Documentary: Why do Power Laws Work so Widely? | Episode 2207 | Closer To Truth

Premiered on 12 Dec 2023

Make a donation to Closer To Truth to help us continue exploring the world's deepest questions without the need for paywalls: https://shorturl.at/OnyRq “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. ▶ Early-release episodes of Season 22 are available now on our website: https://bit.ly/3QwMzIA ▶ For subscriber-only exclusives, register for free today: https://closertotruth.com/register/ Closer To Truth host Robert Lawrence Kuhn takes viewers on an intriguing global journey into cutting-edge labs, magnificent libraries, hidden gardens, and revered sanctuaries in order to discover state-of-the-art ideas and make them real and relevant. ▶ Free access to Closer To Truth's library of 5,000+ videos: http://bit.ly/376lkKN Closer To Truth, hosted by Robert Lawrence Kuhn and directed by Peter Getzels, presents the world’s greatest thinkers exploring humanity’s deepest questions. Discover fundamental issues of existence. Engage new and diverse ways of thinking. Appreciate intense debates. Share your own opinions. Seek your own answers.

Intro

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[Music]

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[Music]

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[Music]

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to seek the foundations of re reality my

0:30

quest is to explore diverse fields of

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knowledge fundamental physics Life

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Sciences philosophy psychology

0:38

Consciousness even claims of the

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non-physical another approach another

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way of thinking is to search for rules

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regularities principles that cut across

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diverse fields of knowledge to seek deep

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structural similarities that may underly

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and unite despite superficial

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differences

1:01

one such candidate I hear is what are

1:03

called Power laws which are mathematical

1:06

relationships expressed by exponents and

1:11

logarithms could power laws be that

1:13

powerful where are they found how do

1:15

they describe the world why do power

1:18

laws work so

1:20

widely I'm Robert Lawrence cun and

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closer to truth is my journey to find

