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Dots and Lines: Hidden Networks
1.2k Plays2 hours ago
In this conversation, Autumn and Dr. Anthony Bonato explore the fascinating world of networks, discussing their significance in various fields, including mathematics, social interactions, and even the spread of diseases like COVID-19 in his new book Dots and Lines. Anthony shares his journey into network science, the importance of understanding networks in everyday life, and how they can reveal hidden connections. The discussion also touches on popular culture references, such as Game of Thrones and Survivor, to illustrate the practical applications of network theory. Ultimately, the conversation emphasizes the need to embrace mathematics and recognize the pervasive role of networks in our lives.
Takeaways
- Networks are fundamental to understanding complex systems.
- The COVID-19 pandemic highlighted the importance of network science.
- Mathematics encompasses more than just numbers and shapes.
- Personal experiences can lead to profound realizations about networks.
- Everyday life is filled with examples of networks in action.
- Game of Thrones and Survivor serve as engaging examples of network analysis.
- The Bacon number illustrates connections in Hollywood.
- Erdős number connects mathematicians through collaboration.
Chapters
- 00:00 The Inspiration Behind the Book
- 03:38 Understanding Networks: A New Perspective
- 06:13 Networks in Everyday Life
- 08:28 The Power of Networks in Society
- 11:03 Real-World Applications of Network Science
- 13:32 Pop Culture and Network Analysis
- 15:38 The Bacon Number and Network Connections
- 21:53 The Bacon Number and Small World Phenomenon
- 26:34 Network Embeddings and Their Applications
- 31:04 Graph Theory: Patterns and Connections
- 35:11 The Importance of Mathematics in Everyday Life
- 36:57 Introduction and Curiosity in Connections
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Transcript
Introduction to Dots and Lines
00:00:00
Speaker
Imagine a world only of dots and lines, no color, no texture, no noise, just connections. Now, imagine that this bare bones world can describe Google's search engine, the spread of a virus, your social circle, and even the structure of the brain. Welcome to the strange, elegant universe of graph theory.
00:00:23
Speaker
I'm your host, Autumn Finaf, and welcome to Breaking mouth In today's episode, we're diving into Dots and Lines, the hidden structure of graphs by mathematics professor, Anthony Bonato, a book that reveals how this simple language of vertices and edges can unlock the mysteries of some of the most complex systems on earth.
00:00:47
Speaker
From mapping the web to tracking diseases, from tracing friendships to understanding the brain's wiring, Graph theory is the skeleton key to hidden patterns that shape our lives. Stay with us as we explore how something so minimal can explain so much and uncover the invisible architecture of the world around us.
Inspiration from COVID-19 Data
00:01:10
Speaker
Hi, Anthony. How are you doing today? i'm great, Autumn. So happy to be here. So happy to have you. Now, out of curiosity, what inspired you to write the book? Short answer, COVID-19.
00:01:21
Speaker
So, of course, all of us, I know maybe short answer, strange answer would start things off. um All of us lived through this global pandemic, very difficult situation. i was fortunate enough, no one in my immediate circle passed away or or had severe COVID.
00:01:36
Speaker
But like I, like everyone else, who was watching the news. And this the thing about this this pandemic that I thought was really interesting was there was an abundance of data. We were inundated with data. Pretty much every newscast, everything we'd read online told us about numbers, about how COVID spread, where it spread from, how fast it was spreading.
00:01:54
Speaker
But one of the things I noticed, given the abundance of data, was there really wasn't a discussion of networks in that, at least explicitly. So by networks, I mean, like title of the book, Dots and Lines.
00:02:05
Speaker
So we know that COVID began in Wuhan, China, and spread from there, obviously internally into the city and then to other parts of China. And then it moved to places like Rome and Tehran.
00:02:16
Speaker
And from there, it spread to New York City. And and then the rest is history. It's spread all over the world. Epidemiologists were on the news. disease Infectious disease specialists were talking. And they were talking about vectors, vectors of transmission.
00:02:27
Speaker
But I never heard the word network. And that really got me thinking, wait a minute, I'm looking at this pandemic. I mean, I'm not like looking at it as a medical scientist, but I'm looking it a mathematician. And I'm not hearing my language. I'm not hearing the language of networks. So I was really drawn to that idea. mean, I've been a network scientist, a graph theorist for almost 30 years.
00:02:46
Speaker
And I really wanted to write a book like this. Given the amount of time that we all had sitting at home, I thought, you know, maybe I could do it now and I could bring in that network perspective onto this pandemic.
00:02:57
Speaker
Now, tell us a little bit more about your background. What actually got you interested in graph theory?
Transition to Graph Theory
00:03:03
Speaker
Well, the story was I was a doctoral student working in logic, but I quickly realized that for, I guess, jobs and for my academic future, I need to do something a little bit more applicable, maybe something that was a little bit more mainstream.
