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Anthony Bonato, "Dots and Lines: Hidden Networks in Social Media, AI, and Nature" (Johns Hopkins UP, 2025)
01:05:42
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Gregory McNiff
Welcome to the New Books Network welcome to the New Books Network. I'm your host, Gregory McNiff, and I'm excited to be joined by Anthony Bunato, the author of Dots and Hidden Networks and Social Media, AI and Nature. The book was published by Johns Hopkins University press in the US in May of 2025. Anthony is a professor in the Department of Mathematics at Toronto Metropolitan University. He is the author of a course on the web, Graph An Invitation to Pursuit of Asian Games and Graph Theory and Limitless Minds Interviews with Mathematicians. I selected Dots and Lines, Hidden Networks in Social Media, AI and Nature because it explains what networks are, how we are surrounded by them, and how they can help us in all areas of life, from literature and medicine to search and rescue and climate change. And candidly, the topic is just very cool even for those of us not steeped in graph theory. And we'll get into this shortly. Hello Hello Anthony. Thank you for joining me today to discuss your book.
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Anthony Bonato
Hi Greg, Happy to be here.
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Gregory McNiff
Anthony, why did you write Dots and Lines and who is the target reader?
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Anthony Bonato
So I'll start with the second question about the target reader first. I would say anyone who has curiosity, really anyone who has. Has some pretty minimal background in math. I'd say even high school math is enough to do this. As we'll get into later, there's not a lot of formulas and jargon in the book, so I think, really, anyone can read it. As to why I read it, well, I've been thinking about writing a book for the general public, focusing on networks for quite a long time. But what crystallized it for me was the COVID 19 pandemic. So I'm sure you did, and many others. We were stuck in our apartments, in our homes. I had a lot of time, and I was, of course, watching a lot of the news coverage and all of the stories on the Internet. What was interesting about this pandemic, as awful as it was, there was also just a huge amount of data that was being thrown around. I'm sure in past pandemics, say, like, for example, polio in the 50s, we didn't quite have that level of that granular upload data that we had available to us on a daily basis, perhaps even too much data. But I listened, like many of us, to epidemiologists and infectious diseases specialists, and what came across was this discussion of how the virus was spreading from person to person. And they talked about vectors and different methods of transmission. But one thing I never heard from these experts was a discussion of networks, which is my bread and butter. So networks measure interactions between objects. And for Covid, what you think in terms of networks would be the people, us being the dots and lines corresponding to close proximity. Like if I'm within, say, two meters apart. So, long story short, I started thinking about how Covid was spread. These long links like we saw originally, from Wuhan, China, to places like Rome and Tehran, spreading the virus and then the rest is history.
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Gregory McNiff
No, that's. That's a great opening. And, you know, I know a number of people have cited Covid as an opportunity to sort of do some research on interesting topics. And I think you and even some colleagues spent some of that time, and we can talk about that, looking into graph theory and developing some ideas around that. Just to start off, I think we're all familiar with networks, whether it's a telecom network of routers and switches or a social network. But you spend some time in the beginning, and it might be chapter zero, and I do want to ask why there is a chapter zero. It's. Or you see that. But what is a network in terms of nodes, edges, heavy tail. Could you give us like a more of a technical description of how you, how you define a network? Sure.
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Anthony Bonato
So networks come with two main ingredients. There are these nodes, which are the dots, which could be really anything you could imagine, and edges or links between the nodes. Okay, so these are the lines. So they're collections of these nodes and edges. And what they do, practically speaking, and also theoretically is they measure interactions between objects. So for example, you could have a bee pollinating a flower. The bee would be a node, the flower would be another node and the pollination would be an edge. Or you could have an account on Instagram and someone follows you and that is another edge. So they, they come up everywhere. And that's one of the amazing things about networks. They appear in every aspect of science and well beyond science too, in things like literature and history.
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Gregory McNiff
Yeah, no, I definitely want to get into that. You have some very interesting examples in current day culture. And just to clarify, networks are a subset of graph theory, which you are a scientist, an academic in. Could you briefly describe graph theory? Sure.
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Anthony Bonato
So graph theory is the mathematical study of networks. It's hard to say if networks are part of graph theory or vice versa. Graph theory stems back to 1736 with the work of Leonhard Euler. He was looking at this problem called the Konigsberg bridges problem, basically in showing that you can't cross these bridges in this town of Konigsberg at the time. But it's really grown over the centuries to a robust area of mathematics, like you'd study geometry or topology, other areas. In fact, one of the themes of the book I've mentioned that graph theory is this third pillar. So I'll back up a bit. Numbers, very essential in mathematics. Obviously it's one of the first mathematical things we learn. And after that, or similar to that, we learn about shapes, geometry and these two things very, very essential and describe a big swath of a lot of modern mathematics. But networks, I feel, are distinct for numbers and shapes. They measure interactions and they're yet another way that we can view the world. So graph theory is this academic field. People are publishing papers on it. There are lots of theoretical questions you can ask about it. And parallel to graph theory is network science, which is very applied. And you have people working in things like ecology, or you have physicists or economists working on elements of graph theory, network science and applying it to their fields.
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Gregory McNiff
Wonderful. As I mentioned, the first chapter is chapter zero. Zero is, I believe, a real number. But was there any significance to why you started it that way.
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Anthony Bonato
Yeah. So there's a bit of a story about that zero, I would consider it the first natural number. The counting numbers 0, 1, 2, 3 and so on. But several mathematicians and even non mathematicians have strong opinions about this. Whether it should start with 0 or 1, I think some people call 1, 2, 3 and so on. The whole numbers different from the natur numbers. But anyway, so the story was I was at a conference with other mathematicians and I have a T shirt that says that zero is a natural number. It actually says it on the, you know, the black background in white, sort of in symbols. And I walk into a room. It was sort of social coffee hour with other mathicians, most of which I didn't know. And I basically started a riot by walking in the room with that T shirt. So I. I had people come up to me say, yeah, I absolutely agree with you, zero is a natural number. And people I didn't know just come up to me and say, you're wrong. But it was great. We had a nice, lively conversation. And it's definitely one of those things. It's sort of a convention. Right. And so for me, again, to underscore that the book is really about math, I started at chapter zero.
