Episode 41: Steven Strogatz on Synchronization, Networks, and the Emergence of Complex Behavior

0:00:00 Sean Carroll: Hello, everyone, and welcome to The Mindscape Podcast. I'm your host, Sean Carroll.

0:00:04 SC: In today's episode, we're gonna tackle a perennial big question in the natural sciences: Trying to understand the sense in which the world is complicated. Now, I mean this in a very particular way. There's a way the world could be, which is completely chaotic, right? Like things just happening at random, there's no order or structure anywhere. There's another way the world could be, which is completely rigid and orderly, right?

0:00:29 SC: The actual world is somewhere in between these two things. There's sort of poles of chaos and order and we balance ourselves in between. That's one feature, but the other is, no one planned it, right? There's not a central designer that says, this is how things should be. The universe somehow organizes itself, and when I say the universe, especially, of course, here on Earth in the biosphere. So today's guest is actually a mathematician. Steven Strogatz has become very well known for his popular books on mathematics, but he's equally successful as a researcher, as I mentioned in the podcast, he's the author of a paper that has well over 30,000 citations, which makes regular physicists like me very jealous. And one of the founders of both the field of studying synchronization, spontaneous synchronization of different physical systems, and then out of that study came the study of networks and a particular small world networks, the way that many systems like the human brain or the internet are organized where it is neither just, you talk to your nearest neighbors nor you talk to everybody, but again, somewhere in between. How does this sort of mixture of chaos and order naturally come about through the working out of math and its equations?

0:01:45 SC: So, Steve's a wonderful communicator and a real pioneer in this field, so I think you're gonna enjoy this conversation. Let's go.

[music]

0:02:09 SC: Steven Strogatz, welcome to The Mindscape Podcast.

0:02:12 Steven Strogatz: Thanks very much, Sean.

0:02:14 SC: You are very well known as a popularizer of math, but also you're extremely well known as a doer of math. I tend to think of the math you do as almost kind of physics, it's not the pure math that is proving esoteric things. You're really hands dirty there in the real world. How did you come to think of yourself as an applied mathematician? That's not something that most people grow up wanting to be.

0:02:36 SS: That's a good point, and I didn't even know it existed. When I was in college, I thought I wanted to be a math major. And they had a whiz kid winning our algebra class for the people who had done very well in math in high school. So I found myself in there and I got crushed in my first semester of college. I got the worst grade that I ever got in college in that course, and it was the kind of course that is supposed to be a... Like to weed out the people who don't have the right stuff to be a pure mathematician. They were trying to decide who's gonna be the mathematician of the future and otherwise people... 'Cause a lot of kids in high school think they're pretty good in math and the idea was correct them very early on. I have to say to this day, I still resent that course, because they put a really quite weak teacher in there and so it was a real sink or swim experience for all of us and I definitely was taking in a lot of water.

0:03:35 SC: This would be a big deviation from the planning of the podcast, but that's okay. Actually, I'll that my best podcasts have involved deviations from the plan, but let me just remark that we live in a culture in physics and in math of this, weed them out, sink or swim. Let's ask them to do impossible things and see who suffers the least. And I personally think that that is not really the way to make the best scientists or mathematicians.

0:04:02 SS: Me, well, of course, I agree with you. As a weed or a would-be weed, almost weeded out. I think we lose a lot of talent with that kind of approach and there's no need to be that discouraging. So yeah, I do kinda wanna go along on this thread.

0:04:20 SC: Yeah. Let's do it.

0:04:21 SS: Because I think it's something a lot of people have experienced, this feeling of having once loved some subject and then getting discouraged in college, and it's not necessarily a bad thing as long as it turns out okay, you go in some other direction. So in my case I thought, well, maybe I'm really meant to do physics. And so, to respond to your earlier question, I took a lot of physics too, which was always a great interest of mine.

0:04:44 SC: I love the idea that physics is where you went after being weeded out, 'cause that is not how physicists think of themselves.

[laughter]

0:04:52 SS: No. Come on, I'm not saying that. I just thought math was my thing because I didn't feel like I had... Certainly I was a disaster in the lab. I knew that already from high school, and I felt like I could... My physical intuition never seemed especially great. I don't know, especially like free-body diagrams, things pushing on... Even action and reaction, I always found that very mysterious so...

0:05:15 SC: Well, it is. Don't feel bad.

0:05:17 SS: Yeah. But the math part of it, when once it was converted to equations, then I felt very secure, so I thought that was my natural strength, but then I started to think otherwise after this one very proof-oriented course, and I had never really had any experience with trying to do rigorous math proofs at that point in my life. So anyway, I enjoyed the physics course, especially enjoyed sophomore... Oh sorry, it was the spring semester of my freshman year, we took a course... We, I think, 'cause these other kids that were in the same cohort, out of the Electricity and Magnetism book by Purcell.

0:05:52 SC: Right.

0:05:52 SS: I know that you would know that book, and maybe some of your listeners will know that book. That's one of the most beautifully written textbooks in any subject, I would say, I think. Do you agree with that?

0:06:02 SC: Yeah, no, we didn't use it when I was undergraduate, but I know it since then. It's very, very good.

0:06:09 SS: What was so interesting to me about it was that it didn't just tell you, for instance, that the force between two charges is gonna be along the line between the charges. Let's suppose it's the electrostatic forces. How much does a proton pull on an electron. That's gonna be some force and in high school I had learned this thing, Coulomb's law, that the force would go along that line between that. But what was so interesting to me and Purcell is that he said, that follows from the symmetry of space.

0:06:40 SS: But that's not just a property of this force, that's a property about space, because if you imagine those two particles by themselves in an empty universe, otherwise empty, there would be, assuming that space is the same in all directions, there is no other preferred direction than this line between them, so it has to be along that line. And whether you believe that argument or not, I just was entranced by the idea that you could make an argument like that.

0:07:05 SC: Yeah, you can see how this would light up the mathematician inside you.

[laughter]

0:07:10 SS: So anyway, I've taken a big long detour from your original question, but the point was that I always liked math, but I did like the math of the real world, or at least the sort of semi-real world that we study in classes, like physics class. Not the real, real world, where I had to catch my clothes on fire or make sparks in the lab or something.

0:07:33 SC: I studied cosmology and quantum mechanics, so I can't make any claims to be too involved with the real, real world, but Applied Maths is an interesting field, right? I guess pure math is the alternative. It's really about proofs, right? Here's some math things. We're gonna prove some things. There's also physicist's math, which is this wonderful growing field, where we say things we think are true and we give some arguments for them but we don't actually prove things. And I think applied math, I have the feeling is somehow in between those poles.

0:08:04 SS: Well, it means many different things to many different people. So some see applied math as... Just to be clear, since we haven't really said about these things. There are these two roughly speaking, two flavors of math. So there's math that looks inward at itself, which is what I think of as pure math, and there's math that looks outward at the world, at the world very broadly construed. It could be oceanography, it could be the universe, it could be biology, whatever. But it's a math in the service of something else. Or sometimes it's math inspired by something else. Like something beautiful that happens in physics can be then the beginning of a new mathematical theory, and some people would still call that pure math, but I would think of that as part of applied math or maybe close to what you just called the physicist's math.

0:08:51 SC: Do you actually, in your daily, in your work, in your papers, are you typically proving a theorem? Or I noticed that one of the wonderful things about your work is that very often, you sort of figure out what the answer to the question is by doing some computer simulations.