00:03:18
Speaker
And graphs were just exciting things to study at the time that we're talking about in the late ninety s And I wrote a thesis that had to do with infinite graphs. It was highly theoretical. But a colleague of mine, Jeanette Janssen, around 2002, got me interested in the topic of applied networks, network science, complex networks like we call them sometimes. Yeah, so at that time in 2002, there was a real surge of interest in networks in the real world.
00:03:44
Speaker
Networks have been around for a long time. Social networks have been around for ages. But around 2000, people like Berbacy and Albert, the physicists, there was a ah very strong interest in networks and how they apply to everyday life.
00:03:58
Speaker
in all kinds of technological networks like the web, in diseases and protein networks and so on. And around that time, I got really excited by models for networks. And the rest is history. I've been working on the topic ever since.
00:04:11
Speaker
I hate to say this to anyone who else is on the podcast. This is actually one of my favorite topics. So the fact that you ended up covering this so elegantly for Dots and Lines, I am actually thrilled.
Networks as a Mathematical Pillar
00:04:24
Speaker
Now, for people who think math is just numbers and formulas, how do networks offer a different way of understanding the world? It's one of the themes of the book that there are two pillars right now in mathematics. I'd say if you want to incredibly oversimplify things with modern mathematics, it has to do with numbers on one hand.
00:04:44
Speaker
Numbers, we learned them as kids. Obviously, they're the foundation of mathematics. And shapes, geometry. Those two parts are the, I guess, mainstream elements. of modern mathematics. And there's ah huge amount of work that's done. All the different fields relate to, you know, mathematics of one of those things in some way.
00:05:01
Speaker
Of course, I'm greatly oversimplifying it, but I'm just trying to put that forward. And also in terms of our everyday experience for non-mathematicians, which is obviously like most of the world, they learn numbers first and they learn about shapes. You know, here's a circle, here's a square.
00:05:15
Speaker
But to me, networks are also as fundamental as networks and shapes. They're just not as discussed as much. right So networks, really what they do is they they measure our interactions.
00:05:26
Speaker
So right now we're having a conversation. In that social network, there's a link between us. Now, I'm in Canada, you're in the U.S., there's like a long geographic link connecting us. In the case of like a pandemic like COVID-19, it was the close proximity to others that spread the virus. But networks appear everywhere.
00:05:42
Speaker
They appear in your garden, just walking around. Or if you're in a park, you can see bees pollinating flowers or various species interacting or some animal eating another animal or something like this. They call them food webs.
00:05:55
Speaker
They're just everywhere. And that's become almost a cliche statement that networks are everywhere. But it really is true. And there have been books written about networks in the past for a general audience. I wanted to update it, do something really quite recent and talk about all the myriad areas. I mean, I'm a mathematician. I'm not a biologist or, you know, an engineer. But there's so many different areas, so many things that I myself learned writing this book.
00:06:18
Speaker
I was very excited to get this into the hands of people. I think it has a lot of very practical pop culture applications in here. You talk about everything from Twitter to Meryl Streep, Taylor Swift.
00:06:33
Speaker
So you describe networks essentially as a lens that reveals hidden connections in everything. Now, what led you to see the world through that lens?
Social Networks and Human Connection
00:06:45
Speaker
The story I told in the preface of the book, I don't like calling a preface because no one reads prefaces, right?
00:06:49
Speaker
But ah the story that I told was i was at the ah Caribbean Carnival in Toronto. It's a huge ah festival. Lots of people from the Caribbean or people who've immigrated to Canada of from the Caribbean celebrate. They have these incredible costumes with feathers. Maybe you've seen this before.
00:07:03
Speaker
ah They do one in Rio as well. I don't know if they do something like that in the U.S., s but... ah In Toronto, it's every August. It just happened last weekend. It's it's really big and bold and and colorful. Lots of flavors, really amazing things to to go to to do on a weekend.
00:07:16
Speaker
So I was there and I was walking around and I noticed I was studying social networks at the time. This was around 2004. And I realized when I was walking around the Caribbean Carnival on the side, I was with a lot of people on the side of the road where the festival was going by.
00:07:29
Speaker
The carnival was going by. I realized at that moment I'm in a network that I have an individual as a node, a dot operating in this network. And there are people all around me forming little clusters of nodes interacting with each other socially. And ah the story I told the the book, I went up and petted a dog, cute golden retriever that i I saw. And its owner smiled at me and we started talking. And I realized i was forging a connection. i was forging one of those lines in that network.
00:07:53
Speaker
So obviously I've been thinking about networks for many years before that, but that particular incident I remember very clearly, and I remember telling like my friends, like, oh, I had this great realization, I'm in a network. And they were like, okay, Anthony, that sounds interesting. But um these were non-math friends, right, people who are not mathematicians.