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Gregory McNiff
No, that's fascinating. What is the Zachary. I'm sorry, the Zachary Karate Club Network.
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Anthony Bonato
Yeah, this is interesting. It actually came up recently with that. That feud between Elon Musk and Donald Trump. Yeah, I'll talk about that a little bit later. I'll just mention first about what the Zachary Karate Club Network is about. So Wayne Zachary was a doctoral student, I think, in anthropology in the early 70s, and he was looking at the social interactions of a karate club. It was a university karate club with about 34 people in it. What was interesting about the study was that a feud broke out. There was an administrator and an owner or instructor, and they had a fight. And what Zachary saw in real time, essentially, was a split, a split of the karate club network into two parts, one centered around the administrator, one centered around the instructor. And what was interesting about this, it may not sound too amazing, but it was the first time really that we thought about the emergence of communities. So communities represent clusters of nodes that are sort of linked in some way, that are more likely to link to each other than outside the community. And this has become a very big theme in network science, defining what a community is and how to detect them. And in fact, this paper by Wayne Zachary, which I think was published a little later in 1977 or so, became one of the most cited works in network science. I mentioned just, I'll say the thing about the Elon, Musk and Trump. So the few that they had, right. Musk posting those things on X. I wrote about this and talked a little bit about it, about how essentially within the Republicans or people who voted for Trump, it may represent a kind of split that we're seeing, similar to the Karate Club, centered around Musk and his idea with the America part.
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Gregory McNiff
Oh, that's interesting. I definitely. You spend a fair amount of time on Twitter accounts, obviously, including Trump and Justin Trudeau, and I want to get there. But you do introduce a concept, the Louvain algorithm. And I think that relates to this idea of grouping nodes into communities faster than humans. But could you briefly talk about that? It seems like that's an important tool for graph theory experts.
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Anthony Bonato
Yeah, yeah, it is. And you're exactly right. It takes a random seeming network and it splits it into parts, into chunks. And these are called communities, like I described. And Leuvain actually is named after a university in Belgium. It's not one of the authors. The Leuven algorithm essentially works by what we call modularity optimization. Very jargony sounding word, but what modularity is about is detecting how clumpy, how clustered your network is. So in a completely random network, you wouldn't expect these communities, these clumps, but in something like social media, where people are interested in various celebrities or themes, they would cluster around these. And so what the Louvain algorithm does is it detects those and it finds an iteratively through an algorithm, does the splitting. It will actually tell you what the communities are in a network. And briefly how it works. Not to get too technical, it starts with every node in the network as its own little community and it moves the nodes around. It says, okay, maybe these two should be in a community. And then it computes modularity. And if the modularity goes up, it's great. Let's keep those two together. If it goes down, it discards it and tries another pair. And it keeps doing this, sort of merging these nodes and eventually getting faster than any human could do. It will maybe thousands of iterations, it will actually start to clump the network into these communities. And as you referenced, like with Trump and Trudeau, we looked at this with regards to their tweets. Back then it was called Twitter when we looked at it and we found interestingly, there are always five communities in the keywords of their tweets.
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Gregory McNiff
No, that's fascinating and that's a great segue into the question of networks and social media. And maybe I'll dive right in there, because I did want to ask you, you cite that 5, I guess that 5 rule. What is the significance of 5, and does that vary or.
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Anthony Bonato
Yeah, it's interesting. We still don't really have a good answer to it. So, again, to back up with most of the accounts we looked at on Twitter now, we looked at Trudeau and Trump first, but we broadened our net to other politicians. In the end, we looked at something like 703 accounts on Twitter, mostly politicians, and scraped it at 562,000 tweets. And I say we, meaning myself and my graduate students at the time. And we found an overwhelming number of them, between four and seven communities, so averaging around five. And I'm not sure exactly why they even have five. It could be that the things people are talking about on Twitter X, they focus on a small number of things. Like, for example, in Trump's case, he often talks about Hillary Clinton. Right. That recurred often throughout his tweets. Of course, he's now on true social mainly. But we found this in other people, other politicians of all stripes in Canada and the U.S. we even did it with a small selection of random celebrities, like people like Kim Kardashian or George Takei. We found similar things as Tech5, and it could be the way language is structured that we really only talk about in a given time. Only a few things like our. You know, if you look at my Twitter account, it's mostly bad math, jokes, things like this. Right. So it's. I talk about sort of math mainly. Right. But if you talk to a celebrity or a politician, they could have various interests, and these can change over time, but it's always clustering around these five communities.
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Gregory McNiff
Yeah, that. That really was fascinating. And I should say you have the illustrations in the book. They're very high quality. They're color, and they present these. We should talk about network visualization at some point. But the book does a nice job of laying out these. I don't want to call them graphs, but illustrations for these Twitter clouds and the comparison. So really nice job there and good job on the publisher. Maybe if you could just talk a little bit more about social media. I think we're all familiar with Facebook. It's a social network. Are there any other particular dynamics around that in terms of. I think you talked about friends of friends, that thing. How do you think about social media as a type of network?
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Anthony Bonato
So it's really interesting to me because social media of course, much maligned these days. It represents an interesting laboratory in a way. Social networks are an old topic. They've been around for decades. But like I mentioned with Wayne Zachary, people actually have to do questionnaires. They talk to people and do the coding and try and figure out the social interactions between them. But what we see on social media, whether it's true or not with bots and so on, but you see a large scale interaction of people, you know, hundreds of millions of people interacting on a site like X or Instagram or YouTube, whatever it is. And so what we've seen in these networks, which we've heard about for decades earlier, really verified, I guess, on the large scale from a big data perspective, is many of these properties that you see, like, so you mentioned friends of friends are more likely friends. This is an old adage, folklore, but it goes back to this thing. I think it was studied originally around the sixties of social balance theory, that basically you have these triads, these triples of nodes, friends of friends are more likely to be friends. Right. So this is one of the adages in social balance theory. The other one is that enemy of an enemy is more likely to be a friend. And these two phenomena, they're kind of like folklore. They definitely drive a lot of behavior on social media. So transitivity, specifically these triangles. Right. You see a lot of these when you look at the network data. Another thing that you see in social media is the small world property, which I know we'll get into. But the small world basically means that there are short distances between short links or paths between different nodes. There's other things too, like high degree nodes. You see a lot of these. Like someone like Elon Musk's account, he has like hundreds of millions of people who follow him. That's not what you see in random networks. So social networks have their own structure. It's similar to other networks, but it's very specific to social networks.