1:28

out

How Power Laws Work

1:35

exponents are taught in fifth or sixth

1:38

grades Primary

1:40

School logarithms in 9th grade Junior

1:44

High

1:45

School how could such simple arithmetic

1:48

have such explanatory

1:50

potency the rather incredible claim is

1:52

that power laws work their magic by

1:54

describing complex systems which are

1:57

found everywhere biology Society the

2:01

economy I go to the pioneering Center of

2:04

complex systems research the Santa Fe

2:07

Institute in Santa Fe New Mexico what an

2:10

ambitious Mission they have to discern

2:13

underlying unifying shared patterns in

2:17

complex physical biological social

2:19

cultural and technological

2:24

worlds I begin with a theoretical

2:27

physicist with a big idea who in en

2:29

Visions Universal scaling laws which

2:33

pervade biology from molecules to cells

2:36

to whole organisms to entire ecosystems

2:40

and even to human constructed entities

2:42

like cities and companies Santa Fe

2:45

Institute pass president Jeffrey

2:48

West Jeffrey you've pioneered the vision

2:52

of power laws and scaling of bringing

2:54

out the hidden order of complexity I'd

2:57

love your personal experiences and how

2:58

it came about the world around us sort

3:02

of has this sort of messiness to it you

3:04

know it doesn't seem to have order you

3:06

know you look at a forest or you look at

3:08

a city or you look around this room at

3:11

First Sight it's hard to see where there

3:13

would be any order in the great

3:15

complexity and diversity of the of the

3:18

world around us but here's what's really

3:20

amazing if you look at it through the

3:22

right lens and I think this is one of

3:24

the great powers of Science and

3:27

Mathematics there is in fact underneath

3:30

that great complexity and diversity

3:32

great order and it's expressed through

3:35

things that we call power laws if you

3:38

look at something that's measurable how

3:41

much food you need to eat each day to

3:42

stay alive so that's called metabolic

3:44

rate and if you look at that as a

3:47

function of the size of the animal and

3:51

if you plot it in the right way

3:53

logarithmically meaning that instead of

3:55

1 2 3 4 5 you plot it uh by factors of

3:59

10

4:00

1 10 100 thousand in terms of the amount

4:04

of food that's that is needed and in

4:07

terms of the size of the animal then

4:10

what is extraordinary is underneath that

4:12

complexity is revealed an extraordinary

4:15

Simplicity because if you plot it that

4:17

way instead of the points lying

4:19

arbitrarily all over the graph which is

4:21

what you might naively have expected

4:24

given the individuality and uniqueness

4:27

of every organism in fact they all line

4:29

up on a straight line it's not only that

4:32

they end up on a line it's a straight

4:35

line so it's it's it's about the

4:37

simplest possible kind of order and that

4:41

is called a power law and what it

4:44

expresses is the the percentage change

4:47

as you change the size of an animal um

4:51

is proportional to the percentage change

4:53

in its size so simple example if you

4:57

double the size of an organism

5:00

you might naively have expected since

5:03

there's twice as many cells you'd expect

5:05

you need twice as much food per day to

5:08

keep it alive but what it turns out is

5:12

true is that you don't need twice as

5:13

much you only need 75% as much and that

5:17

is true every time you double so if you

5:19

look at an animal that is 40 gram

5:22

relative to one that's 20 G it only

5:25

needs 75% more food but if you look at

5:29

one that's 40 kg relative to 20 kg it

5:32

also needs 75% more so there's this

5:36

extraordinary regularity in the

5:39

proportions um as you scale up and turns

5:42

out that phenomenon

5:45

permeates life around you whether you

5:47

look at Rivers buildings genomes and so

5:51

on it doesn't matter what variables

5:53

here's what is also amazing is that the

5:56

slopes of those lines the exponent of

5:58

the power LW

6:00

in

6:01

biology almost invariably is a simple

6:04

multiple of the number one quarter so

6:08

the number four seems to be a magic

6:12

number that is constraining all of life

6:15

around you whether you look inside you

6:18

or you look outside you at the at

6:21

ecosystems and the environment that that

6:24

universality is kind of remarkable given

6:27

this uh belief

6:29

that everything has its own unique

6:32

evolutionary history everything is

6:34

historically contingent and yet somehow

6:38

it's being constrained there's some

6:40

natural laws that work constraining

Why do Power Laws Work

6:44

it it's astonishing who could imagine

6:48

that vastly different phenomena could

6:50

have such similar scaling relationships

6:53

biological properties of animals

6:55

seemingly have nothing in common with

6:57

environmental properties of cities yet

7:00

they do they have a fundamental

7:02

principle in common size determines all

7:07

following the simple mathematics of

7:09

exponents and

7:11

logarithms power laws I want to rebel

7:15

against the claim too simple it seems to

7:18

explain too much but I cannot gain say

7:22

the data where does this leave me what

7:24

kind of world do we have someone has to

7:27

start explaining why why do power laws

7:30

describe complex

7:34

systems I know a theoretical biologist

7:37

who has pioneered self-organization as a

7:40

driver of complexity in biological

7:43

systems a long-term member of the Santa

7:46

Fe Institute this Citadel of complex

7:48

systems Stuart

7:51

calman I got into Power laws because per

7:54

Bach in 1988 came to the S Institute and

7:58

he talked about self organized

7:59

criticality and per and his colleagues

8:02

had a gorgeous model in which they drift

8:04

sand down onto this table they just keep

8:06

drifting it and pretty soon it piles up

8:07

to a big sand pile and the sand pile

8:10

grows to the edge of the table and they

8:12

keep disting sand onto it and you get

8:14

little and large Avalanches of sand

8:16

sliding over the size of the table

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there's a lot of little avalanches and

8:20

not many big ones and if you plot

8:22

logarithm of the size of the avalanche

8:24

on the x-axis and logarithm of The

8:27

Avalanches of that size on the y- axis

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it's a straight line down to the right