0:09:07 SS: That's true. Yeah, I'm a no-holds-barred style. I've definitely been called a physicist by many of my mathematician friends. That they see me as using the spirit of physics, and I do from the physics culture that I absorbed, I learned that it's... That you should try lots of different things. You can guess, you can use intuition, you can use physical reasoning, you can use computers, you could do all kinds of different... Sometimes a proof is great if you can do a proof. But it's not the only game in town. Whereas in pure math, that is the end-all and be-all, is to prove theorems. I never, never, no. So, I'm definitely not one of those. For me it's, there's all kinds of ways of getting evidence and trying to understand what's true. I'm definitely interested in the truth about the equations or the mathematical models that I study, but I sort of actually feel a lot of affinity with artists, I must say.

0:10:07 SS: This could take us in an interesting direction, you have... From some of your writing, although we never met, I get the feeling you'll resonate with this. But I'm really interested in something that you maybe, you could call like mathematical impressionism that is, instead of making a photo-realistic painting of the world or the mathematical equivalent, I like to get inspired by something in nature or in the world, and then do something that's much simpler that's the essence of what's out there. Sort of like the way the impressionist painters didn't try to capture the details. They weren't interested in that. They were trying to, with their, either their dots or their bold strokes, convey the heart of whatever that was. And so I like my models to be, for instance. There was one paper that I did with my colleague, Rennie Mirollo, a mathematician at Boston College, where we were thinking about a certain problem, but it reminded us of fireflies that all flash in sync, all flashing on and off together. And yeah, what we did was not an accurate model of real fireflies. I would never claim that and we were careful in the discussion at the end of the paper to explain why fireflies were quite different from what we were pretending in this paper. Yet they were somehow like the essence of what the fireflies were doing.

0:11:26 SC: Yeah. To a physicist, that just makes perfect sense. It's obviously, there's math involved, but the idea of distilling down some complicated thing to the simplest possible description that captures something vital about it is what we do for a living in our own minds.

0:11:41 SS: So, I've always liked that attitude in physics, and I do feel it has a lot in common with an artistic impulse.

0:11:47 SC: Yeah.

0:11:47 SS: Let's just be clear, it's not like this is something that everyone would agree with. There are quite a few applied mathematicians who say, "That's not good enough. You should really try to make a model that is testable and the ultimate arbiter is reality. Like if your model doesn't do the right things in experiments, or doesn't match observations, even qualitatively, then you've gone too far. Then you've lost the essence."

0:12:16 SS: I sort of believe that. I agree that if you've gone too far, of course, you have lost the essence, but the truth is, if I'm really being honest about it, I care more about math than nature. Okay, that's a big dirty thing to say.

0:12:31 SC: No, go ahead. You're in a math department, it's okay.

0:12:34 SS: Well, that's right, and that's why I'm not a physicist, because I think a physicist ultimately does care about getting the science right, and there's this evil part of me that says, "If it's beautiful but a little bit wrong, I'm okay with that."

0:12:49 SC: It's still beautiful, yeah.

[laughter]

0:12:51 SC: The difference between a fine artist and a photo journalist. Absolutely, right? Or an illustrator?

0:12:56 SS: Well, maybe so. There's an old quote from the physicist...

0:13:00 SC: I'm trying to make this a flattering comparison from your perspective. I'm the illustrator, you're the fine artist.

0:13:06 SS: Well, it's just a matter of what people are interested in. So I think as long as you're honest about what you're trying to do, then I think it should be okay. It's just where you pretend that something is realistic and it's not, then you're gonna get in trouble.

0:13:20 SC: And the example you used is perfect, because I wanted to segue into this whole question of, to an example of what you've been working on through your whole life, I suppose, I guess as an applied mathematician themes come back and forth, rather than just working on one big problem for your whole life, but the fireflies are the example that you used in your book, Sync, to introduce the reader to this wonderful thing that appears all over nature in very different forms, where you had independent agents doing something, and yet they naturally fall into synchronization. Was that firefly paper that you alluded to there the first example of when you thought about that phenomenon?

0:14:00 SS: It's certainly one of our very earliest. I say our because, again, that person I mentioned earlier, Rennie Mirollo, I did a lot of this work with him. He was and still is probably my best friend. And we did a lot of this when we were just starting our career. So I was a postdoc, he was a new assistant professor. But anyway, yeah, it was very early. That was in the early 1990s, and to me it's part of a larger theme, a cosmic theme about order emerging out of disorder. I always found that spooky, almost theological, because in physics, in the simplest case, like, say, in thermodynamics classes, we learn that things tend to disorder if you have a closed system with no energy coming in or getting out.

0:14:42 SC: We do.

0:14:42 SS: And then, the Second Law tells us that system's gonna just come to equilibrium and it'll be disordered sort of maximally as we... In the jargon, the entropy will keep increasing. But yeah, we see fantastically organized structures all around us. We see civilization. We see biology organizing itself into ecosystems, and living things, cells. There's this fantastic fight between the forces that want, not want, but make the universe tend to greater disorder, and yet we see all kinds of what looks like spontaneous order all around us. So that's still somewhat mysterious from a physics perspective, although it's not to say we're contradicting the Second Law, that's not possible, that's a correct law, more or less. We could talk about that, but let's not go there.

0:15:34 SC: No, actually, this is one of my pet peeves. You have not said any of my pet peeve, but there's a pet peeve I have about how people talk about the Second Law, where they say what everyone says, which is that in a closed system, you tend to disorder and then they stop. And then they say, but look at these open systems you get order, but they can't help but telling themselves the order appears in this open system despite the Second Law.

0:16:00 SS: Yeah, no, no. There's no despite.

0:16:02 SC: Yeah, exactly. There is no despite. That's like saying the moon orbits the Earth despite the force of gravity, because you think that gravity pulls things together and the moon doesn't fall, so that's obviously despite the force of gravity. So I just react against that. Now, you didn't do it, you're not guilty of it, but I think that there's a lot of work yet to be done...

0:16:22 SS: Exactly.

0:16:22 SC: In figuring out how open systems do organize themselves and how entropy is actually part of the reason why that happens, not something that's being opposed by it.

0:16:30 SS: That's right. Exactly. Sure we're on the same page with that. That's the challenge of non-equilibrium thermodynamics, that it has to somehow be consistent with what we already know to be true from equilibrium or close to equilibrium case. And yet, the math and the conceptualization, the physics part of it, they're just very difficult, and that's some of the most exciting research going on right now in non-equilibrium statistical physics. So I...

0:16:56 SC: And apparently fireflies are an example of this?

0:17:00 SS: No. I don't approach it from a physics perspective, because first of all, I never took stat mech, statistical mechanics. I never went that far in physics, so I don't really know what I'm talking about here. But I did work at it from the perspective of someone interested in differential equations, which allow us to describe systems that change in time or in space or both. And so, something like fireflies, you can make a little abstract version of them, where you think of, well, what's going on with the firefly? If you have a single firefly that you're keeping alive in a jar that you've captured out in your backyard, then it will flash, and then its flash has stopped, and then it's sort of dark for a while, and then it somehow is building up its readiness to flash until it reaches some, we don't know really what's going on, but there's probably something happening in its nervous system where it's got a little timer, some little circuit, neural... I think literally, it's not known exactly how the flash rhythm works.

0:18:05 SC: Yeah, that's interesting, I was gonna ask, 'cause we really, I don't know, obviously, but it's still not even know, do you think?

0:18:10 SS: Well, we know something about the chemistry of what the substance is that sort of ignites and flashes. We know about the enzymes that degrade it, but I, loosely speaking, it's a bit analogous to something we all learn in the first year of physics where there's a charge between two, on capacitor plates that's building up. It's like in the jargon an RC circuit or a relaxation oscillator. Something is building up toward a threshold and when it reaches that threshold, then you get a sudden discharge, which in this case would be the firefly flashing or in the case of the RC circuit would be the capacitor dumping its charge and then starting to build up again as it charges up. So we have... Rennie Mirollo and I made this little model of, effectively, lots of little relaxation oscillators building up, discharging and every time somebody discharges, it kicks everybody else up a little bit closer to their firing threshold. Or it's...