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Gregory McNiff
Yeah, I absolutely want to get into the small world properties or I'm sorry, the small world networks. That was another really interesting conversation, the way they sort of occupied the space between highly structured and completely random. But before we go there, I just wanted to ask you about what you call the con scores. And you present that in the, I think the context of the Survivor TV show, which if you, Anthony, were to design a TV show, it feels like something you design. I mean, and I mean that because it's just so fascinating from a, I guess a graph theory or network perspective, the way it's structured.
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Anthony Bonato
Yeah. And so far, none of the producers of Survivor reached out to me or my collaborators about it. But again, Survivor, social game show, something like Jeopardy. It's been around for a long time, many, many seasons. I think there's been something like 733 different castaways, people that have appeared on it over 48 seasons in the US alone. And many other countries have their own version. But again, a bit like social media. It's a nice sort of sandbox to do some analysis and experiments. And there's data that you can find about Survivor online. There's a website called Survivor Wiki which has all of the seasons and who voted for who and who is the winner and so on. So myself and some of my collaborators, we looked at this network from a perspective of network science and discovered some interesting things. I think specifically around that thing you call the con score. A con score has a very suggestive sound to it, right. It has nothing to do with cons per se. It's to do with the common outneighbors, common out neighbors. So you have two nodes. If they both point to a third, that's a common out neighbor, right. So they point to that node and both of them share that link. So you can count the number of common out neighbors that a node has and how many pairs there are. And it is actually a metric, it's a measure within a network to tell us something about which nodes are more influential or important than others. And what we found was in Survivor there were a very large number. About 81% of the winners were in the top five Kant scores. And over all the 40 plus seasons that we looked at.
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Gregory McNiff
Yeah, no, that was a really interesting analysis you did. You know, again, going back to graphic theory, you talk about that mainly as doing mathematics for mathematics sake, but there are pure app, I'm sorry, practical applications for networks. And I think one of the most Obvious is Google's PageRank algorithm, which is really their secret sauce. Could you talk about how that, how that works and how they use that ranking to respond to searchers? I think, you know, prior to there, I think you mentioned some competitors, AltaVista, Yahoo was in there. They sort of took a different approach.
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Anthony Bonato
And I'll say that the PageRank was the secret sauce of Google a while back these days using AI. Now they use AI in their searches to narrow down things quite a bit. So PageRank was very popular about 20 years ago and there was a lot of innovation on it. But it's still, even though it's not used so much by Google, we believe it's still useful in network science, even in things like ranking NFL teams. So it's in protein networks, lots of different applications, network science. So the founders of Google, Brin and Page, they wrote a paper, it was around 2000, introducing this PageRank algorithm. And much like the Louvain algorithm, it is that it's an iter process. It calculates a score basically for every node. And the simplest way to think about PageRank is about popularity. So which nodes, which things in your network are popular or not? And if you think about that for a bit, one way to be popular is if a lot of other nodes point to you, right? So like, for example, Adobe Reader downloads page that that has a lot of different points to it, or some website like cnn, there are a lot of people who would link to that. But you could also not have many links to you, but have popular nodes linking to you. So popularity is infectious. For example, like I mentioned cnn, if they wrote a story about me or my book, then I probably have a lot more interest in links. So popularity is infectious. So these two ideas that many notes point to or popular nodes point to are the foundation of PageRank. And really what it is, I mean, I'm talking very general terms. If you want to work it out mathematically, it's a matrix. A matrix is just a chart of numbers, an array of numbers. And you can work this out for any network. You can calculate this. I mean, it's a bit tricky because there's lots of decimal places, and to calculate it requires typically some computer help, but you can do it. And from the matrix you can derive these spores. And there's a rabbit hole there. It gets into things like eigenvalues and the eigenvalue gap between the first and second eigenvalue and so on. The Peron Frobenius theorem, there's all kinds of mathematical machinery about why it works, but at its core, it gives you a score, which tells you the popularity of the page. And this was very important at the beginning of search engines, really, like around 2000, most search engines were using text as their basis for the importance of a page in your search. Like if you were looking up baseball, if baseball was repeated often on the page, like InfoSeek or AltaVista would use this kind of approach with sort of word dominance. But Google used PageRank, which was less sensitive to the words per se and more to the popularity of the page. And that made all the difference from their search at the time. And as we know what happened with Google, it Was became sort of the dominant search engine.
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Gregory McNiff
Yeah, absolutely. And I definitely want to move on to AI later. But you cited popularity. I think you know the Bacon number and I think the mathematical equivalent of the Erdish number. Could you briefly talk about what they are and how they reflect sort of the. The concentration networks or maybe hub and spoke or node and Edge?
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Anthony Bonato
Yeah, so interesting to me. I think the others too, this idea of the Bacon number. So Kevin Bacon, very popular actor in Hollywood, appeared in a lot of different movies. There was a game that came out a while ago where you could name two actors. So say I say Sean Penn and Kevin Bacon. You had to find either a movie that they both co starred in or find a path, a bunch of links of movies. So maybe Sean Penn was with some other actor in some other movie and then links to some other actor and that links to Kevin Bacon. So what's interesting about the Kevin Bacon number? So every actor has a Bacon number, right? If you were in a movie with Kevin Bacon, your Bacon number is one. Kevin Bacon himself has number zero. But it's interesting, amongst the hundreds of thousands of actors on IMDb that are listed there, almost all of them have Bacon number four or less. And when I say four or less it makes it sound that it could be four or so, but even two or three. Very, very popular. Even Albert Einstein has baked a number three. He was in some movie in the 1950s, had a cameo role. So it's a very. I think it's really interesting how you really have the small world property within the actor network, within people appearing in Hollywood. And as you mentioned about Paul Erds, not an actor per se, but a very well known mathematician, also very prolific. He wrote about 1500 math papers in his lifetime, which is a huge amount and had about 500 co authors. So you can develop a similar kind of number for Paul Erds, where Erds is number zero. His collaborators are Erds number one and so on. And again, most people have small Erds number. I'm Erds number two. I wrote a paper with someone who wrote a paper with Erds, but I didn't write a paper with Erds. Sadly, he passed away when I was doing my doctoral studies. I'm sure if I met him today, I probably would have written a paper with him. Yeah. So Erdos number, very similar again to Bacon number. You see small Erds numbers. And again underscores this idea of the small world within collaboration. And I'll just mention briefly there's the Erdish Bacon number. I don't know if you've heard of that, Greg.
