VIDEO
“Power laws” are mathematical relationships expressed by exponents or logarithms—why are they so powerful in describing the world? Why are power laws found in diverse complex systems in biology and society? How could simple arithmetic have explanatory potency?
Featuring interviews with Geoffrey West, Stuart Kauffman, and Aaron Clauset.
to seek the foundations of re reality my quest is to explore diverse fields of knowledge fundamental physics Life Sciences philosophy psychology Consciousness even claims of the non-physical another approach another way of thinking is to search for rules regularities principles that cut across diverse fields of knowledge to seek deep structural similarities that may underly and unite despite superficial one such candidate I hear is what are called Power laws which are mathematical relationships expressed by exponents and logarithms could power laws be that powerful where are they found how do they describe the world why do power widely I'm Robert Lawrence cun and closer to truth is my journey to find exponents are taught in fifth or sixth School logarithms in 9th grade Junior School how could such simple arithmetic potency the rather incredible claim is that power laws work their magic by describing complex systems which are found everywhere biology Society the economy I go to the pioneering Center of complex systems research the Santa Fe Institute in Santa Fe New Mexico what an ambitious Mission they have to discern underlying unifying shared patterns in complex physical biological social cultural and technological worlds I begin with a theoretical physicist with a big idea who in en Visions Universal scaling laws which pervade biology from molecules to cells to whole organisms to entire ecosystems and even to human constructed entities like cities and companies Santa Fe Institute pass president Jeffrey West Jeffrey you've pioneered the vision of power laws and scaling of bringing out the hidden order of complexity I'd love your personal experiences and how it came about the world around us sort of has this sort of messiness to it you know it doesn't seem to have order you know you look at a forest or you look at a city or you look around this room at First Sight it's hard to see where there would be any order in the great complexity and diversity of the of the world around us but here's what's really amazing if you look at it through the right lens and I think this is one of the great powers of Science and Mathematics there is in fact underneath that great complexity and diversity great order and it's expressed through things that we call power laws if you look at something that's measurable how much food you need to eat each day to stay alive so that's called metabolic rate and if you look at that as a function of the size of the animal and if you plot it in the right way logarithmically meaning that instead of 1 2 3 4 5 you plot it uh by factors of 1 10 100 thousand in terms of the amount of food that's that is needed and in terms of the size of the animal then what is extraordinary is underneath that complexity is revealed an extraordinary Simplicity because if you plot it that way instead of the points lying arbitrarily all over the graph which is what you might naively have expected given the individuality and uniqueness of every organism in fact they all line up on a straight line it's not only that they end up on a line it's a straight line so it's it's it's about the simplest possible kind of order and that is called a power law and what it expresses is the the percentage change as you change the size of an animal um is proportional to the percentage change in its size so simple example if you double the size of an organism you might naively have expected since there's twice as many cells you'd expect you need twice as much food per day to keep it alive but what it turns out is true is that you don't need twice as much you only need 75% as much and that is true every time you double so if you look at an animal that is 40 gram relative to one that's 20 G it only needs 75% more food but if you look at one that's 40 kg relative to 20 kg it also needs 75% more so there's this extraordinary regularity in the proportions um as you scale up and turns permeates life around you whether you look at Rivers buildings genomes and so on it doesn't matter what variables here's what is also amazing is that the slopes of those lines the exponent of biology almost invariably is a simple multiple of the number one quarter so the number four seems to be a magic number that is constraining all of life around you whether you look inside you or you look outside you at the at ecosystems and the environment that that universality is kind of remarkable given that everything has its own unique evolutionary history everything is historically contingent and yet somehow it's being constrained there's some natural laws that work constraining it it's astonishing who could imagine that vastly different phenomena could have such similar scaling relationships biological properties of animals seemingly have nothing in common with environmental properties of cities yet they do they have a fundamental principle in common size determines all following the simple mathematics of logarithms power laws I want to rebel against the claim too simple it seems to explain too much but I cannot gain say the data where does this leave me what kind of world do we have someone has to start explaining why why do power laws systems I know a theoretical biologist who has pioneered self-organization as a driver of complexity in biological systems a long-term member of the Santa Fe Institute this Citadel of complex calman I got into Power laws because per Bach in 1988 came to the S Institute and he talked about self organized criticality and per and his colleagues had a gorgeous model in which they drift sand down onto this table they just keep drifting it and pretty soon it piles up to a big sand pile and the sand pile grows to the edge of the table and they keep disting sand onto it and you get little and large Avalanches of sand sliding over the size of the table there's a lot of little avalanches and not many big ones and if you plot logarithm of the size of the avalanche on the x-axis and logarithm of The Avalanches of that size on the y- axis it's a straight line down to the right with a slope of minus 1.5 that self-organized criticality and they became quite famous for it literally the day that I heard it per and I and some others from The Institute went off for lunch and I said per I wonder if there's large and small uh Extinction events in the record I wonder if if the bias were self-organized critical so a bunch of us then got involved I made models in which it turns out that the size dist of Extinction events really is a power law or very close to a power law it turns out that all of the critters get to to peaks of Fitness when there's a phase transition another edge of chaos where the system is self-organized critical and you get a power LW distribution of Extinction events and you get a power LW distribution of the lifetime of a creators as well it turns out that the lifetime distribution of Genera is in