8:30

with a slope of minus 1.5 that

8:33

self-organized criticality and they

8:35

became quite famous for it literally the

8:37

day that I heard it per and I and some

8:40

others from The Institute went off for

8:41

lunch and I said per I wonder if there's

8:44

large and small uh Extinction events in

8:46

the record I wonder if if the bias were

8:49

self-organized critical so a bunch of us

8:52

then got involved I made models in which

8:55

it turns out that the size dist of

8:57

Extinction events really is a power law

8:59

or very close to a power law it turns

9:01

out that all of the critters get to to

9:04

peaks of Fitness when there's a phase

9:06

transition another edge of chaos where

9:08

the system is self-organized critical

9:10

and you get a power LW distribution of

9:12

Extinction events and you get a power LW

9:15

distribution of the lifetime of a

9:16

creators as well it turns out that the

9:18

lifetime distribution of Genera is in

9:20

fact a power law and then we also used a

9:23

cousin of that model to to wonder if the

9:25

economy is self-organized critical and

9:28

the extin events are in fact uh sharian

9:32

gilds of creative destruction that means

9:35

that in in an economy certain

9:37

Technologies or companies would go out

9:39

of existence because other things would

9:40

come in when the car comes in it

9:42

replaces the horse with the horse goes

9:45

the the Saddlery and the Corral and the

9:47

buggy whip but with the car comes the

9:50

opportunity for uh gas industry paved

9:54

roads uh fast food restaurants motels

9:57

and suburbia that's a big creative

9:59

destruction so these models give us a

10:02

power law distribution of sharian Giles

10:04

of creative destruction and a dream that

10:07

may not be a dream is that the biosphere

10:09

is self-organized critical and that the

10:12

economy is self-organized critical they

10:14

both have power laws what would be the

10:16

significance of that to have a theory of

10:20

how things evolve where the things are

10:23

so radically different which is the

10:25

biological evolution of of the biosphere

10:28

versus the the the economy so beautiful

10:31

there are plants and and fungi where the

10:34

the fungus fixes nitrogen and the plant

10:36

makes sugar and they the two Critters

10:39

exchange them the exchange

10:42

ratio uh given the metabolic cost to

10:45

make them and an advantage for getting

10:46

them is exactly the analog of price it's

10:49

not an analog it is price so the baser

10:51

has price new ways of making living in

10:54

the biosphere often come with mutualists

10:57

of price and the form of price the same

11:00

thing is going on in the economy where

11:02

new new New Economic niches emerge and

11:05

and pric is right there to allocate

11:08

resources uh in the ways that we

11:10

understand so new goods and services

11:12

come into the economy new species come

11:14

into the bias a one of the stunning

11:17

things about about the economy is that

11:19

50,000 years ago there was maybe a

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thousand goods and services and there's

11:23

now billions it's about the number of

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goods in the economy in the next period