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Gregory McNiff
Yeah, I think you cite just in the book, there are some academics who have been in newbies and I think Kristen Stewart might actually bake an Erdish number. That, that and the fact that Harvey Keitel is, quote, more bacon as a lower bacon number than bacon.
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Anthony Bonato
Right. Some people tried to figure out who's the most popular actor who's in the center of the Hollywood universe. And Harvey Keitel, I just watched A Taxi Driver recently, he was in that. He has like a average KEITEL Number of 2.9. Right. So maybe Kevin Bacon is not the center of the Hollywood universe. But I'll just mention the Bacon Erds number is the sum of the two. You take your Bacon number and your Erds number. Now, mine's infinite because I've never been in a movie. But as you mentioned, Kristen Stewart, she had a paper on AI a couple years back, AI applied to the movie industry and I think her Bacon Erris number is seven. So there's, there's a small handful of people who have a finite Bacon heritage number. We're really getting into the weeds now.
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Gregory McNiff
Yeah, no, it was, that was very interesting as well. Now I do want to dive a little more into this topic of small world networks because it seemed very cool. I mean, you talk about Proteo, Proteomics and this idea of wow, space. But could you tell us what small world networks are and what's fascinating about them for you?
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Anthony Bonato
Sure. Small rural networks go back a while. There was a study in 1967, I think it was Stanley Milgram, who did basically a chain letter experiment, found that between random people in the US So the idea behind the chain letter experiment, you had to send a, you were in one place and you had to send a letter to someone you didn't know many states away. So you had to send the letter to someone you think may know that person, and then they had to forward it to someone else who may know that person and so on. And what I think Milgram found was that on average about 5.5 links, 5.5 letters were needed to connect these random people. Now, Milgram's study, highly cited, a little controversial, not everyone believed it at the time. But interestingly, Facebook did a study much more recently with the huge amount of data they had, and they found similar things. I think it was like under. Under 4 links, about 3.5 or so Links between random people and Facebook. So this idea of the small world, meaning between two nodes in a network, you'd expect these short number of links connecting them. So I may not know Barack Obama directly, but there is a short path between me and him on a network like X. So small world networks, very pervasive, they come up not just in social networks, that's one of the main areas that they're studied, but also, like you mentioned, proteomics. Now, your listeners probably are familiar with genomics, study of genes and their structure, but proteomics, it's study of proteins, which are proteins, are really important macromolecules in the cells. Really, it's the next thing, proteomics and more generally, omics research. And you'd see again, within a cell, these proteins as nodes and their interactions, their biochemical interactions as links. You see these small paths, these small links connecting them. And this really is surprising in a way. And in a way it isn't. Because, like we saw with Kevin Bacon, I mean, even within proteins, there are sort of very key proteins in a cell. I'm not a cell biologist, but there are key proteins in a cell that sort of govern and really are in charge in a way and control a lot of the function of the cell. So, yeah, they're pervasive. They come up in lots of different directions. But it's interesting, when you think about the small world property, you say, well, okay, if there's short links between me and Barack Obama, I should be having coffee with him tomorrow. Which I'm not. Right. It's because essentially there's a geometry underlying social networks. And in the book I refer to it as Blau space. Sociologist Peter Blau, who discovered this. So the nodes are points in space, two dimensions, three dimensions or higher. And their nodes are close if they have similar characteristics. So blouse space really tells us that even though in the network side, myself and maybe Barack Obama are close, there's a huge social gap between us. Right. We don't walk in the same circles. Right. And that's because in the underlying Blouse space, which you don't see, that's kind of hidden, right. There's these long paths connecting us. So there's some caution when talking about small world, it's pervasive, but there's a lot of other things going on like Blau space and so on.
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Gregory McNiff
And one thing that's interesting and you sort of alluded to this patterns. And I want to read you a quote. You talk about comparable, comparable results. And I think you mean the average distance between nodes for the power grid and the neural networks of nematode worms. And that struck me because, you know, it's. Everyone's fascinated with patterns in nature, but the fact there's some kind of pattern or relationship or something unique or connecting the power grid network and the neural network of worms, is that random or is that something. Were you, were you surprised by that or was that just random coincidence?
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Anthony Bonato
I was surprised about it when I learned a couple decades about it ago. So like when I first learned about it, I was really amazed. Now I'm, I'm more comfortable with the idea, but I would definitely see anyone learning about this for the first time. Small world networks, very pervasive everywhere. You see them in things like you mentioned, like within nematodes. Right. But also in things like the cosmic web. The different galaxies are nodes.
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Gregory McNiff
Absolutely, yeah.
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Anthony Bonato
And if two nodes, like two galaxies are interacting by gravity, you have connections between them.
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Gregory McNiff
Yeah, that's almost. I'm sorry, I was going to ask you, was that an underlying law, like a mathematical law or principle? I mean, it's just odd the way it shows up so frequently in different environments.
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Anthony Bonato
Yeah, I think it is kind of a characteristic of most networks in the world. I mean, there are networks that are not small world. So for example, you can take cities, right, different cities in the US or Canada, and link them by roads. There are some long paths there. It's not necessarily a small world. But if you look at something like the airline network in contrast, where again, you have cities as nodes and whether there's a flight between one or another, you have these hubs, these high degree nodes, many links like Atlanta or Toronto, New York City, and they make it small world. So there's this universality of networks I feel is one of the most kind of enchanting things about them. You can see within different areas. Like I'm not a biologist, I don't have a lot of knowledge and say something like ecology. But in these different areas which people may view as quite disparate, using different tools and approaches, networks appear and they appear again and again. And really I feel it's the next wave. I mean, networks have been around for a long time. But this approach of using networks in different areas of science and again, even beyond science. It's really like the next approach, I think.