00:08:09
Speaker
But it it was really like a light bulb that went off in my head. I mean, many things in mathematics you study from an intellectual point of view just as an abstract thing. But that moment I realized, you know, being physically present, I was in this network.
00:08:22
Speaker
And then one thing led to another. and I'm starting to see networks everywhere. Maybe this is sort of a character flaw now. i'm I'm seeing them all over the place. I wouldn't necessarily call it a character flaw. Once you start to see this, especially as a mathematician, even an artist, if you take a minute, step back, you will realize those little connections. And that's also some of the beauty of the world that we live in So think of even LinkedIn, right?
00:08:48
Speaker
You make a connection with somebody new and you would have never had that connection before. So carnival to making it part of your everyday life and seeing it through that lens.
00:09:01
Speaker
Maybe it's just a way of reflecting about the world that we live in and maybe part of our jobs. I think so. And I, you know, mathematicians often feel like they're divorced from reality, that they're not part of the larger conversation. But I think networks as this third pillar, I talked about numbers and geometries being these two core pillars of mathematics.
00:09:20
Speaker
Yes. i think I think networks are this emerging third pillar. they're They're related to numbers, they're related to geometry, but they're thing on their own. They're another way, another lens that we have to view the world. Wonderful.
00:09:33
Speaker
Now, your book moves between science, technology, and everyday life. How do you make network science relatable to a general audience?
Relating Network Science to Real Life
00:09:46
Speaker
That's the easiest thing and the hardest thing to do. So with a book like this, and I could fill it up with formulas, ah there's some jargon in there. I Some people have said that I use too much jargon and others say I don't use enough. I mean, I tried to find a happy medium where I had some mathematics. There's very little formulas there, if you noticed, right? I tried not to bog the book down full of formulas. You can put up pick up one of my other books or books by so many other authors and have that.
00:10:09
Speaker
I really wanted to explain network science on a personal level and try and explain it in a way that is relatable to people. It's very easy to do that in the sense that, you know, it's not very mathematically challenging to write something simple, right?
00:10:24
Speaker
But at the same time, to make it understandable to people. That is the the key thing. And that's something that I worked really hard to do. We're often surrounded by these invisible networks. Can you give us a couple examples in our everyday lives that people might overlook? An example that came up, I think it was during the 2016 election, there was the Cambridge Analytica scandal, if you've heard of that.
00:10:48
Speaker
So was a company called Cambridge Analytica that put out a pop quiz on Facebook, maybe dating myself saying I'm on Facebook. I'm not quite a boomer. I'm a Gen Xer. but So this quiz, it just asked innocuous questions about yourself, like your favorite color or something like this. And what people didn't know when they signed on to this quiz, it was giving access to your friends and their information, their demographics.
00:11:11
Speaker
So just to put it in context, Facebook has about a billion users. Your friend group may be, you know, a couple hundred people, maybe a hundred other friends that you have. But the friends of your friends, that could be much, much larger. It could be in the order of thousands, tens of thousands of people.
00:11:26
Speaker
And you don't know all those people, right? This is a a property in networks we call transitivity. Friends of friends are more likely friends. So this is something that definitely they exploited. And from, i think, around 270,000 people taking the quiz, there were about, think it was 87 million people that they had access to.
00:11:44
Speaker
And they took that information and they used it for, um you know, political marketing. So that was an example that really, I think, brought home the idea of the power of social networks and the small world network that I talk about in the book, right? Small world networks.
00:11:59
Speaker
You know, we are connected by short paths of links. That's something that is part of social media. It's part of our everyday life. And we don't normally think about that, but it's definitely has this importance and power.
Ecological and Fictional Networks
00:12:11
Speaker
Another example I'd give would be something as simple as looking in your garden, or if you don't have a garden, if you're walking in the park or in a forest. What you see are trees, plants. In my part of the world, do you see a squirrel, or maybe you have a dog, and birds. Where I live in Toronto, I'm about 10 minute walk to something called High Park.
00:12:30
Speaker
which is the largest park in the city. And I think there's a couple hundred species just living in the park alone, including coyotes. You have to watch yourself when you go for walks in my neighborhood at night, especially if you have a a dog or small dog. There are coyotes that come out sometimes.
00:12:43
Speaker
So anyway, this is an ecological network. It's a network of species, everything from flowers, grass, trees, squirrels, chipmunks, coyotes, insects that are interacting with each other.
00:12:55
Speaker
And one of the things I learned when I was writing the book is how big an impact network science is having on the field of ecology. Directly through these ecological networks, you have networks of predation. the ecologists give it a very clean sounding word like exchange of carbon.