11:27

and the basic idea is that we create new

11:30

things by recombining them in all

11:31

possible ways like like jury rigging

11:33

with junk in your garage it's easier to

11:36

jury rig with a lot of things in your

11:38

garage than only one thing in your

11:39

garage so so there an exponential

11:42

component to it yeah it's more than

11:43

exponential it's hyper exponential and

11:46

uh I wrote down an equation and I looked

11:49

at it and it looked like what would

11:50

happen is that nothing would happen for

11:52

a long time and then all of a sudden it

11:54

would it would just hockey stick upward

11:56

well it does we call this the tap

11:58

equation in the tap roses where tap is

12:01

the theory of the adjacent possible I

12:03

think this is the growing Juggernaut of

12:06

the global economy it's crashing upward

12:09

and it's now impacting the planet

12:11

everywhere we've got global warming even

12:14

worse we've got massive Extinction

12:16

events the expectation the UN says is

12:19

that a million species 20% of of species

12:22

will go extinct by 2050 we are

12:24

overwhelming the planet with the

12:26

Juggernaut of our own creativity

How do Power Laws Work

12:30

Stu is a big thinker linking the

12:32

biosphere and the economy with power

12:35

laws defining each's complex system and

12:38

self-organization when Stu says it's so

12:41

beautiful I can almost feel the

12:45

beauty power laws have such astounding

12:48

explanatory

12:50

potency but while the theory has been

12:53

General and expansive the data has been

12:55

specific and targeted how would power

12:58

laws applied to large sets of

13:02

data I meet an external Professor

13:05

visiting the Institute and expert in

13:07

network science and computational social

13:10

science Aaron

13:13

clet the thing that makes power law is

13:15

interesting is that they have this

13:17

property called scale invariance if you

13:19

examine the size of a system using one

13:21

kind of a ruler and then you change the

13:24

size of that ruler to look at a system

13:26

at a larger scale you see the same kind

13:28

of structure

13:29

regardless of which kind of ruler you

13:30

use so an example of this would be

13:33

something like coastlines so if you try

13:35

to measure the length of a coastline uh

13:37

coastlines are actually fractals and so

13:39

as you zoom in to a smaller part of the

13:43

coast of Great Britain for instance it

13:45

has the same sort of statistical Wiggles

13:47

that the entire coastline does if you zo

13:49

in some more you see the same Wiggles

13:51

again so if I measure the coastline

13:53

using a ruler the size of a kilometer or

13:55

a meter or a centimeter I see kind of

13:57

similar structure

13:59

that's the notion of scale and variance

14:00

that if I zoom out from a system the

14:03

same Properties or the same rules govern

14:05

the shape of that system it gives us the

14:07

idea that we could sort of tie together

14:09

phenomena that seem completely different

14:11

from each other using a single kind of

14:13

physics or rules and if we can find a

14:16

system that exhibits this property of

14:18

scale and variance then we have some

14:20

sense that there's something really

14:21

fundamental about it that we've

14:22

understood so power law is a way of

14:26

describing a system uh that has this

14:29

scale and variance okay but certainly

14:32

not all distributions are power laws

14:34

yeah so a lot of the quantities that we

14:35

see in everyday life like heights and

14:37

weights and things they don't follow

14:38

power law distributions so if I tell you

14:40

that the average height of an American

14:42

citizen is you know 5 foot seven or

14:45

something like that then that tells you

14:46

a lot about the whole distribution

14:48

because there's nobody who's twice that

14:50

height um but Power laws are different

14:52

they're they sort of confound our

14:53

intuition because the largest examples

14:56

of things that come out of them are

14:58

orders of magnitude sometimes larger

14:59

than the average so if I tell you that

15:01

the average size of American city is

15:03

somewhere between 50 and 100,000 people

15:05

that gives you no inclination that

15:07

there's a New York City or an LA lying

15:09

out there in the tail of the

15:10

distribution so when we think about the

15:12

scale and variance we're really thinking

15:13

about how do we get to bigger and bigger

15:15

and bigger things and so that is

15:17

thinking about the tales of the

15:18

distribution the rare events that happen

15:20

to be extremely large and if the

15:23

quantity really does follow a power law

15:25

then that tells us something very

15:27

powerful about well how life is it to

15:29

see a city that is a hundred times as

15:31