- [33:02]
Gregory McNiff
Yeah, I think that really comes across in the book. I do want to get into that where we could talk about proteins, but just a. The fact that the relationships helping us understand it and then the predictive capabilities as well. Even in the case of literature, for example, when two characters might meet before we've actually read the book, the network construct there could predict that. But as I said in the beginning, there isn't much math. But the math you introduce is key. You introduce a concept called a burning number as it relates to the optimization problem. Could you briefly talk about why that's important?
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Anthony Bonato
Yeah. So do you know what a meme is?
- [33:40]
Gregory McNiff
I do. My 18 year old nephew is probably laughing because I've mispronounced it as a mem.
- [33:45]
Anthony Bonato
Yeah, my mom's a seven. She says mimi, which is probably a good way to pronounce it actually. But I mention it because a grad student once asked me, like I talked to a student who was 25 and said, do you know about memes? They said, well, of course, I'm 25. Right. So people our age, maybe we are not as familiar with meme culture, but it's a big thing on social media. But memes come from Dawkins, I think it was 1976, right, Richard Dawkins. And this idea of memes. So it's some sort of behavior or image or dance or whatever it is that people replicate. So it's a replicating idea. It could be in culture or other contexts. So burning really talks about memes. It kind of models what memes do. So what do memes do? They start somewhere, right? They start at some node in a network and they spread. Right. So you see a meme on your timeline, it sort of reached you and it can reach all of your followers from that, if you like it, or if you retweet it or something. And then also it'll appear randomly in other parts of the network. So it starts somewhere, it's spreading, but then it'll appear in different, seemingly random parts of the network and spread as well. And that's really what burning is meant to model. So I introduced burning with some collaborators back in 2014 or so. We were all excited, we proved lots of things about it, wrote several papers about it. What we didn't know, burning actually came up in the math literature in the 90s. Well known mathematician and computer scientist Noga Alon actually was the first to write a paper about burning. Although he didn't refer to it as burning and didn't develop it. But burning is a mathematical model. I talk about it in the book. That simulates. Right. It's a simplified model for how memes spread. And on the other side, not just from the point of view of memes, you can look at contagion. So that could be social contagion in something like Facebook, but it also could be real contagion, like influenza or COVID 19. Like with the COVID 19 pandemic, you saw the epidemic start in Wuhan, China, and it spread locally there. Someone presumably took a flight from there to Europe and I think in Iran as well. Tehran had an outbreak early on, but then it. It went from Europe to New York City. And then after that, as we know, it just went everywhere, pretty much everywhere in the world. And so what you saw with the spread of COVID 19 is similar to what's happening with burning. There's a spread between different nodes, but then it appears in random places. In this case, from, you know, plane travel was spreading it throughout the world. And pretty soon, eventually, as it spreads, it's going to spread and infect everything. That's what happens with memes. Most everybody see the meme after some time. But also like with contagion, like influenza or COVID 19, pretty much all of us at some point were exposed to the viruses.
- [36:39]
Gregory McNiff
I mentioned practical applications earlier with Google's page rank. You also spent some time discussing how understanding networks can protect our art museums from theft as well as improve search and rescue missions in caves. I think you use. I know you're a member of a group and it's not spelunking. I can't remember the actual.
- [37:00]
Anthony Bonato
I have an ex member now. Well, I just didn't renew my membership. So I was a member of the National Speleological Society for this book. So you know the authors go to no length, right. Like we do anything to get our material for our books. So the long story was, I'll make it short. I needed a book. It was on networks applied to caving. This is a thing, Speleological Society, they do this. And I couldn't get it through Amazon, I couldn't get it through their website unless I became a member. So I became a member of the National Speleological Society. They sent me badges. It was nice. So in caves, you can think of the different locations of the caves as nodes. There could be like a room or a corridor that is a node and linking adjacent corridors or rooms are forming edges. Right? So two close Rooms that you can walk from one to another that would form an edge between those two nodes. And caving. These people, they go into these. I would never do this personally, but they go into these unknown caves and they search around and they try and map out the cave. And unfortunately, as we know, remember back there was that case of those students and an instructor who got lost, I think, in Thailand in a cave. There have been many cases over the years either with cavers or miners who've been lost in these caves. And there actually is applications of network science to help them. Again, I don't know if people lost in a cave are going to pull out a pad of paper and a pen and start working out the math, but there's this idea of path width that I talk about in the book. So pathwidth is. It's a technical thing from graph theory, but in essence it measures how close your network is to something linear. A path is like a linear lineup of nodes. You have nodes going from left to right. Now, most networks don't look like that. That's very simplified. But path width is some measure of how close you are to being a path. So someone said if, you know, if you squint and look at a node with network with small path width, it looks like a path. So as it turns out, to search these caves, if you're looking for a lost caver, is very much highly dependent on this notion of path width, which is a number. You can calculate it for any network. And if you do that, you know how many searchers to deploy. Sort of the optimal number of searchers you need.
- [39:19]
Gregory McNiff
No, that's a great example. And I think you do something similar for art museums. I'm forgetting you cited theft. So another interesting fact is somebody had the Mona Lisa. Right. For two years in their apartment.
- [39:30]
Anthony Bonato
Yeah, right. Someone sold the Mona Lisa a long time ago.
- [39:33]
Gregory McNiff
Yeah, I don't know about this, but similar concept for protecting art museums with cameras.
- [39:39]
Anthony Bonato
Right.
- [39:39]
Gregory McNiff
Like how many cameras need. Yeah.
- [39:42]
Anthony Bonato
So you can take an art gallery, could be a wild shape, doesn't have to be just a square room. Split it up into triangles. Put cameras in the centers of these triangles and they can see all around within the triangle. And effectively guarding an art museum boils down to a problems, problem in networks. Something to do with coloring. Like you have different labels of the vertices of the triangle, different colors like red, blue and green. And this allows you to look at, view an entire art gallery, make sure that no one's stealing the Mona Lisa with some specified number of cameras. Which the math tells you.
- [40:18]
Gregory McNiff
Yeah, no, definitely an important application. We've talked about helping networks understand relationships better. I want to move a little bit into the predictive powers. And here, and I think it may be a graduate student at your lab, one of them, developed, I guess, an algorithm for either Twilight or Game of Thrones. And, you know, the characters met in chapter two, but you could predict that they, I'm sorry, were introduced in chapter two, and you could predict they would meet in later chapters. Could you talk about networks and literature?