0:19:09 SC: They get excited or something like that.

0:19:10 SS: Yeah, that's right. And that part we know is correct qualitatively. That fireflies not only flash, but they see the flashes of others and adjust the timing of their flash rhythm in response to the flash of others. We know that from experiments where you can use a pen flashlight to be like an artificial firefly, and if you periodically flash your flashing flashlight at this firefly... For instance, you can play tricks on it. You could flash a little faster than its normal rhythm, and then you'll see that firefly sort of scurrying to keep up, not literally moving its feet, but it flashes a little bit faster to stay in sync with your abnormally fast flashlight, up to limits. If you drive it more than like 20% faster than it wants to go, it can't keep up and then it falls out of sync.

0:20:00 SC: This is not helping the reputation of mad scientists and statistic experimenters on the animal kingdom, I'm afraid.

0:20:05 SS: They do a lot of worse things than this.

[laughter]

0:20:09 SS: Well, anyway, so we do know that there are interaction rules that can be measured through the kind of experiments I just measured, and so the hypothesis was always that what's happening in a congregation of fireflies is they're all both flashing and receiving flashes from others and adjusting in accordance with some rules. And so then the big math problem is... And it's something that transcends biology, they're not trained to solve a question like this. If every firefly is individually obeying the rules that you can measure, then is it the case that synchrony will emerge automatically every time in such a population? And that turns out to be quite a hard math problem because of the discontinuities in it.

0:20:53 SS: I said when there's a flash then... First of all, the flash is like a sudden impulse. Just a quick on and then quick off. And then also we imagine that the... To continue that capacitor analogy, that the discharge is very fast so the firefly sort of, so to speak, instantly goes back to its baseline before it starts charging up again. And so when you have jumps in any kind of system that you're trying to describe with math, unfortunately, the apparatus that we use in math assumes everything is smooth and continuous and doesn't jump around discontinuously. So it jumps...

0:21:29 SC: When you say the apparatus, these are all human constructions, right? Like the math we're used to doing is comfortable with this. So this is why you get paid the big bucks, to sort of generalize that way of thinking a little bit.

0:21:42 SS: Yeah, thank you. I didn't mean apparatus in a laboratory. I meant the mental tools that we use. Specifically, I'm avoiding using the word calculus, but that's really what I wanna say. That calculus is predicated on the idea that things can change, but they only change smoothly. They don't jump discontinuously from one place to another. Whereas these fireflies sort of act like they do, at least with these sudden flashes and the sudden response to flashes. And so what made the problem hard and why it was a research problem for us was that, how do you accommodate something that is on the one hand continuously charging up towards its threshold, but then at the same time, it can suddenly flash and suddenly discharge and go back? So it was this mixture of continuity and discontinuity that made the math abnormal and weird and challenging.

0:22:32 SC: And this is all started by, in a very scientific way in the sense that we saw the fireflies doing this. This had been known for a long time, the fireflies got into sync somehow, and I guess as a human being your first guess is that, well, there's a boss firefly. There's a clear leader that is telling them what to do. There's a conductor for the orchestra, but that didn't seem to pan out empirically. So your question was, could it be self-organizing? This coming into synchronization.

0:23:01 SS: Yes. Exactly, right, right. I did get a little ahead of myself there with talking about the math. It is a great story that people that lived in Thailand or Malaysia had known about this. They'd seen it forever, but the first Westerners to come to those parts of the world, like Sir Francis Drake from England in the 1500s, there are ships logs that we can read of his sailors saying, as we go down these rivers, like the river to Thailand, there are these strange creatures that live in the mangrove trees that flash, and they call them lightning bugs.

0:23:36 SC: That's what I called them growing up in Philadelphia, so that's still lightning bugs.

0:23:40 SS: Exactly, so many... Yeah, lightning bugs, fireflies, whatever you wanna call them, they... But what was spectacular is that there would be tree after tree for miles along the river bank filled with these fireflies, thousands of them, it seemed. And like you could visualize it almost like a Christmas tree with Christmas lights on it, except that these are beetles that we think of... They're not... They're these little beetles that have this strange flashing property, and the whole tree will ignite and get dark simultaneously and it goes on all night long and it's just an amazing spectacle.

0:24:16 SS: But the question was, as you said, how is it possible? These are not the most ingenious creatures. They're just little bugs. And at the beginning of the night, they're not doing that. When they fly into the trees as the sun goes down, they're totally disorganized. If you get there at dusk, you don't see the phenomenon. It builds up over the night and it takes a while, but by the middle of the night, it's perfectly organized. So that was always the question. Is it that there's, as you say, is there some kind of master firefly, like a conductor for the orchestra and they're all following the lead of that one? That's not gonna be a very robust biological way of doing it 'cause if a bird eats that particular maestro, you know, what, then it's not gonna work that night?

0:25:02 SC: So also, who appoints the head firefly, that doesn't seem biologically sensible.

0:25:07 SS: On the other hand, you have to keep in mind that there are things in biology organized like that. Like there is a queen bee who is different from the other bees. So it's not unthinkable that there is a king firefly, but no one ever found one. And so the prevailing view today is that it's done, this spontaneous synchronization is an emergent phenomenon that as you say, it self-organizes. You don't need an external signal like lightning in the jungle. You don't... Those were the old theories that there was... Look, it's in the tropics, there's gonna always be some lightning storms and all the fireflies get startled by the lightning flash and that kind of pre-synchronizes them all, as they respond in shock to that lightning bolt. But that's ridiculous because it happens every night even if it's not raining so...

0:25:57 SS: Anyway. But they were, like I say, it was a very old problem, and it was only by around the 1960s that those experiments that I mentioned with pen flashlights demonstrated that the fireflies are both reacting to flashes as well as emitting them. And the conjecture was always that they somehow, through that process of interaction, come in to sync, but no biologist could figure out how to demonstrate that because they didn't have the computers that could do... It's not the way that they think. They have different training. So you really need a physicist or a mathematician to try to work on this.

0:26:35 SC: And the mechanism is just that the fireflies are looking at each other, and somehow what they see is affecting their rate of firing.

0:26:42 SS: That's right, exactly. So yeah, if they see a flash when they were not expecting to see one, it might tell them, "Oh, God, I'm a little bit late." And then not consciously, but through just some unconscious processes in their nervous system, 'cause we don't think of them as having much consciousness, they just automatically adjust their timing, something in their nervous system changes in response to seeing this flash such that on the next time, they will be more nearly in sync with that flash.

0:27:14 SC: And it's not an especially complicated form of order, especially sophisticated, but still, it's self-organizing in the sense that there's no teleology, right? There's no goal, there was no idea that we're gonna get together and do this. It's just everyone's doing their own individual thing and suddenly, or not so suddenly but yeah, ultimately they're all doing the same thing. I saw you do the same thing at a TED talk, where you ask everyone to clap in the audience and very quickly, they're all clapping in rhythm.

0:27:41 SS: Yeah, that's an interesting comparison, 'cause there the audience knows it's trying to get in sync and... But yeah, you're right. That the fireflies don't have any particular goal, and there are different theories about why they're doing it. I should clarify, it's only the male fireflies that are doing this.

0:28:00 SC: Ah. I didn't know that.

0:28:00 SC: So it's definitely something going on with mating. It's not just all fireflies. The females are flying around and they're looking for males to mate with and it seems that... Well, so this is where the biologists haven't quite figured out what they wanna say. They used to say the idea was that all the fireflies of a certain species would get in sync because it would send their message the farthest possible distance. 'Cause you would make a very bright signal that could escape the darkness of the jungle. You could sort of... The females could find them from far away 'cause there'd be this beacon. Like here's where all the boys are. But still, from a Darwinian perspective, how is it to my advantage if I look just like my neighbor? Maybe she'll come and...