big as New York City at some point in

15:32

the future of civilization um if the

15:35

size of the Cities really do follow a

15:36

power law forever then knowing the

15:39

scaling variants exist tells us how

15:40

often those things should occur well the

15:42

excitement about power laws is really

15:43

because we see these heavy tale

15:45

distributions everywhere it it almost

15:47

doesn't matter what kind of modern data

15:48

set you look at you know you could you

15:50

could grab data from Book Sales you

15:53

could grab data from um earthquake sizes

15:55

you can get data on just about anything

15:57

in the world now if it's related to a

15:59

social or biological or economic complex

16:02

system um it'll exhibit these heavy tail

16:04

distributions and so the excitement is

16:06

the idea that maybe we can learn

16:07

something about how these complex

16:09

systems really work by starting at these

16:11

large scale patterns and working

16:13

backwards to the mechanisms you've

16:15

applied power laws to terrorist attacks

16:18

the idea here was to look at the sizes

16:21

of terrorist events and by size I simply

16:23

mean the number of deaths since that's a

16:24

very easily measurable thing and the

16:27

data sets that exist for uh uh terrorism

16:30

worldwide are pretty large in fact they

16:32

span the past 40 years or so they

16:34

include 30,000 events 911 is of course

16:36

largest event in any of these data sets

16:38

it killed approximately 3,000 people the

16:40

next largest event in the data set that

16:42

I've been working with uh is only about

16:44

500 deaths and so already you can see

16:47

there's a big gap but the important

16:48

thing to notice is that in these data

16:50

sets is that the typical size is roughly

16:52

around four or five deaths so 911 was

16:55

almost a thousand times bigger than the

16:57

typical size of of a terrorist event

16:59

somewhere in the world and the analysis

17:02

of this data set showed that this

17:04

distribution of terrorist event sizes is

17:06

plausibly a power law suggesting that

17:09

there may be a scale and variant

17:11

mechanism underneath that explains how

17:13

it is that such disperate phenomena that

17:16

go into the production of a terrorist

17:17

event the planning the ideology the

17:19

technology could all be somehow wrapped

17:22

up in this nice clean pattern what would

17:25

that imply if if that were the case if

17:28

there really is some sort of scaling

17:29

variant mechanism it would imply that

17:31

there's something very fundamental about

17:33

the production of political violence in

17:35

this context of terrorist events and

17:37

that would allow you to make forecasts

17:39

for instance about how often you might

17:41

expect to see events of a certain size

17:44

in the next 10 years and that would be

17:46

useful for planning purposes for

17:47

instance if we knew the true risk of a

17:49

large terrorist event then we could

17:51

Marshal our resources better than simply

17:53

freaking out about it in the

The Hidden Labyrinth of Natures Secrets

17:56

media to air in power laws are

17:59

revelatory because these heavy tail

18:02

distributions or diverse data sets with

18:05

many rare events occur everywhere

18:08

complex systems abound in biological

18:10

social economic domains and because

18:13

power laws operate similarly can we

18:16

infer mechanisms for how complex systems

18:19

actually

18:20

work is this then the final step the

18:24

hidden Labyrinth of Nature's Secrets

18:27

what could be the actual mechanism can

18:30

we have an underlying theory of complex

18:33

systems would this answer my question

18:36

why do power laws work so

18:39

widely I return to scaling Pioneer

18:43

Jeffrey

18:44

West Jeffrey I've appreciated how power

18:47

laws work in the biological realm you go

18:51

way beyond that and and and bake that

18:54

into the fundamental structure of our

18:56

social systems one of the things that

18:57

was so EXC exciting actually about the

19:01

biological work was actually extending

19:03

it into the kind of socioeconomic realm

19:06

and that was motivated from the

19:10

underlying theory of why we have all

19:13

these scaling laws in biology why do we

19:15

have all these extraordinary Universal

19:18

scaling laws that cover every aspect of

19:22

physiology and life history no matter

19:25

what organism at what scale it's kind of

19:28

amazing and that theory was based on the

19:33

fundamental idea that um any complex

19:37

system that meaning a system that has an

19:41

enormous number of constituents like

19:44

ourselves for example requires in order

19:48

to for its sustenance requires a

19:51

network and whether it's so in our case

19:55

surcy system our respiratory system our

19:58

renal system even our bones um our

20:01