00:13:09
Speaker
So when one animal eats another, they're exchanging carbon. So this is ah these are food webs. These are really powerful and interesting tools to study ecology. ah But there are also other networks that have a role. Example, something like um ah insects or birds pollinating flowers and the interaction between one species, say the bird or a bee and a flower, um a plant, there's an invisible line there.
00:13:34
Speaker
right So every time you walk now in nature, mean, think about that. You're surrounded by this invisible web. And that web is is ancient and it's really delicate also. It can be easily disturbed.
00:13:45
Speaker
And so networks play a role in understanding that, especially as we face down things like climate change and its impact on ecosystem. Now, just thinking about that, do you have a favorite sort of network that you've either discovered or ah topic that really lights you up?
00:14:03
Speaker
ah Can I give two examples? Absolutely. So one is the Network of Thrones, which I didn't discover. I didn't start working with at first. ah Have you seen Game of Thrones? Of course. Yes. Or you read the books?
00:14:14
Speaker
Yes. One of the biggest shows on television. This is ah dating ourselves a little bit because it's a few years ago, but I think most people know about Game Thrones, either through the books or you know through that the new one, which is called House of the Dragon or something like this. Yes.
00:14:27
Speaker
So anyway, ah in Game of Thrones, you have thousands of different characters. A bit like the Lord of the Rings, you have this very large cast of characters and they're all interacting. It's very much an ensemble television show or book.
00:14:39
Speaker
And um a friend of mine, a mathematician named Andrew Beveridge, with a student of his, looked at the social network of Game of Thrones. specifically in the books. Later on, he it became very popular and he went into all the books and all of the shows, the scripts and so on.
00:14:54
Speaker
How we form the network so the characters in the network are the nodes and two characters are linked if they're a short space in the text. So like I think it was 15 words apart. So what you do with that is form this network and it has a structure. You can understand its properties, analyze it through various sort of mysterious sounding terms like betweenness centrality or closeness or eigenvalue centrality. And from that, make predictions. And at the time, i think what Andrew did he took the third book, Storm of Swords or something like this. I forget the exact name of the book.
00:15:25
Speaker
And at that point, not all the characters were fully developed. But no spoilers, so i think it's an older book and series, but um at the time, people like Jon Snow and Tyrion Lannister, big characters, Daenerys Targaryen, Cersei.
00:15:38
Speaker
So through Andrew's analysis of that third book, Sansa Stark emerged as an important character, specifically because of the math. The math was saying that she's a character to look out for.
00:15:50
Speaker
And again, no spoilers, but if you've seen the series, Sansa emerges... Very prominently later in the series as a very critical character up to about the third book. She's a pawn. She's being married off between these great houses.
00:16:01
Speaker
But later on, she emerges as like an extremely important character. And it was networks, network science that that gave that intuition. and again, you know, this software that they ran was not necessarily reading the book.
00:16:12
Speaker
from the point of view of its meaning and its semantics. It was just analyzing network properties. And that's one that emerged. So that's become a favorite of mine. it's It's very visual. i have Andrew gave me permission to use it in the book. It's a very beautiful looking network with all the characters. And you can see the main characters sort of prominent with the lesser characters fanning out from them. So you're asking of my favorite networks. The other one that i'd I'd say also comes from pop culture and it's to do with Survivor. Okay. Tell me more.
00:16:39
Speaker
Now, Survivor is one of those game shows. Now, i'm I'm also dating myself here when I say at the time in 2000, it was the biggest show on television. The finale of the first season of Survivor had 50 million viewers. So it's it's changed dramatically. It's still a popular show. it's I'm thinking of it almost like Jeopardy.
00:16:56
Speaker
Jeopardy or you know Wheel of Fortune, these are still um ah popular shows, um but they they don't have as as much popularity as they once did. But nevertheless, Survivor is still cultural phenomenon. It's been with us for many, many years. There's been— 50 seasons? 48 seasons? 48 and about over 700 castaways. Wow.
00:17:15
Speaker
over seven hundred castas Wow. Yeah, so it's a lot of people. So you can take a given season, say, for example, the latest one. I don't remember the name of it, but they go to different locations around the world. They have to go to an island or some remote part and fend for themselves in the wilderness. And you can form a voting network from that.
00:17:32
Speaker
So what I mean by that is the individual castaways are the nodes. And every, if you've seen Survivor before, every week they have a tribal council and players try and vote each other off. So what you get is we call a directed network. You have lines between the nodes, but there's ah an edge with an arrow pointing. So if I voted you off Autumn, I would have an edge to you, right? Or vice versa.
00:17:54
Speaker
You're voting me off already. I won't vote you off, I promise. But we'd have an alliance. But in Survivor, what you find, I mean, obviously the beginning of of the season, there and there are no edges because no one has voted. But over time, those edges start to grow. and And what we did...