0:28:42 SC: Do the females flash at all?

0:28:47 SS: They will flash. Yeah, sometimes they will flash. They actually do a little dance of light where they flash and then if the male is flashing in the right way, that is, they wanna mate with someone of the right species because if you go with the wrong species, you could get eaten.

0:29:01 SC: Yeah, okay, good.

0:29:01 SS: But they do. They send signals to each other, and if the flashing is done in a way that they find attractive or from someone of their species, then they'll actually be drawn toward it. So yeah, the females are flashing too. There's a lot of signaling to find each other and make sure everybody's on the same page.

0:29:21 SC: And your mathematical result, was it that this kind of spontaneous ordering by a synchronization is inevitable, or it happens under the right circumstances? Is it rare and fragile or generic or what?

0:29:35 SS: Well, so yeah, that's... The particular model that we made was the simplest possible thing, where we imagine every firefly could see every other firefly, and that isn't really true. Really, they would mostly see the ones that are near them in a tree, and they wouldn't be paying very much attention to one that's a mile down the river. But we ignored that, so we did what in physics would be called the infinite range approximation, where every firefly is imagined to be able to interact with everyone else out to infinite distance away from them. The reason for doing that is that when you're studying something that consists of a million interacting, complicated... They're sort of complicated for the reasons I said with this RC circuit analogy.

0:30:20 SS: So if you're trying to study the collective behavior of a million of these little tricky things, you don't wanna think... Have the additional complication of who exactly is talking to who. Because we didn't even know how to solve the simpler problem. So yeah, in the case that we did, with everyone interacting with everyone, we proved, actually this is one of the few times we did actually really prove a theorem. We proved that for this model, they would always synchronize in the sense that... It's not that there wasn't a possibility of something else happening, it's just that there was zero chance of it happening. So in the jargon, it's like this, if I... What I wanna say is, it had probability zero. It's not the same... So what's a good analogy for this?

0:31:09 SC: I don't know. For a physicist, it's gonna happen. You're a mathematician, you're trying to be more careful, don't bother, it's gonna happen.

0:31:14 SS: Yeah, zero probability doesn't mean that there aren't possibilities of it happening, it's just that you'd never see them in practice. To give a loose analogy, if I throw a pencil in the air, there is some possibility it could land on its point and balance. That is possible, it just... You're never gonna see that.

0:31:31 SC: That's right, it's possible that the popular vote for president would be exactly a tie, right? These things could happen but your mathematical notion of measure zero is even less than that. This is just not very likely.

0:31:44 SS: That's the idea.

0:31:44 SC: The fireflies are gonna come in to synchronization, that was a great triumph and then... But one of the wonderful things and one of the wonderful things consistently about good math is that it finds application all over the place. In the brain and how we sleep and things like that, similar notions come to rise, right?

0:32:04 SS: That's true. There are so many different examples of spontaneous synchronization that are important in science and in medicine and nature, so... Like you mentioned in the brain. So a lot of neuroscientists will tell you that synchronous oscillation of neurons in the brain is related to phenomena like attention and memory. You can sort of see sometimes when, if you're looking under an FMRI machine, people call them brain scans or they'll talk about what part of the brain lights up when someone's doing a certain task or whatever. Very often the way the brain will sort of get itself... Okay, so here's an example. Suppose I'm looking at an apple, and I can recognize that it's an apple. You might think that's kind of obvious, of course you can recognize it, but not everyone can. There are people with brain damage who can't recognize simple objects anymore. So what... There's actually a miraculous thing happening in our brains when we see an apple on the table and recognize it as an apple and not something else. We...

0:33:09 SS: Think about what's involved. You have specialized neurons that detect color, there are others that are looking at shape, there are some that are thinking about other qualities of the apple, and we somehow put all those different qualities together to recognize one whole coherent object, the apple. And what you observe in the brain when that's happening is that the parts of the brain that are noticing color are actually firing electrically at the same frequency as the parts that are noticing shape or whatever. So synchronous oscillation is the brain's way of, the biologists call it the binding problem. How do you bind all the different features of an object into a coherent single object to recognize that it's not just a bunch of different things happening in your brain all at once?

0:33:56 SC: Interesting, so it's not just that there's a part of my brain saying I'm seeing something red and another part saying I'm seeing something apple-shaped, but they're saying it in synchrony with each other and somehow that lets the brain or conscious perception say that is an apple.

0:34:11 SS: Yeah, and it says that, right, exactly. Those separate things that are all oscillating in sync are all... Meanwhile, other parts of the brain that are not paying attention or that are thinking about, or interested in other aspects, they're out of sync and so they're ignored. It's the brain's way of telling itself what's all part of one object, or one sensation. And this is... Go ahead.

0:34:35 SC: I was just gonna say, I'll be very honest and confess that I didn't actually read this chapter of your book, I just remembered the chapter titles so I wanted to ask you about it. But does this imply that there's a separate part of the brain that is keeping track of the frequency of oscillations to say, oh, yes, this is a coherent thing?

0:34:53 SS: That's an interesting question. Yeah. It might be, I think the thalamus is often regarded as a relay station. That it takes in signals from different parts of the brain and it might do some of the binding of what's happening elsewhere. I'm not positive I've got that right, so some of your listeners may correct you.

0:35:13 SC: Well, I'm asking that they don't correct me, the neuroscientist and the physicist, so you're allowed to say that you're not an expert on...

0:35:20 SS: That part I'm not sure, but I am pretty confident that if you ask most brain researchers now they'll tell you the binding problem is solved by synchronization. Or at least that's their best current guess. Now, I should say, you don't always want things in sync in your brain. Epilepsy is a famous disease, you can picture someone having those, during a seizure the most famous symptom is these rhythmic convulsions. And the rhythmic twitching and convulsions is that many millions or billions of neurons are discharging in perfect step when they're not supposed to.

0:35:55 SC: Yeah, that I did know about, 'cause it's a problem. This is always one of the problems with organization, that you want the right amount of it.

0:36:02 SS: Yeah, exactly. So yeah, you can have pathological synchrony too.

0:36:07 SC: And where does sleep come into this? I literally just yesterday was having a conversation with a bunch of scientists in different areas about, why do we sleep? And the answer is often, well, there's this chemical that puts us to sleep, but there's the higher level question, what is the purpose of falling asleep? Is this kind of research relevant there?

0:36:25 SS: It could be. That question, wow. That takes me back. My PhD was about sleep research. I worked on human sleep and circadian rhythms, those 24-hour rhythms that we have in hormone fluctuations and body temperature and lots of other internal rhythms. But nobody has ever really, believe it or not, figured out why we need to sleep. It might sound ridiculous, like my mother when I was a kid, she actually, she had a theory. She said because you have too much sleepy gas.

[laughter]

0:36:57 SS: And it's like, what?

0:37:00 SC: That's a very mechanistic theory. I like it.

0:37:01 SS: Yeah. She said when you're awake, the longer you're awake, the more sleepy gas you build up and then eventually when you get so much of it, then you have to go to sleep and then it goes away. So she was actually doing the RC circuit model that I talked about earlier, where something builds up to a threshold and then she imagined it gets degraded and chewed up when you're asleep. But actually, that isn't so far off from what... One of the leading theories of what's going on is that there are sleep substances that are measurable, there are peptides that can be detected. For instance, there's a classic experiment from a long, long time ago, maybe about on the order of 100 years ago, where researchers kept sheep, the animal, the farmyard animal. You keep a sheep awake, how would you do that? You could probably keep bothering it and disturbing it and poking it, I don't know. So they did something to keep the poor sheep awake for way past its bedtime so it was getting...