fact a power law and then we also used a cousin of that model to to wonder if the economy is self-organized critical and the extin events are in fact uh sharian gilds of creative destruction that means that in in an economy certain Technologies or companies would go out of existence because other things would come in when the car comes in it replaces the horse with the horse goes the the Saddlery and the Corral and the buggy whip but with the car comes the opportunity for uh gas industry paved roads uh fast food restaurants motels and suburbia that's a big creative destruction so these models give us a power law distribution of sharian Giles of creative destruction and a dream that may not be a dream is that the biosphere is self-organized critical and that the economy is self-organized critical they both have power laws what would be the significance of that to have a theory of how things evolve where the things are so radically different which is the biological evolution of of the biosphere versus the the the economy so beautiful there are plants and and fungi where the the fungus fixes nitrogen and the plant makes sugar and they the two Critters exchange them the exchange ratio uh given the metabolic cost to make them and an advantage for getting them is exactly the analog of price it's not an analog it is price so the baser has price new ways of making living in the biosphere often come with mutualists of price and the form of price the same thing is going on in the economy where new new New Economic niches emerge and and pric is right there to allocate resources uh in the ways that we understand so new goods and services come into the economy new species come into the bias a one of the stunning things about about the economy is that 50,000 years ago there was maybe a thousand goods and services and there's now billions it's about the number of goods in the economy in the next period and the basic idea is that we create new things by recombining them in all possible ways like like jury rigging with junk in your garage it's easier to jury rig with a lot of things in your garage than only one thing in your garage so so there an exponential component to it yeah it's more than exponential it's hyper exponential and uh I wrote down an equation and I looked at it and it looked like what would happen is that nothing would happen for a long time and then all of a sudden it would it would just hockey stick upward well it does we call this the tap equation in the tap roses where tap is the theory of the adjacent possible I think this is the growing Juggernaut of the global economy it's crashing upward and it's now impacting the planet everywhere we've got global warming even worse we've got massive Extinction events the expectation the UN says is that a million species 20% of of species will go extinct by 2050 we are overwhelming the planet with the Juggernaut of our own creativity Stu is a big thinker linking the biosphere and the economy with power laws defining each's complex system and self-organization when Stu says it's so beautiful I can almost feel the beauty power laws have such astounding potency but while the theory has been General and expansive the data has been specific and targeted how would power laws applied to large sets of data I meet an external Professor visiting the Institute and expert in network science and computational social clet the thing that makes power law is interesting is that they have this property called scale invariance if you examine the size of a system using one kind of a ruler and then you change the size of that ruler to look at a system at a larger scale you see the same kind regardless of which kind of ruler you use so an example of this would be something like coastlines so if you try to measure the length of a coastline uh coastlines are actually fractals and so as you zoom in to a smaller part of the coast of Great Britain for instance it has the same sort of statistical Wiggles that the entire coastline does if you zo in some more you see the same Wiggles again so if I measure the coastline using a ruler the size of a kilometer or a meter or a centimeter I see kind of that's the notion of scale and variance that if I zoom out from a system the same Properties or the same rules govern the shape of that system it gives us the idea that we could sort of tie together phenomena that seem completely different from each other using a single kind of physics or rules and if we can find a system that exhibits this property of scale and variance then we have some sense that there's something really fundamental about it that we've understood so power law is a way of describing a system uh that has this scale and variance okay but certainly not all distributions are power laws yeah so a lot of the quantities that we see in everyday life like heights and weights and things they don't follow power law distributions so if I tell you that the average height of an American citizen is you know 5 foot seven or something like that then that tells you a lot about the whole distribution because there's nobody who's twice that height um but Power laws are different they're they sort of confound our intuition because the largest examples of things that come out of them are orders of magnitude sometimes larger than the average so if I tell you that the average size of American city is somewhere between 50 and 100,000 people that gives you no inclination that there's a New York City or an LA lying out there in the tail of the distribution so when we think about the scale and variance we're really thinking about how do we get to bigger and bigger and bigger things and so that is thinking about the tales of the distribution the rare events that happen to be extremely large and if the quantity really does follow a power law then that tells us something very powerful about well how life is it to see a city that is a hundred times as big as New York City at some point in the future of civilization um if the size of the Cities really do follow a power law forever then knowing the scaling variants exist tells us how often those things should occur well the excitement about power laws is really because we see these heavy tale distributions everywhere it it almost doesn't matter what kind of modern data set you look at you know you could you could grab data from Book Sales you could grab data from um earthquake sizes you can get data on just about anything in the world now if it's related to a social or biological or economic complex system um it'll exhibit these heavy tail distributions and so the excitement is the idea that maybe we can learn something about how these complex systems really work by starting at these large scale patterns and working backwards to the mechanisms you've applied power laws to terrorist attacks the idea here was to look at the sizes of terrorist events and by size I simply mean the number of deaths since that's a very easily measurable thing and the data sets that exist for uh uh terrorism worldwide are pretty large in fact they span the past 40 years or so they include 30,000 events 911 is of course largest event in any of these data sets it killed approximately 3,000 people