00:18:08
Speaker
myself and some of my grad students, we analyzed these voting networks. You can so find all the data on Survivor Wiki from all the past seasons. So you can download all that data. You can analyze it. We started to see some really interesting patterns when we looked at the voting networks.
00:18:22
Speaker
For example, this new metric came out. it's It's probably not completely new, but in the context of voting networks, it was. It's called the CON score or common out neighbor score.
00:18:34
Speaker
Which is this idea, it measures basically how much in sync a node is with other nodes in terms of their voting. So if you vote the same way as a lot of other people in the Survivor Network do, you're going to have a high con score.
00:18:46
Speaker
If you vote differently, then you're going to have low con score. Con score was very much correlated with people winning. I think 60% of the seasons we analyzed, the top five people had the highest con score.
00:18:59
Speaker
So con score, high con score meant that you were more likely to be in the finale or very close to the top. and also alliances. So these these groups of people, like I mentioned earlier, alliances, this idea that you have people who are not likely going to vote each other off.
00:19:12
Speaker
That's really, in essence, what the one way of thinking about an alliance. An alliance just means, in common of vernacular, just like-minded people who have some common goal or interest, like in a political party, say, or something like this.
00:19:23
Speaker
But in the context of Survivor, you have people who make an alliance, often hidden. They try not to advertise they're in an alliance. And And one of the key things about alliance is that you're not voting each other out. You're voting people outside the alliance off.
00:19:35
Speaker
But the fascinating thing is you can look inside that alliance and do math. You can analyze how many times yeah people in the alliance vote for each other. So there's this notion I talk about in the book of a weakness of an alliance.
00:19:46
Speaker
And it's a mathematical measure. It's essentially the ratio of the number of votes inside to the number of possible total votes inside. ah So that gives you a number between zero one and you can measure weakness. And what we found, again, is that The winners of Survivor tended to be in in alliances which had low weakness, i.e. strong alliances. So this is, it was fascinating to me at the time. There's been a lot of other work done on it. No one from Survivor's ever contacted me or my research team to talk about it.
00:20:11
Speaker
We don't claim to know how to, you know, predict the winners of Survivor. I mean, i think that's a little bit too far-fetched with the math that we have now. But we do have a lot of really interesting ideas, I think, and theories about how the game works. And interestingly, I'll say with Survivor, I mean, you can say, oh, a Survivor, that's just a silly, you know, social game show on television.
00:20:30
Speaker
But you can look at things like food webs from the same point of view. yes You can look at things like animal dominance networks. So like a pack of mongrel dogs living in Rome, Italy, and how they interacted. They gnash teeth or they bark at each other, something like this. And another thing I learned in the book is there are people actually spend their whole life analyzing ah Dogs like this, they will actually be in person or have cameras and be watching how dogs are interacting in Rome and write papers about it, animal behavioral scientists. It's fascinating to me.
00:20:57
Speaker
But there is a ah dominance network from that. And again, in in food webs, and animal dominance networks, even in networks of conflicts between groups in the world, you can find similar characteristics of survivor with regards to conscore and alliances and so on. Wonderful.
00:21:10
Speaker
Now... I have a couple of other examples that I saw when going through the book that might interest people.
Small World Phenomenon
00:21:19
Speaker
One is called the Bacon Number. Yes.
00:21:23
Speaker
But some may call it the Bacon Erdos Number. Tell us a little bit about that. Okay, so there are actually two numbers, really three numbers you can get out of that. So Kevin Bacon, actor in Hollywood, maybe you've seen his movies. He's well known for having done a lot of movies with a lot of different actors. So there was a game that came up a while back, probably two decades ago or so.
00:21:44
Speaker
It's called The Bacon Game. And you come up with an actor, like I mentioned, Viola Davis, and you have to find either a movie that they start together or find a ah path going from Viola Davis to Kevin Bacon.
00:21:57
Speaker
So maybe Viola Davis was in a movie with someone who was in a movie with someone who was in a movie with Kevin Bacon. So this idea of a Bacon number, you know, Kevin Bacon is number zero, right?
00:22:07
Speaker
People who've been in movies with Kevin Bacon are Bacon number one. If you've been in a movie with someone who's been a movie with Kevin Bacon but not directly have being in a movie with Kevin Bacon, you have Bacon number two and so on. So you get the idea. it gets larger and larger. There are hundreds of thousands of actors in Hollywood, at least that have been recorded in terms of IMDb and others.
00:22:26
Speaker
And beyond Hollywood. And interestingly, almost all of them have bacon number three or less. Almost all actors. Actually, i tried when I was writing the book to find people who had bacon number four or higher. There are some claims online that said you can even go as high as 10.
00:22:40
Speaker
I found some Russian actor in a silent movie from the 20s, I think, who had bacon number four. Even Einstein has bacon number three. Albert Einstein was in a movie in the 50s. And just a small cameo role. He was just talking on the telephone.
