --> 00:17:25,620 So they're very intricate things. 509 00:17:25,620 --> 00:17:27,345 These are just two dimensional 510 00:17:27,345 --> 00:17:29,522 patterns. But of course, it looks a little 511 00:17:29,522 --> 00:17:31,587 bit three dimensional. 512 00:17:31,587 --> 00:17:33,231 You can also do these things in 3D 513 00:17:33,231 --> 00:17:35,311 but it is much harder to vision. 514 00:17:35,311 --> 00:17:39,469 This one, is from electrical engineering 515 00:17:39,469 --> 00:17:40,630 It's a circuit simulation. 516 00:17:40,630 --> 00:17:46,131 So we also call this the porcupine. 517 00:17:46,131 --> 00:17:48,212 And this is a close up 518 00:17:48,212 --> 00:17:49,876 It's not a super high resolution image, 519 00:17:49,876 --> 00:17:52,020 but it gives you an idea. 520 00:17:52,020 --> 00:17:54,507 This is another linear programming problem 521 00:17:54,507 --> 00:17:56,755 That comes from some sort of optimization 522 00:17:56,755 --> 00:18:00,170 problem. And I forgot which one this is. 523 00:18:00,170 --> 00:18:03,118 And this color is often the strength of 524 00:18:03,118 --> 00:18:06,549 connection. You can also use it in also 525 00:18:06,549 --> 00:18:07,899 different ways. 526 00:18:07,899 --> 00:18:09,897 And my friend, Tim, who created this, 527 00:18:09,897 --> 00:18:12,602 uses colors so the pictures look really 528 00:18:12,602 --> 00:18:16,827 nice. (audience laughter). 529 00:18:16,827 --> 00:18:17,997 So we can play with these colors because 530 00:18:17,997 --> 00:18:19,102 there is lots of different ways to color 531 00:18:19,102 --> 00:18:19,864 this. 532 00:18:19,864 --> 00:18:21,414 I have to admit, this is pretty nice. 533 00:18:21,414 --> 00:18:22,280 I'll show you some others. 534 00:18:22,280 --> 00:18:29,249 This one, is part of my field, it's 535 00:18:29,249 --> 00:18:38,188 a matrix of an ocean, of shallow water. 536 00:18:38,188 --> 00:18:40,358 So where the depth of the water is much 537 00:18:40,358 --> 00:18:42,372 less than the width of the area. 538 00:18:42,372 --> 00:18:44,569 Now, this doesn't nearly look as nice. 539 00:18:44,569 --> 00:18:45,757 But the matrix that comes out 540 00:18:45,757 --> 00:18:49,568 is very unstructured, but still, 541 00:18:49,568 --> 00:18:52,738 it almost has the same feel to it as 542 00:18:52,738 --> 00:18:57,383 water. That's of course why we made it 543 00:18:57,383 --> 00:18:58,970 green and blue. 544 00:18:58,970 --> 00:19:02,002 Now this is a close-up. 545 00:19:02,002 --> 00:19:05,028 This one is another linear programming 546 00:19:05,028 --> 00:19:07,582 problem. It's my favorite. It has quite 547 00:19:07,582 --> 00:19:09,783 beautiful structures. 548 00:19:09,783 --> 00:19:12,905 And this one is actually a social network 549 00:19:12,905 --> 00:19:16,812 Which we have labeled, the poppy. 550 00:19:16,812 --> 00:19:24,761 And here, where you see poppies 551 00:19:24,761 --> 00:19:27,788 of flowers, they're really clusters 552 00:19:27,788 --> 00:19:29,447 of friends. 553 00:19:29,447 --> 00:19:31,756 They are strongly connected friend 554 00:19:31,756 --> 00:19:34,147 networks, within this large social 555 00:19:34,147 --> 00:19:37,331 network. So you can have a lot of fun 556 00:19:37,331 --> 00:19:42,755 with this. Right? 557 00:19:42,755 --> 00:19:46,271 We played with something else as well. 558 00:19:46,271 --> 00:19:49,730 And I brought a poster. 559 00:19:49,730 --> 00:19:58,086 Because Allison was going to show her 560 00:19:58,086 --> 00:19:59,250 beautiful work, so I wanted to show 561 00:19:59,250 --> 00:20:01,595 that we also do nice things. 562 00:20:01,595 --> 00:20:05,561 Here is my artwork. 563 00:20:05,561 --> 00:20:11,768 Sometimes I ask people, what are they 564 00:20:11,768 --> 00:20:14,500 looking at, and they say I'm looking at 565 00:20:14,500 --> 00:20:16,896 some sort of network or graph. But 566 00:20:16,896 --> 00:20:20,337 this is the LCSH. 567 00:20:20,337 --> 00:20:24,525 Library of Congress Subject headers. 