the next largest event in the data set that I've been working with uh is only about 500 deaths and so already you can see there's a big gap but the important thing to notice is that in these data sets is that the typical size is roughly around four or five deaths so 911 was almost a thousand times bigger than the typical size of of a terrorist event somewhere in the world and the analysis of this data set showed that this distribution of terrorist event sizes is plausibly a power law suggesting that there may be a scale and variant mechanism underneath that explains how it is that such disperate phenomena that go into the production of a terrorist event the planning the ideology the technology could all be somehow wrapped up in this nice clean pattern what would that imply if if that were the case if there really is some sort of scaling variant mechanism it would imply that there's something very fundamental about the production of political violence in this context of terrorist events and that would allow you to make forecasts for instance about how often you might expect to see events of a certain size in the next 10 years and that would be useful for planning purposes for instance if we knew the true risk of a large terrorist event then we could Marshal our resources better than simply freaking out about it in the media to air in power laws are revelatory because these heavy tail distributions or diverse data sets with many rare events occur everywhere complex systems abound in biological social economic domains and because power laws operate similarly can we infer mechanisms for how complex systems work is this then the final step the hidden Labyrinth of Nature's Secrets what could be the actual mechanism can we have an underlying theory of complex systems would this answer my question why do power laws work so widely I return to scaling Pioneer West Jeffrey I've appreciated how power laws work in the biological realm you go way beyond that and and and bake that into the fundamental structure of our social systems one of the things that was so EXC exciting actually about the biological work was actually extending it into the kind of socioeconomic realm and that was motivated from the underlying theory of why we have all these scaling laws in biology why do we have all these extraordinary Universal scaling laws that cover every aspect of physiology and life history no matter what organism at what scale it's kind of amazing and that theory was based on the fundamental idea that um any complex system that meaning a system that has an enormous number of constituents like ourselves for example requires in order to for its sustenance requires a network and whether it's so in our case surcy system our respiratory system our renal system even our bones um our neural system and uh these systems primarily transport energy or information from usually Some central place to the the cells and within cells or are networks that distribute them within cells and the idea was that the origin of these scaling laws is in the generic of networks that the networks themselves obey general laws that transcend the evolved design so it does doesn't matter that it's a tree or human being or an insect they're all networks and it is that mathematics transcends that design and then you can transport that Network to social system ex so the idea was so when we did that we put that into mathematics when you put that the those ideas these Universal properties of networks into mathematics and start to calculate all the multiple properties of and remarkably comes out these quarter power scaling laws these and the origin of the number four and even the origin as to why it is that the number of heartbeats should be the same for a shrew mouse a giraffe and a human being and a whale so we have this IDE idea that we have these scaling laws we understand them because of the network structures that permeate all of biology so and you realize of course that if you look at instit social institutions and in particular cities but companies and uh universities whatever they are also Network systems I mean cities obviously are there's Road networks and uh gas lines and water lines but also social networks that is doing what we're doing here talking to each other exchanging information but the idea fundamentally that there are networks and that mathematics then gives rise to the idea the suggestion that my God cities too probably scale with respect to one another that despite appearances New York maybe is a scouted up Los Angeles which is a scouted Up Chicago which is a scout up Cleveland which is a scout up Santa Fe despite their different histories geographies and even cultures so just as in Biology we have to look at the multiple metrics of a city and ones that you can actually make measurements of and get good data on and there things as mundane so to speak as the length of all the roads the length of all the electrical lines the volume of buildings but then ones that are much more interesting fundamentally really and that is to do with socioeconomic quantities like wages the number of patents produced The Innovation that's being produced in the city disease number of AIDS cases number of flu cases all these multiple networks where we could get data and what was fantastic and so exciting was that when those data were plotted in the same way as they were in biology in this logarith mic fashion increasing things by factor of 10 and using as a size for the city just population and when we plotted that data sure enough it looked just like in biology straight lines on uh log logarithmic plot the the one caveat to that the data is much cleaner in biology much more variance in cities but we can understand that because biology life organism have been around for not just hundreds of thousands but in many cases hundreds of millions of years lots of time for the system to hone itself and get more become more optimal so to speak cities hundreds of years nevertheless what we saw was that there was this beautiful scaling laws for all of these metrics power laws Astound me on its surface power laws are a basic mathematical expression in its depth power laws reflect deep structural similarities among vastly diverse complex systems that compose our worlds power laws portray extraordinary Simplicity underlying great complexity power laws and gender self-organizing criticality a deep principle of the biosphere driving Evolution and also of the economy destruction parall describe a system that is scale invariant zoom out zoom in at whatever scale You observe with whatever ruler you measure the map looks the same many rare instances are common from metabolic and heart rates of animals to sizes of cities and earthquakes to frequency of terrorist deaths complex systems are built with networks a tree an insect a human a city all are networks and the mathematics of networks is clear that's the answer that's why power laws work so widely secret the implications are awesome power laws reveal regularities at the foundations of reality that's closer to Truth for complete interviews and for further information please visit closer
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