0:37:57 SC: Let it read Twitter, that'll be enough.

[laughter]

0:38:00 SS: Yeah, you can... So anyway, you could do sleep deprivation on an animal. And then the experiment was take a little bit of the blood of that animal, that is sleep-deprived, and inject it into another sheep and that other sheep falls asleep right away.

0:38:13 SC: Okay, so that certainly seems to imply to my scientific mind that there is a chemical tracer in the blood that's saying, "Dude, you should go to sleep."

0:38:21 SS: Yeah. And we've, over the years, isolated candidates for that. For a long time, it was thought to be something called muramyl peptide that was the alleged sleep substance. I think others have been found since then. So it's possible that that's one of the things that's happening. That just being awake and active produces biochemical by-products of activity that give you the subjective sensation of feeling tired, and that that is, you need to restore yourself back to having less of that stuff. So sleep is partly for that. But it seems there's a lot more going on with sleep. There are ecological reasons to sleep, depends on if you're a predator or prey. If you're prey, you wanna be in your burrow, especially when it's not favorable for you, like an animal that wants to be out in the daytime, then you better be hiding when it's night time, when all those nocturnal animals are looking for food.

0:39:20 SC: You can see how synchronization gets involved with all of these different things. Certainly, I'm about to fly to Europe and I know that jet lag is something that hits me very hard and apparently that's in part because different circadian rhythms inside your body get out of sync with each other.

0:39:36 SS: Right. Exactly, that is what jet lag is. It's a funny thing, a lot of people get confused about it and they'd say... I've heard so many people say, "Oh, I don't get jet lag. I just stay up all night and then I sleep it off the next day and then I'm better." But what they're not noticing is that jet lag just is not only about sleep and sleep disruption. There are all these other rhythms that we're not so aware of that are inside of us like... If you think about it, you're aware of when do you wanna go to the bathroom?

0:40:06 SC: Yeah.

0:40:06 SS: When do you feel hungriest? When are you most alert? Those are internal rhythms. And then there are others that, like I say, body temperature is going up and down. Even if you lie still in bed and people have done experiments like this, just keep someone awake in bed all day long, you can measure their temperature and it's going up and down like a nice sine wave. So there are internal rhythms of temperature, of alertness, and... What happens during jet lag is that even if your sleep gets on to the local time, your internal rhythms are still back at home time. That's the lag in jet lag.

0:40:41 SC: Yeah. Every individual person has a lot of things going on inside them that can be in or out of synchronization. It really does matter.

0:40:41 SS: That's right. That's right. And so there are things in the outside world that help to re-synchronize us. Most important is sunlight but food is another one. The timing of your meals will affect. And melatonin, of course. People have heard now of this brain hormone melatonin that you can use as a pill for a sleeping pill or for a circadian rhythm restoration pill. But I have to say, I've always been skeptical of melatonin. Not having used it. So maybe those people out there who are listening and do use it and swear by it, I'm not saying don't use it, but I can tell you that in experiments, picograms of melatonin are enough to be biologically active. That is... That'd be a one follow... That's like 12 zeros.

0:41:29 SS: A picogram is a tiny, tiny, tiny amount [chuckle] of melatonin has a biological action so I can't imagine how much is in a pill. I think it's like a trillion times more than your brain wants. So I feel like, "Really? You're taking such an ungodly amount of melatonin?" I would be scared to do it but I don't hear of anyone having trouble with it so I'm probably wrong.

0:41:52 SC: I'll confess, I think melatonin is a miracle drug for me. It could be entirely psychosomatic but... 'Cause I don't take sleeping pills and I generally have no trouble sleeping and even if I take NyQuil, I wake up feeling groggy and I just don't want to do that. But when I travel and I wanna get to sleep, it's something like a local bedtime, I take melatonin. It puts me out and I feel no after-effects the next day. It's really...

0:42:16 SS: Well, I gotta start doing it. This is a case where I'm too much of a theorist. [laughter]

0:42:21 SC: Yeah. Well, [chuckle] it could be killing me long-term, I don't know, but... [laughter]

0:42:24 SS: Hope not.

0:42:26 SC: Yeah. Hope not. Okay, you've raised this issue and I really wanted to leap in at the time, but I knew we had other things to talk about so I didn't. You raised this issue with the fireflies of the approximation where every firefly is seeing every other firefly. Now, obviously, that's not true but maybe it's a good enough approximation to get what's going on. In something like the brain, it's even obviously less true and it's kind of super important that it's not true, right? Every neuron is not talking to every other neuron. And I think, correct me if I'm wrong, that this kind of consideration led you to think about the phenomenon of networks and how things were connected, which eventually resulted in a paper that has so many citations that my jealousy is overwhelming.

[laughter]

0:43:13 SS: Well, that's right. It was a desire to move away from that very unrealistic infinite range approximation and to just start paying attention to whatever new phenomena would occur if we tried to be a bit more realistic about the networks that really do occur in so many different phenomena that got us thinking about what came to be called small-world networks. The phrase small-world is supposed to make you think of that experience that we all have when we get on a plane or go to a cocktail party and you meet somebody and you start talking, and then you realize, "Oh, yeah. My cousin went to that summer camp." [chuckle] So then people say, "Oh, it's a small world." Because it seems like, how is that possible? How can we be so... Or there's also the counterpart. The other phrase that you hear all the time is six degrees of separation.

0:44:06 SC: Yeah.

0:44:06 SS: That I know someone who knows someone and we can connect ourselves to anybody, it seems, through just a very small number of mutual acquaintances or chain of acquaintances. Which then became a popular game, right? That Kevin Bacon game with actors.

0:44:21 SC: I'll mention that this morning, Jennifer, my wife, who's a science journalist, said, "Do you know a guy named Erik Winfree?" And of course yes cause he's a professor at Caltech, but also I realized, he I think is the nephew of one of your collaborators or mentors in the synchronization game, right?

0:44:39 SS: He's actually the son.

0:44:41 SC: He's the son? Okay.

0:44:41 SS: Yeah. He's Art Winfree's son and Art Winfree was my closest mentor of my career.

0:44:46 SC: It's a small world. [laughter]

0:44:47 SS: Yeah, it's a small world. When I went to work with Art Winfree at that time, he had a 12 or 10-year-old son, Erik, who has gone on to be a great scientist and is your colleague at Caltech but who also, notable about the two of them is that they're one of the few father and son double MacArthur Award winners. They both got MacArthur prizes.

0:45:10 SC: That's something to tell your dad. I got my own MacArthur now. Yeah.

[laughter]

0:45:13 SS: So two geniuses in the family. And Erik was a very smart little boy. I'm not surprised he's turned out to be a very smart professor too.

0:45:21 SC: And that's actually not surprising, 'cause we're all academics and scientists and so forth but if you did a very simple view of the world where everyone knew who ever lived within five miles of them and nobody else, then it would take a huge number of steps of separation for me to get from someone in another continent. And the small-world network phenomena says, "No. It's really just not like that at all."

0:45:48 SS: Yeah, that's right. That's what's so counter-intuitive, because we do have this strong sensation that I only know the people that live near me geographically or that are in my department at work or go to my church or whatever it is. So we feel like we move in small circles and so that's why we are always so surprised and say, "Oh, wow, that's weird. It's a small world." And yet, if we were statistically minded, we should be thinking, "This small-world thing keeps happening to me. It must not be that rare." [laughter] Because everybody experiences it. So there must be some explanation.

0:46:23 SS: And one of the things that Duncan Watts, who was my grad student at the time, was interested in. His father had said to him, "Do you know that you're only six handshakes from the president? That you could find some... You may not have ever met, let's say in this case, Donald Trump but you might know someone who knows someone and within six handshakes, you'd know someone who knows Trump."

0:46:44 SC: Yeah.