- [40:50]
Anthony Bonato
Right. So it's a very fascinating area. Kurt Vonnegut, I guess, is one of the first people to talk about this. In the 40s, he had a thesis that was rejected, actually, about the shape of stories based on the sentiment of the story. In a given chapter, whether in Cinderella, things start low, but then they go high and then back down and then go back up again. Or Oedipus, where things start low but then go down steadily in the story. So the analysis of literature from the point of view of networks and so on, this is not necessarily a new idea, but it crystallized, I think, a couple years back, a colleague of mine, Andrew Beveridge, looked at Game of Thrones and looked at the characters there and analyzed it from the point of view of networks. So the study that you're referencing, it was one of my students who looked at characters appearing in the text. So you have two different characters in an early chapter, and they may not be in the same scenes. And that's crucial. It was in Harry Potter first we looked. So I think, for example, we took the example of Ron, Ron Weasley and Baltimore, who, you know, critical characters in the books, all the books of Harry Potter, but they don't appear in a scene early on, but they're mentioned. They're mentioned different parts of a given chapter, say Chapter two, for example. But what we could do using AI and applications of machine learning networks, we could predict how many chapters ahead that they would link together. And I think it was seven chapters on average. So you'd have different characters introduced. And what the AI told us, using principles from network science, is that seven chapters over like 100 pages or so, we could predict that they were actually going to appear together in the same scene. And we did this for all of Harry Potter, all the books there. But we also looked at all the books of Game of Thrones and Lord of the Rings. We chose these big fantasy novels because they tend to be long and there's multiple books in the series. Right. So we had a lot of data points, a lot of different characters. Like in Game of Thrones, There are about 2,000 different characters that appear in all the books. So we did it across the board, all these. And statistically, it said, yeah, always seven chapters ahead. So way to spoil the story, I guess, right?
- [42:59]
Gregory McNiff
No, it's another rule that is multidisciplinary or cross, you know, environments. It was another, you know, interesting learning. Speaking of learnings, you introduced this concept of embedding and how it could reveal hidden relationships. And I think you referenced either eharmony or chatgpt there. Could you briefly talk about what embedding is?
- [43:22]
Anthony Bonato
Sure. So the analogy that I use in the book, I think is helpful is, is looking at the night sky, because it's something we can all relate to, right? So you look up at the night sky. There are no clouds, and you can see stars. You can see dots in the sky. But what our brains do and what we've done over the centuries is to create the constellations. So we see the dots and our mind, or various people would connect those dots to form the constellations, like the Big Dipper or something like this. So it looks like those stars that are in a given constellation are somehow related or connected, but they're not really. It's an interpretation that we've made on the night sky. In reality, those stars are in space, and they could be millions, billions of light years apart. And we're seeing the light from them. It takes many, many years to hit the earth, hit our retinas. And our brains are mapping essentially these networks, these constellations onto them. So that is a quick way of thinking about embeddings. So it takes a network, which is just a random tangle of nodes and lines, and it maps it into space, maps into some dimensional space. Now, the dimension could be three dimensions, two dimensions, like the night sky, three, four. In some cases, people even go to hundreds of dimensions. And what we hope to see from that is the different nodes, which are kind of in random position. Starting off, when you embed them, they can get close. And this idea of embedding is very powerful. Like I said, it was used to predict character interactions in things like Harry Potter, but practically, it's also very useful in things like Recommender Systems. So Netflix, Spotify, even Uber Eats, they talk about how they use graph neural networks and graph embedding to predict the kind of things that you would like to see from a given restaurant that you order a lot from or even suggest different restaurants. Right. So this idea of putting nodes in space can tell us a lot about the nodes. It's called lake prediction. And tell us maybe these two nodes, they're not linking right now, but maybe in the future they will link.
- [45:25]
Gregory McNiff
Yeah, yeah. That's another great segue into what my opinion is that it's a great book and the highlight is that chapter 8 cells interlink and here you discuss a little more about proteomics and protein to protein interactions, which you reference as PPI throughout the book. And what's fascinating is just how a number of disciplines, but particularly math and this graph theory, is coming together to map these protein relationships and really help us better understand medicine and even the brain. I, I know you have a concept, I think it's neurocomputing, I'm sorry, neuromorphic computing there. But could you talk about how networks are helping us understand, I guess our, our own brain and our own proteins in our body and medicine better?
- [46:12]
Anthony Bonato
So I'll start talking about a paper that Joel Cohen wrote 2004. It's an interesting one called Mathematics is Biology's Next Microscope, only Better. And Biology is Mathematics Next Physics, only Better. And in this paper he puts forward the idea that mathematics is going to revolutionize biology as it did physics, and we're really seeing the outcome of that now. There's a huge amount of interest in all aspects of biology using networks and just generally mathematics and data science and machine learning as well. But the example you mentioned of PPIs or protein protein interaction networks, these are really fascinating. I'm not a biologist myself, but I'm surprised to learn the importance of proteins in our cells. You can think of proteins essentially as cells like factories and the proteins as workers sitting inside the cell and they're doing various functions. And if there's some problem with their job or if a lot of the proteins fall off or die, you can get disease or the death of the cell. So understanding these protein networks, very important from the point of view of things like drug targeting and other things. And I talk about in the books and things like domination networks and pruning algorithms and how they've been used to find essential proteins. But it's really interesting the connections in biology. I mean, I'll just talk a little bit about them. I mean, again, I'm not a biologist, but I'm fascinated by the connectome which has our neurons as the nodes. A hundred billion neurons in our three pound brain and there are a hundred trillion connections there. So it's a massive network and it's the most complicated computing thing that we know in the universe, right so whether you think of the brain as a computer or not, everything we are, everything we know, the way we view the world, is all a function of our brains. And networks are increasingly playing an important role in connectonomics, the field which looks at this connectome, these neurons and how they interact. There are studies that talk about how network science can tell you more about brain disorders like Alzheimer's and Parkinson's and epilepsy. And there's studies that show that taking Parkinson's patients and having them exercise over a period of weeks, they did MRIs of their brains and their neural networks, and what they found over some period of weeks, regular exercise, would rewire their brains. So there's definitely an impact there in terms of the exercise, and you can see it within the network. So networks can help us sort of measure and analyze our brains and how they're interacting. And beyond that, in ecology. I was very surprised to learn about the extent of which network science has applied biology, or rather ecology. So in ecology, you could have different species as the nodes and how they interact. Either they call it exchange of carbon, like if one species eats another predation, or it could be just pollination. I mentioned earlier in the interview, there's lots of ways that species could interact. And there's all kinds of fascinating topics like connectance and nestedness and studied. And I was aware of a little bit about the things are called food webs, but not to the extent that I saw in the literature. There's a huge interest in networks in ecology.