00:22:54
Speaker
So most actors in Hollywood have this really small Bacon number. And then there's another number that mathematicians look at specifically called the Erdős number, Paul Erdős. So Erdős, Hungarian mathematician, well known for many reasons. Many books and movies have been made about him.
00:23:09
Speaker
Fascinating guy. He has something like... um 1500 papers to his name with about 500 co-authors. So someone came up with this idea of a number similar to the Bacon number around Erdős. So Erdős has Erdős number zero.
00:23:25
Speaker
People who've co-authored with Erdős have Erdős number one. If you've written a paper with someone who's written a paper with Erdős, but not again directly with Erdős, you're two and so on. So I'm Erdős number two.
00:23:36
Speaker
I've written a paper with Peter Cameron, who wrote a paper with Paul Erdős. But around the time I was doing my PhD, Erdős died. I think it was 93 at the beginning of my PhD. So I never got a chance to write with Erdős.
00:23:48
Speaker
Apparently Erdős should write a paper with almost anybody he'd strike a mathematical conversation up with. Right. So if he was around today is very good chance I probably would have Erdős number one. So anyway, these two numbers, they seem very different, but they they speak to this small world property, a bit of jargon, right? Small world property.
00:24:03
Speaker
And that's this idea going back quite a long way. Stanley Milgram, the social psychologist, these these chain letter experiments he did in the 60s, that was something that that was really interesting to a lot of network scientists, still is. at The work of Watts and Strogatz around the turn of the millennium, um they had a very, very highly cited paper about small world property.
00:24:23
Speaker
But it's this idea that between any two nodes in in most networks that we know of, between two nodes, there are these short connections, short chain of links. So between two random actors in Hollywood, you're going to get this short path. Right. There is even I'll mention the Kevin Bacon, Paul Airdish number.
00:24:39
Speaker
Right. The the Bacon Airdish number, which is a little admittedly a little bit more arcane. Yes, it is. it is it's the sum of your bacon number and your erdish number now i think i'm pretty representative of this number i have ah bacon errorish number infinity because i've never been a movie so have no connections No path at all to the bacon node.
00:25:02
Speaker
My bacon number is infinite. My Eerdus number is too small, but my bacon number is infinite. So unfortunately, I'm not even on the charts. i know where to be But most people in the world are in that boat, right? I mean, most people haven't written a math paper. Most people haven't been in movie. But there are people like um Kristen Stewart, for example.
00:25:20
Speaker
I think she has Bacon Erdős number seven. I don't know she's been a movie with Kevin Bacon, but she wrote a paper. She was a co-author and of a paper on artificial intelligence and its applications to the movie industry a couple of years ago.
00:25:32
Speaker
So she's one of the most prominent examples of small Bacon Erdős number. There's a math physician you may know, Jordan Ellenberg, written lots of great pot math books. He has a finite Bacon Erdős number.
00:25:44
Speaker
But not many people do. it's ah It's a bucket list thing of mine. So, you know, any producers... Do you want a list item? We can check one off. Okay. We're going to be collaborating on a paper. Awesome. Awesome.
00:25:54
Speaker
How's that? I don't even know about it. ah We don't know yet, but we'll figure it out. Okay. So we'll be on both charts. yeah You'll have finite Anthony number. I'll have finite Autumn number. So that'll be awesome.
00:26:06
Speaker
Yes. See, there you go. It seems to be like a thing in Hungary, like just yeah studying math. I mean, studying math here, a little bit less common, right? yeah Especially for sort of people who are native to our countries, like newcomers and so on often do.
00:26:21
Speaker
and found But in Hungary, it just seems to be like a national pastime, right? so Yeah. But if we ever collaborate, we will both have numbers. Oh, wow.
00:26:32
Speaker
For Bacon and Erdős. Out of curiosity, was there anything surprising that you were discovering while writing this book? I have a ah chapter on network embeddings, which are really powerful tools, specifically in industry. Network embeddings come up a lot.
00:26:50
Speaker
And I was thinking of a way to introduce the chapter. And what i really strove for in the book is to find a way to do that with examples either from my personal life or or something relatable to people.
00:27:01
Speaker
And the example that I landed on ah was it was a bit of an aha moment when I i thought of it. It was in terms of the constellations. So we look up in the night sky. it's i think everyone can relate to this. Anybody who's old enough has looked up at the sky and seen stars in the sky.
00:27:16
Speaker
And what humans have done is we put these artificial lines on these stars. Like we look at the Big Dipper, or we see, you know, like this cup or something. Of course, those there's no real cup out in space, but it's it's a representation we have, an organization of these dots in the sky.