0:46:47 SS: That struck Duncan as an interesting thing because at the time he was trying to study, not fireflies but crickets. There are crickets that can chirp in unison. It's the analog of the fireflies flashing in unison. So the biologists speak of choruses of crickets that are all chirping at the same time. And we have this particular species of those crickets, snowy tree crickets as they're called, that live in Ithaca. And so we thought, we could do experiments on these crickets and see if some of the mathematical models of synchronization actually predict what the crickets are really doing. That would be new because the case of the fireflies in Thailand, much harder to go all the way to Thailand and capture them, but the crickets are right here in the orchards in Ithaca, New York where I teach at Cornell. And so we have mass... Like the grandmaster of sonic synchronization right here, [chuckle] the snowy tree crickets. So we thought that would make a nice experiment for Duncan to do for his... A project for his PhD work and... So he learned how to capture the crickets and keep them alive and we were starting to put them in little sound-proof boxes so that we could control how strongly they could hear each other and try to set up an experiment with the help of a bioacoustics expert named Tim Forrest.

0:48:02 SS: Anyway, it was in the course of doing these cricket experiments that Duncan started to think about, I wonder, when the crickets are out there in the orchard, who's actually listening to who? Do they all hear each other? That can't be right. Maybe they only hear the ones right next to them. And so he got to thinking about connectivity in general and then through a brainstorm, he remembered this thing his dad had said about six handshakes from the president. And so he came into my office one day and said, "How would... If things were connected in this small-world or six degrees of separation way, would they synchronize better than if they were just connected to their neighbors or would they... How would it work?" And I thought, I don't know. I mean [laughter] nobody... I don't even know how you do this six degrees thing.

0:48:49 SS: And so we realized, there's a whole big math problem. What explains this small world? And not only that but how would it affect synchronization? And Duncan said, "It's much bigger than synchronization." It would... Because anything that's connected like that, you'd think it would make a big difference because everyone is so close to everyone in the sense of the small world. Just a few hops away from everyone else. So, like how would that affect diseases spreading?

0:49:13 SC: Yeah.

0:49:13 SS: At the time, actually, people were talking about HIV a lot. It doesn't get discussed as much any more but during the height of the AIDS epidemic, you would hear people say if you sleep with someone you're not just sleeping with them, you're sleeping with everyone that they slept with and everyone that that person slept with. So the idea was out there that you feel like you're only interacting with a low-risk group, but actually you're only a few steps away from the virus, let's say.

0:49:40 SC: I think I would like to try explaining these two concepts that you talk about in the book that are relevant to characterizing these networks. The idea of the shortest path between two people in the network and the separate idea of the clustering, right? Like how many people are connected to overlapping friends. Is that something you're willing to try to put in unassailable terms?

0:50:05 SS: Sure. Totally, yeah. It's not very hard. The idea of path length is just the idea that we were talking about with six degrees. That if, let's say you and I, we've never actually shaken hands, so we haven't met but... And I don't actually know what our shortest path is but we could start naming physicists and mathematicians that we've met. Okay, I'm gonna guess. I'm guessing you've shaken Brian Greene's hand.

0:50:31 SC: I have. He's been a previous podcast guest, so... [chuckle]

0:50:34 SS: Okay. And I know Brian Greene because I've known him since he was a high school student. So that would make... And I don't think there's any faster route. That's one handshake for me to Brian and one handshake from him to you. So we're two degrees of separation apart and that's our shortest possible path. That's our path length. So path length is just what's the shortest route from one node in a network to another. Okay, so that's one thing you can calculate for a network, you'll get all the shortest paths between any pair of points in the network and then that average is what we would call the average path length in the network, which is basically a way of quantifying this idea that everyone's about... The number six is not important. It's just that it's a small number of steps from any point to any other point, even in a very big network like something the size of the world with 7 billion people...

0:51:27 SC: But if your network were more like a lattice, where you only knew your nearest neighbors, then the average path length would be enormous.

0:51:34 SS: That's right. If you picture a checker board, like think of the nodes in the network as being the squares of a checkerboard, on a say... Literally a checker board which is eight by eight, if you wanted to go from one corner to the diagonally opposite corner, the fastest way you could get there... Well, you could go down the diagonal, I guess, but that would still be eight squares in between.

0:51:57 SC: Yeah.

0:51:58 SS: And so there, if the world starts getting big, that shortest path is also getting very big. It would not be like in the world... If there were seven... Well, seven billion's sort of hard to take the square root but... [chuckle]

0:52:11 SC: Let's do the 10 to the 5.

0:52:13 SS: Okay, so let's say... Yeah, let's say if it was 10 to the 10th. So we had 10 billion people on Earth, which we will pretty soon. If we're 10 billion people on Earth and they were standing in this big square checkerboard pattern, that would then be as you said 10 to the 5th. So that's 100,000 people on each side of the square.

0:52:33 SC: 100,000 degrees of separation. [chuckle]

0:52:35 SS: I would say six degrees of separation. They'd say 100,000 degrees of separation and no one would say it's a small world 'cause it wouldn't be. The point being that worlds don't have to be small. The checkerboard world is not small and yet our world is small. So that was a question that Duncan and I wondered about. What does it take to make a world small? The other question, though, is the commonsense answer to this what does it take to make the world small comes from an old idea that like, suppose I know 100 people or whatever number you wanna pick, it could be 1,000 but say it's 100. I know them well enough that they would lend me money. Maybe I have 100 close friends and contacts and relatives. Okay. So if I know 100 people and each of them knows 100 people, then naively, I can just figure out how small the world is by saying 100 times 100 or in your scientific notation, [chuckle] that's 10 to the 2 times 10 to the 2, that's 10 to the 4th. And if I wanna get to 10 to the 10th, I have to do this five times so it'd be five degrees of separation, if the world were such that I know 100 people and everyone else knows 100 people. Except not the same 100 people.

0:53:51 SC: That's the trick.

0:53:51 SS: In other words, if there's no overlap, then the simple multiplication of 100 times itself, doing that five times, that will be enough, but that's the trick, as you say. The problem is that of the 100 people I know, when they know 100 people, a lot of those are the same people because we live near each other or because we're in the same profession or whatever. And so that's what we're calling clustering. The fact that it's not just a random choice of 100 people anywhere on Earth but... So we have a more precise definition of clustering which is we imagine... You could think of it this way, think of two people you know, and now ask do they know each other. Maybe they do or maybe they don't. Like maybe someone you're thinking of right now is someone you went to high school with and someone else is somebody that you know now from work and they've never met and they don't know each other. Okay, so those two people don't know each other even though they both know you. But you could also think of two of the people at work, maybe they do know each other.

0:54:53 SS: So the concept we had for clustering is pick any two of your friends and ask what's the probability that they're also friends of each other.

0:55:02 SC: Yeah. Okay.

0:55:03 SS: It's a number from 0 to 1. And in some types of worlds, that number will be very small and close to 0 and in other types of worlds it would be very... Also, we used expressions. Like we imagined kind of fraternity world where... [laughter] The only people you know are the people in your fraternity so of course, two of your friends will know each other 'cause they're in the fraternity too.

0:55:27 SS: Okay. So in that kind of world, the clustering coefficient, as we called it, will be very close to 1. That's almost a certainty that your friends will know each other, whereas if you picked your 100 friends at random on the surface of the Earth, there's one in Ethiopia and one in Indonesia. And then each of them pick their friends at random, 100 friends, somewhere else on Earth, there's very little chance that two friends of yours would know each other. The odds are way against it that they would happen... There's no reason they'd pick out of their 100 people out of 10 billion, why would they pick those same friends?