568 00:20:24,525 --> 00:20:27,274 So this is the library system that we use 569 00:20:27,274 --> 00:20:29,073 in almost all libraries of the world 570 00:20:29,073 --> 00:20:31,773 And what you are looking at are 571 00:20:31,773 --> 00:20:36,503 library main categories, and their sub 572 00:20:36,503 --> 00:20:38,293 categories, and how everything 573 00:20:38,293 --> 00:20:40,525 is linked together. 574 00:20:40,525 --> 00:20:45,786 Now the LCSH asked us in 2005, me 575 00:20:45,786 --> 00:20:47,555 and a group of my students, to help 576 00:20:47,555 --> 00:20:50,234 them understand the structure. 577 00:20:50,234 --> 00:20:53,527 So here are cataloggers who have 578 00:20:53,527 --> 00:20:57,173 worked on this categorizing 579 00:20:57,173 --> 00:21:00,753 for decades. But they have never seen it, 580 00:21:00,753 --> 00:21:11,819 they just put it the data. So we 581 00:21:11,819 --> 00:21:14,938 used exactly the same kind of program 582 00:21:14,938 --> 00:21:19,805 that you just saw. We call it the galaxy. 583 00:21:19,805 --> 00:21:26,321 I will make sure they get the link 584 00:21:26,321 --> 00:21:27,736 so they can play with it. You 585 00:21:27,736 --> 00:21:29,652 can go in and zoom in and click on 586 00:21:29,652 --> 00:21:32,058 the node and it will jump up 587 00:21:32,058 --> 00:21:34,436 and show connected notes, and this 588 00:21:34,436 --> 00:21:36,689 is a way to browse. What is also really 589 00:21:36,689 --> 00:21:40,412 funny here, is you see this whole mess 590 00:21:40,412 --> 00:21:43,104 in the middle, because it is very strongly 591 00:21:43,104 --> 00:21:50,718 interconnected stuff, we but in words 592 00:21:50,718 --> 00:21:57,917 where the connections were strongest 593 00:21:57,917 --> 00:22:02,871 But the library also used it to find 594 00:22:02,871 --> 00:22:07,100 lazy cataloggers. Because look at this, 595 00:22:07,100 --> 00:22:09,543 we call this a Supernova. This is 596 00:22:09,543 --> 00:22:14,301 Japanese Antiquites. And it has one main 597 00:22:14,301 --> 00:22:17,360 category, and a whole bunch of sub 598 00:22:17,360 --> 00:22:19,813 categories. But the catalogger did 599 00:22:19,813 --> 00:22:21,360 not interconnect the sub-catagories. 600 00:22:21,360 --> 00:22:23,491 So we have Japanese Antiquities, 601 00:22:23,491 --> 00:22:25,540 and 140 things attached to it, 602 00:22:25,540 --> 00:22:28,753 but how they related, he or she did 603 00:22:28,753 --> 00:22:30,130 not relate it. So the library can look 604 00:22:30,130 --> 00:22:34,051 at it and say, we really should do 605 00:22:34,051 --> 00:22:36,444 something about this. Because the more 606 00:22:36,444 --> 00:22:38,592 messy it looks, the better it is for 607 00:22:38,592 --> 00:22:40,609 browsing purposes. But we just 608 00:22:45,218 --> 00:22:47,703 thought it was a beautiful picture. 609 00:22:47,703 --> 00:22:49,539 This took a long, long time to create, 610 00:22:49,539 --> 00:22:52,272 we had to think about the colors, 611 00:22:52,272 --> 00:22:58,962 and how the nodes. 612 00:22:58,962 --> 00:23:00,337 And now we are looking at ways 613 00:23:00,337 --> 00:23:02,078 to put this in 3D, so you can 614 00:23:02,078 --> 00:23:03,270 truly fly through the galaxy. 615 00:23:03,270 --> 00:23:04,746 And you can do the same with 616 00:23:04,746 --> 00:23:07,854 Wikipedia, and other networks that you 617 00:23:07,854 --> 00:23:12,767 have. You can say let's go on a flight 618 00:23:12,767 --> 00:23:15,790 through my social network. 619 00:23:15,790 --> 00:23:17,871 And you can see who is stronger connected 620 00:23:17,871 --> 00:23:20,889 than you are. Anyway it is a lot of fun 621 00:23:20,889 --> 00:23:22,555 to play with. So I'll make sure you have 622 00:23:22,555 --> 00:23:24,247 this link, and I'll take any questions you 623 00:23:24,247 --> 00:23:27,376 may have.