00:27:30
Speaker
And I realized what was happening was three-dimensional space where these stars occupy. projecting the light down into Earth and our eyes, our retinas picking them up, our brains are actually doing a network embedding.
00:27:42
Speaker
We're taking this three-dimensional group of stars. In many cases, they're light years apart. They're incredible vast distances apart. But in this the two-dimensional representation we have in the sky, they look very close. So what our brains are doing is we're taking these collections of dots and we're assigning constellations to them.
00:27:58
Speaker
And that that's something that we do, humans do. We look at random things and we create patterns. But again, the the novel thing was this idea of an embedding. So from three-dimensional space down to two-dimensional. And what happens more generally with network embeddings is you take some network that could be very complicated, very large, thousands of nodes, and you represent every node by a vector. If you don't know what a vector it's just an array of numbers, it's a list of numbers.
00:28:21
Speaker
And these are different coordinates that you put together and you get a vector, right? Whenever you talk about dimensions, it sounds like some sort of science fiction thing, but it's really not a big deal. You just assign some number of vectors, some number of coordinates to every node in your network in this case, and you have a de dimension.
00:28:36
Speaker
So what network embedding does is it takes ah arbitrary network, And it assigns it to ah collection of dots and lines in space. could be twodimenional three-dimenal It could be It could be 32, 128-dimensional, any number of dimensions. And but so that example really surprised me. I mean, network science has been talking about embeddings for a long time.
00:28:55
Speaker
But ah the power of embeddings, definitely. i was surprised to learn also that Uber Eats uses network embeddings. That's something they talk about in their blogs online. I was a bit surprised. They use something called graph neural networks, actually, with network embeddings. So yeah, it's really, really fascinating topic. Now, I know that this is a little bit of a, I would say almost like side
Social Media and Network Dynamics
00:29:14
Speaker
tangent. It's been a theme ah that I've, we've had, I've noticed in the podcast first time that I brought this out.
00:29:21
Speaker
We've been dealing a lot more with social media, especially with networks. Are there any hacks or tips that you have boosting scores or rankings? with no If I knew that, I'd have a million followers on X. But yeah, no, I mean, there are some common themes. So transitivity is a big one in social networks. That is one of the most powerful properties that you have in social networks. And what that is, is this old adage of friends of friends are friends. yeah So if you're friends with someone, you know, Autumn is friends with Anthony, Anthony's friends with Bob, then Autumn and Bob are more likely to be friends. so
00:29:54
Speaker
This drives a lot of behavior on social media. So having ah a good diverse friends group is maybe one hack you can do. um You know, definitely also high PageRank.
00:30:05
Speaker
So PageRank goes back to Google back in 2000. It was discovered it's basically a Markov chain or a random walk on a graph. But practically speaking, it's just a matrix that you can calculate for a network. And it's fairly easy to calculate that matrix.
00:30:19
Speaker
To analyze PageRank, you probably need a computer. It's lots of decimals and someone to play with. But PageRank, in a and nutshell, measures your popularity. So you can get popularity in two ways with PageRank. You can have a lot of links to you.
00:30:32
Speaker
So something as innocuous as like a Adobe Reader download page would have a lot of links to it, right? um Or, you know, um Oprah Winfrey could write a great review in her magazine in my book and then suddenly my book has lots of attention, right? that's So if you're if you're friends or you're getting links from popular people, so popularity is kind of recursive. If you're getting links from popular people, that makes you also popular.
00:30:55
Speaker
So maybe if you're trying to increase your social media ranking, get some popular friends. It's not all about having millions of friends, but maybe have a few influential ones. So one additional thing is graph theory. I would be remiss not to mention this as someone who calls himself a graph theorist. You see it on my T-shirt. The listeners can't see it. I have the Peterson graph there with the coloring on it.
00:31:14
Speaker
Yes. Graph theory is an ancient and robust field of mathematics with its own set of tools and methods, much like group theory or analysis or topology or under any other area of
Advanced Graph Theory Concepts
00:31:25
Speaker
mathematics. And when I was writing this book, I thought, well, I'm going talk about, you know, survivor and, you know, memes and things like this.
00:31:32
Speaker
But I realized that near the end of writing, i'd better include a chapter on and graph theory or right my colleagues are going to kill me. Right. so So I did. i mean, there are huge topics in graph theory itself that people maybe are not aware of.
00:31:45
Speaker
One big one has to do with graph coloring. So graph coloring is an assignment, as it sounds, of colors to the nodes of the network. You color every node. Sometimes you color edges, but say you color the nodes.
00:31:56
Speaker
And the the rule is, it's coming from maps, right? If you have two countries which are adjacent, which have have a border, you don't have them the same color or else they look like one country. So the rule for what we call proper coloring is that two nodes in a network that are adjacent, have a link, have to have different colors. And maps, a good example, you don't you need at most four colors to do that.