0:56:03 SC: And these two examples, these are sort of the classic kinds of networks that people have been thinking about. Either everyone knows everybody else so there's lots of clustering and the path is also very short. Or nobody knows anybody else and so the path can be short but there's no clustering, is that right?

0:56:20 SS: Well, let's see if we got that right.

0:56:24 SC: Nah. I think I didn't get it right.

0:56:25 SS: I think not quite. Like the checkerboard world, or actually, we often don't use a checkerboard. We used to like to think of it as people standing in a circle. If all 10 billion people were just standing out there in a big circle, where the 100 people they know are the ones right next to them in the circle. So I have 50 friends on my left and 50 on my right and I don't know anybody else. And the same for everybody else in the circle, the same thing, that's a very big world, 'cause for me to get a message, let's say, to someone diametrically opposite me, I have to go leap-frogging around in steps of 50, and it's gonna take a long time to get over to 5 billion people away.

0:57:04 SC: So path length very long, but clustering very high.

0:57:06 SS: Yeah, there is clustering. Yeah, high clustering because of my 50 friends on either side, they will overlap a lot as I move to my friends. So those worlds, that kind of world has very high clustering but very long path. And the other kind of world, the random world, is very small but it has no clustering. And so what we thought was this kind of paradoxical thing is that our lives, it feels like our lives are very clustered. Most of our friends do know each other, or at least many of them do, much more so than if the world were random. But yet the world seems small, almost as small as if it were random. And so that wasn't obvious. Could you have both? Because the real world somehow does have both. Could you make a model that has both? And what Duncan and I realized, but mainly him, was that what was really important, I mean, one way to do it, and we thought this was the way it was probably done, is if people had a certain number of far-flung connections.

0:58:10 SS: To give you an example of what I'm talking about, I used to play a lot of chess on the internet and I got to be friends with a guy in Holland and... I mean, I would say he was really my friend. I knew how many kids he had, and I knew about his life, and I never met him actually face-to-face, but I feel like he was my friend. But the point being, if I wanted to get a message to somebody in Holland, I could use my link to him and there'd be a chance he would know that person, or he'd know someone who would know the person. In other words, there was this bridge where I was suddenly connected to somebody I had no business being connected to except that we both like to play chess on the internet. And that bridge not only made me closer to everyone in Holland, but everyone that I know is also now much closer to everyone in Holland, although they don't realize it.

0:58:57 SC: Right, right.

0:58:58 SS: Because they know me and I can take the bridge.

0:59:01 SC: And likewise, more conceptually, if you had a friend from high school who is now a classical pianist, suddenly you have a whole bunch of connections, it's a short distance to you to everyone in the classical music world.

0:59:13 SS: Exactly, right. And so, what's interesting about this mechanism, we called it shortcuts, is that the shortcuts really make the world very small very quickly, though you don't sense it, because you don't realize you're connected to everyone in the classical music world.

0:59:28 SC: Right.

0:59:29 SS: Because it doesn't sort of operate in your consciousness, but you are. And, so this shortcut mechanism, we showed that very, very few shortcuts were enough to make the world incredibly small, and that seemed to be a very generic phenomenon that because it took so few of them, it seemed like most networks would have this property because you kind of, in order to avoid it, you had to scrupulously avoid having any shortcuts in a network and... So we made a prediction back in... Well, I guess it was... Now, I'm spacing out, when did that paper come out?

1:00:06 SC: It was a while ago. Yeah. I always hate thinking about these things.

1:00:09 SS: It just was, it was about the 20th anniversary. I think it was...

1:00:11 SC: Okay, '98? Yeah?

1:00:13 SS: Yeah, that sounds right, yeah, 1998. The paper came out in 1998. And so we had said the brain, for instance, will turn out to be a small world network when we can measure all the connection... We haven't yet measured all the connections between all the neurons, there's estimated to be trillions of neurons in the brain, tens of trillions, I think. So we don't know what the connectome, as they call it, is for the brain. But we do know the connectome of a tiny worm. That's the one nervous system that's been completely mapped out, it's called C-Elegans.

1:00:45 SC: Oh yeah, my favorite little tiny worm.

1:00:47 SS: Yeah, so we know every neuron in its body, there's about 300 of them. It's only a few... Something like on the order of a thousand cells in the whole creature, but by looking at the nervous system of this worm, we show that it actually satisfied our criteria for a small world. That it was much more clustered than a random world, but it also had path length about comparable to a random world. So it was as small as it could be, while much more clustered.

1:01:17 SC: And the internet and a whole bunch of other networks in the real world...

1:01:20 SS: Yeah, a bunch of other networks, lots of real world networks turn... And since then, in the 20 years since then, it's been abundantly documented that our prediction was right. Lots of naturally occurring networks will be small world. And you could ask, "Well, okay, so what?" But what's interesting is that small worlds allow for very fast propagation of information through these shortcuts. So anything that needs to coordinate itself or act as a coordinated unit but is very big, a small world mechanism is a really good way to do it, but also it's dangerous, in the case of like the HIV example. Anything that can spread for good or for bad, will spread much faster on a small world than it would on, let's say, on a lattice or a ring.

1:02:03 SC: And also typically, a wonderful example of self-organization. Nobody planned it out. You said naturally occurring. There's mechanisms that typically give rise to networks just like this.

1:02:15 SS: Right. They're found all over the place. And as you say, there's no central plan or there's no need to design it, it just sort of happens on its own. I did wanna mention one thing, earlier, actually, that I sort of I had in my head but didn't say out loud, which was that there's another way of making the world small that you could imagine, which we deliberately did not imagine, which is you could imagine that there's somebody that everybody knows. Right? If everybody knows Lois Weisberg. I use that example because Malcom Gladwell wrote an article a long time ago called Six Degrees of Lois Weisberg, that she was this person in Chicago that seemed to know jazz musicians and she knew newspaper people, and just everybody knew Lois Weisberg. Anyway, that's one way that the world can be small, if everyone knows these super-hubs, these connectors.

1:03:07 SC: Right.

1:03:08 SS: Then of course everyone's just a few degrees away, 'cause they're all close to Lois Weisberg. But we didn't think that was really gonna be... First of all, that felt like cheating. Of course, that will make the world small if you have that, but more to the point, we thought that's not really gonna be how nature will use it. That's not the mechanism because, for one thing, it's hard for the Lois Weisbergs out there to maintain all those connections. It's a lot of energetic cost, a lot of... It's just hard.

1:03:37 SC: And you don't need it, right?

1:03:39 SS: Well, you don't need it. I mean, of course, you might have it, but you don't need it. The shortcut mechanism, and you need only a very few of those, doesn't cost much and that will do the job for you but... So we deliberately ignored... Also, I would say that I had been told by my friends in neuroscience that every neuron in the brain connects to about, that is make synapses with, about 10,000 others.

1:04:03 SC: Okay.

1:04:03 SS: Which might sound like a lot, but then when you remember that there's a trillion neurons in the brain, or something like that, or maybe it's 100 billion, but it's way, way more than 10,000.

1:04:13 SC: I think it's 85 billion, yeah. But it's a lot.

1:04:15 SS: Yeah, okay, right. So, it's on order of a 100 billion. So, that's right. So if it's 10 to the 11th, but there's only 10 to the 4th synapses per neuron...

1:04:25 SC: Yeah.

1:04:25 SS: That's a factor of 10 to the 7th difference, so it's a very sparse network in that sense. You're not connected to anything like the whole network. And there are no Lois Weisbergs in the brain. So, we deliberately didn't wanna pay attention to those networks. And the reason I'm harping on this is because we missed the boat. It turns out that there are a lot of networks that use hubs, the most obvious being airports. If you think of the airport system and for air travel, you fly to a hub, and the hub has a lot of routes going into it. So there are some networks that use this hub mechanism to make the world small. And so, that was studied by other people and we didn't... Like I say, it's an interesting thing, like in the history of science, that sometimes you'll have a prejudice about what you think the way the world should be, and you deliberately don't let yourself entertain another possibility 'cause it strikes you as ugly, or too simple or irritating in some way. And so I'm...