- [49:35]
Gregory McNiff
Yeah, I know. You give an example. If a prey disappears, does the, you know, the. I guess the. Those who depend on the prey. I'm blanking on the word. Could also be impacted, that relationship between the two.
- [49:47]
Anthony Bonato
Yeah, very much. Yeah. Like there was a study with Martu people in Australia, the indigenous people there, and how they impacted the ecological networks actually had a positive impact on the ecological networks. So even humans. There's been much talked about how humans are having a negative impact on species and our ecosystems. But we also are part like humans, you know, we eat other animals and we interact with other animals. We have a part to play within those ecosystems. And I believe network science is going to help show us more about that connection and the impact that we're having.
- [50:19]
Gregory McNiff
Yeah, and I know you talk about climate change towards the end of the book. I want to get there. I just want to reiterate for anyone who's listening, chapter eight makes the book worth the price alone. And hopefully Anthony will, that chapter will be in the New Yorker at Some point. It really, it is just fascinating. I know you start to book out saying, I'm just in love with mathematics, it may or may not have practical applications. And chapter eight, you're thinking, oh my gosh, you know, we'll find we'll be able to map these interactions and really understand the relationships between the two. And it, yeah, it's just like looking at the forefront of medicine, I guess. But on that note, no conversation about networks is complete without referencing AI. You do cite AI in the book and I know you talk about moving from models, training the algorithm to training, I'm sorry, moving from I guess, rules based models to training the algorithm on real world networks for its predictive power. Could you talk about briefly the. The role of networks in AI?
- [51:17]
Anthony Bonato
It's massive. So I think like every other area of science, we're seeing a revolution right now with the applications of AI to network science and how we can predict things about the networks, whether links will appear, node classification, trying to figure out like which nodes are important, and so on. I mentioned a few different examples in the book. I've referenced one of them, the character relationships in fiction. And there are also applications within network science to things like recommender systems. So I touched on this earlier through embeddings. So there's an algorithm called node2vec. Node2vec, more jargon, I guess. But it was introduced a while ago, not too long ago, 2016 or so. I'm not sure exactly the date, but it's introduced by some researchers in Stanford and it's become the de facto algorithm. There are many embedding algorithms, but it's a machine learning algorithm. It uses AI princ. But essentially this idea of embedding networks in space and trying to figure out how that embedding works. So I give you a random network, just like a big tangle of nodes and lines. How do we do an accurate embedding and AI? I mean, much has been written about AI and how it works. One thing that AI does really well is comparisons. You can give it a huge amount of data. You know, the example of like the, I guess the blueberry muffin and the chihuahuas, right? So that if you look at pictures of Chihuahuas and you look at blueberry muffins, they do look kind of similar, right? If you put enough of them together. But this is a classic example of the sort of the challenges with training data and AI. But essentially what AI does at a highly simplified level is just it looks at a huge amount of data and it makes comparisons and says okay, well this is, you know, this is a chihuahua and this is blueberry moth. And trying to make those decisions like our brains do that, right? We do this, but AI does that in this machine based algorithm way. So node2vec, getting back to that, it uses these AI principles to do these embeddings and it's found a ton of applications and there's, you know, within embeddings themselves there's lots of applications. Like I mentioned the things like Recommender systems and things like Spotify and Netflix, but also in things like biology trying to determine, you know, like protein folding. People have won the Nobel Prize about, you know, research related to AI and biology. So there's, there's lot, lots of applications we're seeing not just within network science, but within all of mathematics and science.
- [53:41]
Gregory McNiff
Yeah, no, Ed, it was fascinating the way you set it up and what we could expect in the future with the combination of AI and our understanding of networks. But perhaps the only thing bigger than the popularity of AI and all consuming is Taylor Swift. Could you. The Taylor Swift economy and the network. I guess the network effects in that.
- [54:04]
Anthony Bonato
Yeah, so I talked about that related to this paper from 1928 by Frank Ramsey. So there's a field of research in mathematics called Ramsey theory, named after Ramsey, basically saying that complete disorder is impossible. So if I have a large system, it will emerge with some order. Like again, like looking up at the sky, you see constellations. If I have a Jackson Pollock painting, I look at the dots and the lines there, I might see some pattern. That's what our brains do. But mathematically speaking, you tend to see these patterns always. And from the point of view of Ramsey theory, what you see are these cliques. Cliques are collections of nodes that all link to each other. And anti cliques, the opposite, where you have a collection of nodes but no links at all. What Ramsey's theorem says in a nutshell is that you will always, if your network is big enough, you're always going to see these cliques or these anti cliques. So what does this have to do with Taylor Swift? I know your listeners are wondering right now, what is the connection here? So trading bracelets. So if you're familiar with the friendship bracelets with the Swifties. So before the concert, after the concert, they have a large number of these bracelets and they trade them, right? They like some bracelets someone else has, you can trade. So if you take six Swifties, say before a Taylor Swift concert. I know she's on hiatus, but she has a new album coming out. So you never know. There's going to be maybe another tour. You take six Swifties, you're always going to see three people who will trade the bracelets or three who don't. None of them trade with each other. So a clique of order three or anti clique of order three, you're always going to see that if you have six or more swifties. And this is because The Ramsey number R6 is exactly 3. So Ramsey numbers, a very big topic in theoretical graph theory, has lots and lots of interesting applications of them and much that we don't know about the Ramsey numbers, even R5. So how big does a network have to be? So you'll see a clique or an anti clique with five nodes. No one knows the exact value of that. In fact, it's somewhere between 43 and 46. It seems like a small number of cases. Right. We should be able to figure out. But even the largest computers and the most powerful algorithms cannot be applied to solve this. Just combinatorial explosion. Too many cases to analyze. So there you go, from Taylor Swift to cutting edge graph theory.