00:32:17
Speaker
So with any map you can possibly think of, you need at most four colors. Sounds intuitive, sounds simple. We don't really have a proof of that. We have a proof of that using a computer. So there's a lot of math behind it, but then at some critical point, you need a computer to do thousands of cases. No human can do it.
00:32:30
Speaker
So we don't have a fully logical proof of it. So the four-color theorem essentially, in a way, remains unproven, though we we believe it to be true. But Graph coloring, very active area, lots of people working on it. Another one that I mentioned in the book is is Ramsey theory. a Frank Ramsey, I wrote a paper in 1928, turned out to be one of the most influential papers in all of mathematics and science.
00:32:49
Speaker
It's this idea that if you have a large collection of objects, you will find patterns. Much like looking at the night sky, you see a mess of of stars, you will see patterns and constellations. I use the example in the book about Swifties trading bracelets.
00:33:02
Speaker
Maybe I'll get some pushback on that. People think it's not serious mathematics, but... But, you know you know, Taylor Swift, huge phenomenon. And, you know, her bracelets, people trading bracelets, that's a massive thing.
00:33:14
Speaker
So if you have a group of six people trading bracelets, you're always going to find three people who trade with each other or three people who don't. And that's because the Ramsey number R3 is equal to six. So the Ramsey number has to do, I won't get into all the technicalities, these cliques and anti-cliques, but it's about finding these large patterns. And again, very active area of research. There was a breakthrough a couple years back.
00:33:35
Speaker
on upper bounds for these Ramsey numbers. And we mentioned Paul Erdős already. ah Eridus and Zechras 90 years ago proved an upper bound of the Ramsey number rk, where k is just some positive integer, to be at most 4 to the power k. So an upper bound meaning you can't get bigger than that. Like any Ramsey number as big as it could be won't be bigger than 4 to the power k. Well, some mathematicians recently proved that you can do, I think it's 3.993 the So 0.0007. Did I get the right number of Yeah.
00:34:06
Speaker
so point zero zero zero seven they get the right number of zeroes and 007. So it's four versus 3.993. Yep. You can improve the bound by from four to the K to 3.993 to the K. And I tell people this and they're like, so what? Right. It's 0.007. But it does matter. It took 90 years to do that.
00:34:31
Speaker
Right. And I mean, it it doesn't sound very exciting, but if you have a large enough K, there's a big difference between 4 to the K and 3.993 to the K. And so even now, there's there's work done in Ramsey theory that's really groundbreaking and and very inspirational to lot of graph theorists. So...
00:34:47
Speaker
So that's definitely one of the things I want people to take away from the book and just in general about networks. There is a huge interest and application of the of networks to everyday life, but there also is a theory of dots and lines, graph theory, which has a lot of really beautiful mathematics. It does.
00:35:03
Speaker
Now, what is one big thing that you hope readers walk away from after reading your
Engaging with Mathematics and Networks
00:35:11
Speaker
book? Be a little less scared of math. I think definitely we live in an age where math and science may be not the top of people's minds right now.
00:35:19
Speaker
And there's a lot of questioning about the utility of math and science, but I feel like it's more important than ever. And mathematics is one of those things that's so foundational and fundamental just to human life. Some would argue it's even fundamental to the way we flourish as a society or civilization. I would definitely be in that camp.
00:35:33
Speaker
So i'm I'm not trying to be as pretentious to say my book does that, but I would say that to have an open mind, i i did not write this book for mathematicians. I mean, I'm happy if mathematicians read it, but I did not write it for them. I wrote it for the general public. I mean,
00:35:47
Speaker
Have some high school math, whatever. I don't really get into calculus or linear algebra, you don't have to worry about anything beyond that. But just have an open mind and be aware that, you know, I mean, the cliche saying, it's true, math is everywhere.
00:35:57
Speaker
And networks play this critical role that has been overlooked a lot. it's like I've described it as this third pillar. It's another way of looking at the world. And I think the story of networks is far from over. There's a lot more than networks can reveal.
00:36:10
Speaker
So that that's, I guess, probably my main takeaway. I think, so this is my personal opinion, if there's anybody who actually wants to explore this, even if it was for introductory college seminar, just to see without the complexity of what you're going to learn, this has so many more practical applications than I've seen in most textbooks.
00:36:35
Speaker
As a more condensed version, that speeds up networks into the modern age for Like I would say high school to early college audience for a reader.
00:36:49
Speaker
High schooler could read this book, definitely. And that was one of my goals. And I think it's just so relatable. Thank you so much, Autumn. Anthony, thank you so much for coming on Breaking Math.
00:37:01
Speaker
And for listeners, until next time, stay curious and be mindful of all of the different connections that you have in the universe. You might be one or two connections away from somebody famous.
00:37:16
Speaker
Until next time.