1:05:25 SC: The world is very irritating 'cause it keeps doing different things in different circumstances in different models. It's full employment, but still, it's a little bothersome sometimes.

1:05:34 SS: Anyway, those kinds of networks with the hubs were published... An analysis of them came out the year after our paper on the small world. Réka Albert and her advisor Laszlo Barabási wrote a paper about what they called scale-free networks that had not only hubs, but also a distribution of... The jargon is degree. That's how many connections any individual node has. How many friends in our earlier social network analogy, so if you ask how many people have, say, 10 friends, how many have 100 friends, how many have 1000 friends, you can find people with more or less big rolodexes. And it obeyed a power law in that there was the probability of having a certain number of friends went down like the number of friends to some power. It was an inverse, like 1 over X to some power. But close to 3 or so. 1.2 something. 2.2.

1:06:31 SC: So that technically would be a small-world network but it has this extra feature, that there are also Lois Weisbergs.

1:06:38 SS: There were Lois Weisbergs. And also sort of like things a little bit less than Lois Weisbergs. You had sort of Lois Weisbergs at all scales of degree. They were small worlds, they didn't have the right clustering properties, actually, at least in the first models. So we missed the hubs, Barabási and Albert missed the clustering. And since then people have figured out that real networks are more complicated than the models either of us proposed, which was no surprise, we were both putting out really idealized, simple models. More like for thought experiments.

1:07:12 SC: Because this is within the realm of self-organization, these kind of networks, we can speculate a little bit, it's late in the podcast, these are at the heart of how complexity and complicated interconnected systems arise in a world that is ultimately governed by the Second Law of Thermodynamics and things run down.

1:07:35 SS: Well, all I can say is that both of these are common network themes, the small-world theme, and the scale-free. There are a lot of other things going on. We've learned a lot more about networks in the past 20 years. So they were very stimulating early ideas, I think we could say that. I wouldn't claim that either one of them is in kind of... The small-world law is pretty much a universal law that almost every real world network is gonna be a small world by our criteria.

1:08:02 SC: Yeah.

1:08:03 SS: Scale-free has not held up as well, but it's still a pretty common theme to say if you... The power law part is the part that's not reliable. If you just ask, "Is the distribution of degrees something that obeys a heavy tail?" Meaning it doesn't look like a bell shape curve with a exponentially damped tail, but it has something that's got a lot more of these Lois Weisbergs, than you would think.

1:08:25 SC: Yeah.

1:08:26 SS: That does seem to be true.

1:08:27 SC: Yeah, a heavy tail distribution has a lot more things you might think are unlikely or improbable than you would get by doing the sort of traditional bell curve-like probability distributions.

1:08:37 SS: That's right.

1:08:37 SC: And so the world is full of things like that, and we're still learning to deal with that. Maybe earthquakes or solar flares are examples of these, which means that terrible disasters can happen a lot more frequently than we might guess.

1:08:52 SS: It's true, there are a lot of those heavy tail distributions in natural disasters. Yeah, with floods, wildfires. The statistics of those are often very heavy-tailed.

1:09:07 SC: So, okay, clearly we have enough room for a whole another podcast down the line. That is good, but I do want to give you the chance, 'cause you mentioned right at the beginning, how the very favorite tool of every mathematician, which is calculus, isn't an obvious fit to the kinds of studies you wanted to do with the fireflies where there seemed to be some kind of discontinuous jump. And nevertheless, your most recent book is about calculus, so I think many people think of calculus as, there's nothing new there. We've done calculus a long time ago, and in fact, my memory of it is, one might say, a terrible class I had in high school. What is it that you think makes us need another book about calculus right now?

1:09:48 SS: Well, I would sing the praises of calculus. I think it's one of the greatest achievements in the history of humanity. This is something that took about, more than 2000 years to develop, starting from the days of Archimedes up through Isaac Newton and Leibniz, and I'm taking a very broad view of what I mean by calculus, but let's say, roughly speaking, the systematic use of infinity to solve hard problems.

1:10:14 SC: Right.

1:10:15 SS: That's the thrilling idea in calculus, that you can take a hard problem and chop it up into infinitely small... That is infinitely many, infinitesimally small pieces, and then, those turn out to be easier problems to solve, those small ones. And then, if you can figure out a way to put them back together, which is what we call integral calculus, or integrating a differential equation, then, you can do the kind of math that has changed the world. That's the math that let Maxwell predict the phenomenon of wireless that's letting us talk to each other right now, or...

1:10:47 SC: Yeah. But I always try to gloss calculus as saying, it's the statement that a medium-sized thing can be thought of as an infinite number of infinitely small things.

1:10:56 SS: Yeah. That's a great, great insight, and it literally changed the world. We can't... Without it, we wouldn't have radio, we wouldn't have television, we wouldn't have turned the tide in the fight against HIV. So, in the book, I'm trying to make... It's called Infinite Powers, and I'm trying to make the case about just how revolutionary calculus was when it was invented, and how it's still giving us gifts today. And I feel that there's a need to do it, while it's true, as you say, it's not the newest thing under the sun. A lot of people, in fact, like a million, more than a million, students in the US alone, take calculus every year, and for them, it's something that they do the advanced placement test, and they don't know what the heck they'd... Why did I do that? [chuckle] It's often just taught as a lot of... There's a lot to learn. There's a lot of technical things to learn about how to do this kind of integral, or that kind of derivative, but I feel like the great human story of this fantastic idea that I would rank right up there with evolution and quantum theory, this is a fantastic thing that I want people to understand just how rich the story is, and how world changing it was, and still is.

1:12:06 SS: So that's why I... To me, it's not another calculus book. It's a book that's... We know it, those of us who are professional scientists know these things, but I don't think that the typical high school student, or even someone who might be teaching it in high school, they may not realize this very broad context of what calculus has done for the world.

1:12:25 SC: Well, Jennifer wrote a book about calculus called The Calculus Diaries, and the gimmick was that a 40-something-year-old English major learns calculus and learns to apply it to the world. So, I was helpful as an experimental test subject when we went to Vegas, or drove a car, and just noticed all the different ways in which you could think about the phenomena using calculus. So, I completely agree with you that people just don't quite appreciate how absolutely universal it is, and if you really get what calculus is trying to tell you, your view of the world changes in a profound way.

1:13:01 SS: Perfect. [chuckle] Right. Exactly. So, I feel like that needs to be better known, that this shouldn't just be for insiders. I really... I feel it's a beautiful thing, it's an inspiring thing. And so, I know there are people out there who would like to know this, and I'm... I should say, I've written the book for this sort of person that, actually, that you and I both like to write for, the educated person who's curious, but who is not a professional physicist or mathematician, or may not have even taken these subjects in college, maybe was happy to be done with them in high school.

1:13:35 SC: Yeah. Well, I think that's what makes it interesting, because we, probably... There's a pre-existing resonance with the word calculus in many people's minds, and it won't always be positive. So, you don't have a blank slate to deal with, you're trying to push up against some resistance, and that's a fun challenge to take up.

1:13:53 SS: Yes, it is. Yup.

1:13:55 SC: Alright, Steven Strogatz, thanks so much for your time.

1:13:57 SS: Thank you, Sean. This was great fun.

1:14:00 SC: I'll definitely be recommending all your books, and I wanna read more about small worlds and scale-free networks and things like that. And maybe we'll have you on the podcast again to dig into them further.

1:14:08 SS: Oh, okay. My pleasure, I hope I will.

1:14:10 SC: Alright. Thanks a lot.

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