- [56:32]
Gregory McNiff
Theory, another great section in the book. And I think you say we're still looking for that, I guess the higher end or trying to figure out is it the ceiling on the Ramsey number or a proof for the Ramsey number.
- [56:45]
Anthony Bonato
Yeah, the bound. So there's, you know, we can't calculate what the Ramsey numbers are exactly like. We know R3 is 6, R4 you can figure out to be 18, but we don't know R5 or any of the higher Ramsey numbers exactly. So in math, when you don't know what something is, you try and approximate it. And that leads to this topic of bounds. Bounds, they're kind of like fences saying that, you know, this Ramsey number could be big, but it doesn't get bigger by this, bigger than some other function that we know well. So the upper bound that was known for a long time, that was the best bound, was due to Paul Erds and zegras from over 90 years ago. And it was 4 to the power K. So K RK, the KTH Ramsey number. I know mathematicians use K and N a lot, but RK is at most 4 to the K. And there was a recent breakthrough, it was 20, 23 that improved 4 to the K to 3.993 to the K. And you tell people this and they say, well, you know, big deal. Or it's not a big improvement. But it took 90 years and a great deal of work and innovation to get from 4 to the K to 3.993 to the K. And this is the state of the art now. We don't know any other better upper bounds on the Ramsey network.
- [58:01]
Gregory McNiff
Thank you. We've talked about how networks can help us understand our environment and predict certain behaviors or relationships. As I alluded to. You end the book on climate change. Could you talk about how networks can help us understand and make the right decisions around the climate?
- [58:18]
Anthony Bonato
So this is a nascent area of a network science. So there are many applications of networks to climate science. One that I'll talk about is teleconnections. So tell the connections, what is it? So El Nino La Nina probably heard of this or maybe the listeners have heard about it. So what happens is you have a warming of the waters of the, I think in the Pacific Ocean, for the case of El Nino, around the equator, and that has an impact on the weather in far off places like in New York City or where you live. In Toronto, where I live, you can have a milder winter from El Nino. And that speaks to teleconnections. So what are teleconnections? You have regions of the earth as nodes, as the dots, and if they have some sort of interaction with another adjacent region, either through weather patterns, temperature, atmospheric pressure, something like this, you have edges. And what people are just beginning to understand, we don't really have the full picture yet with the teleconnections, is how this network of teleconnections impacts the climate. So right now we're experiencing climate change, you know, huge topics, everyone's interested in it. What is the impact? What is the impact of humans and so on. We're still trying to gauge this. You know, teleconnections play a critical role in this as we see through things like El Nino and La Nina. And this is, like I said, a, you know, emerging area of study within meteorology. But another great application of network science.
- [59:45]
Gregory McNiff
Before we end the interview, could you talk a little bit about your own research? You alluded to the burning number and I, I think you also contributed to a paper on connecting color to localization and network. Maybe about your own research.
- [59:58]
Anthony Bonato
Sure. So I work in many different areas of graph theory and network science. I'll mention two. So one is a recent paper generalizing burning. So as you mentioned, I was one of the first people to study burning. And there's another idea you can pull out of burning called cooling. So cooling is like the complement, the complementary process, instead of we're burning, where you're trying to spread this meme or contagion as fast as possible. In the case of cooling, you're trying to slow it down. Same rules of how it spreads, but you're trying to slow it down. So cooling is much slower than burning. And you can study it. You can study something called the cooling number of a network. But some recent work we're really excited about is called liminal burning, which is in between cooling and burning. And I actually referenced this in the book. I didn't talk about liminal burning. I said, it'll be fascinating to study. And, you know, actually after the book came out, we figured out a model, figured out a way to study this. And liminal burning. There's a lot of things we can say, but many, many unknowns. Even with something like a path, like a linear collection of dots and lines, we don't fully understand liminal burning. Right. So there's a lot of things left to do there. And another thing that I'm working on, we have a paper that will come out soon, is about applications of network science to anti money laundering. So I'm not a banker, but I had a student who worked at a bank and had a bit of a hiatus from his bank to come and do graduate work with me. So he wanted to work on banking networks. And we found some data that was really amazing. We were able to get access to some data, different accounts, and the transactions between the accounts. It was all anonymized and we couldn't publish it, you know, publicly. It was. There's some rules about how we could use the data. But what we did was we looked for these cycles, cycles within the data. So in money laundering, if people don't know, it's putting money into the system, into banks, and sending it to various accounts, maybe it's gotten from, you know, whatever gangs or some nefarious purpose, but you're trying to essentially clean the money by moving it through various accounts that have it come back to you. And what that does, effectively, it creates a loop, a cycle, a directed cycle, as we call it, inside of a network. So within this banking network that we had this data, we analyzed it, there were millions of nodes and millions of edges, and we narrowed it down to about a couple hundred nodes which were suspicious that were in these directed cycles, like directed cycles of length, around three or four. Now, we don't know what we call the ground truth labels. Are these really people doing money laundering? We have no idea. But it suggests a method using this idea. First we broke into communities and then we looked at the networks. And some recent work is using AI and machine learning to even get a finer analysis. But again, using networks in this context of banking, it's never been done quite that way before and we think it might have some application, might have some use.
- [62:55]
Gregory McNiff
Yeah. And you touch on this in the book. It sounds like you've done some research since then. But you do touch on money laundering and banking in the book, right?
- [63:04]
Anthony Bonato
The first paper. Yeah. The second one coming, we'll have a little deeper analysis.
- [63:07]
Gregory McNiff
Yeah. Well, that concludes our interview. Again, the book is Dots and Lines, Hidden Networks and Social Media AI feature by Anthony Bonato. Highly recommended. You will definitely have a greater appreciation for network in your life as well as how they impact our lives and how they can improve it. Anthony, thank you so much for your time and thanks for writing a great book.
- [63:28]
Anthony Bonato
Thank you so much for having me.
- [63:30]
Gregory McNiff
Absolutely.