neural system and uh these systems

20:04

primarily transport energy or

20:07

information from usually Some

20:10

central place to the the cells and

20:14

within cells or are networks that

20:16

distribute them within cells and the

20:18

idea was that the origin of these

20:23

scaling laws is in the generic

20:27

mathematics

20:28

of networks that the networks themselves

20:33

obey general laws that transcend the

20:38

evolved design so it does doesn't matter

20:40

that it's a tree or human being or an

20:43

insect they're all networks and it is

20:46

that mathematics transcends that design

20:49

and then you can transport that Network

20:51

to social system ex so the idea was so

20:54

when we did that we put that into

20:56

mathematics when you put that the those

20:59

ideas these Universal properties of

21:01

networks into mathematics and start to

21:03

calculate all the multiple properties of

21:07

organisms amazingly

21:10

and remarkably comes out these quarter

21:14

power scaling laws these and the origin

21:16

of the number four and even the origin

21:20

as to why it is that the number of

21:22

heartbeats should be the same for a

21:24

shrew mouse a giraffe and a human being

21:26

and a whale so we have this IDE idea

21:28

that we have these scaling laws we

21:30

understand them because of the network

21:31

structures that permeate all of biology

21:34

necessarily

21:35

so and you realize of course that if you

21:38

look at instit social institutions and

21:41

in particular cities but companies and

21:44

uh universities whatever they are also

21:47

Network systems I mean cities obviously

21:49

are there's Road networks and uh gas

21:52

lines and water lines but also social

21:56

networks that is doing what we're doing

21:58

here talking to each other exchanging

22:01

information but the idea fundamentally

22:04

that there are networks and that

22:07

networks obey Universal

22:09

mathematics then gives rise to the idea

22:14

the suggestion that my God cities too

22:18

probably scale with respect to one

22:20

another that despite appearances New

22:23

York maybe is a scouted up Los Angeles

22:26

which is a scouted Up Chicago which is a

22:28

scout up Cleveland which is a scout up

22:30

Santa Fe despite their different

22:33

histories geographies and even cultures

22:36

so just as in Biology we have to look at

22:39

the multiple metrics of a city and ones

22:44

that you can actually make measurements

22:45

of and get good data on and there things

22:48

as mundane so to speak as the length of

22:50

all the roads the length of all the

22:53

electrical lines the volume of buildings

22:56

but then ones that are much more

22:58

interesting fundamentally really and

23:01

that is to do with socioeconomic

23:03

quantities like wages the number of

23:05

patents produced The Innovation that's

23:07

being produced in the city disease

23:10

number of AIDS cases number of flu cases

23:13

all these multiple networks where we

23:15

could get data and what was fantastic

23:19

and so exciting was that when those data

23:24

were plotted in the same way as they

23:26

were in biology in this logarith mic

23:28

fashion increasing things by factor of

23:31

10 and using as a size for the city just

23:33

its

23:35

population and when we plotted that data

23:38

sure enough it looked just like in

23:41

biology straight lines on uh log

23:45

logarithmic plot the the one caveat to

23:48

that the data is much cleaner in biology

23:51

much more variance in cities but we can

23:53

understand that because biology life

23:57

organism have been around for not just

24:00

hundreds of thousands but in many cases

24:02

hundreds of millions of years lots of

24:04

time for the system to hone itself and

24:08

get more become more optimal so to speak

24:12

cities hundreds of years nevertheless

24:16

what we saw was that there was this

24:19

beautiful scaling laws for all of these

Conclusion

24:25

metrics power laws Astound me on its

24:28

surface power laws are a basic

24:30

mathematical expression in its depth

24:33

power laws reflect deep structural

24:35

similarities among vastly diverse

24:37

complex systems that compose our

24:39

biological and social

24:41

worlds power laws portray extraordinary

24:45

Simplicity underlying great

24:47

complexity power laws and gender

24:50

self-organizing criticality a deep

24:53

principle of the biosphere driving

24:55

Evolution and also of the economy

24:58

driving creative

25:00

destruction parall describe a system

25:03

that is scale invariant zoom out zoom in

25:07

at whatever scale You observe with

25:09

whatever ruler you measure the map looks

25:12

the same many rare instances are common

25:16

from metabolic and heart rates of

25:18

animals to sizes of cities and

25:20

earthquakes to frequency of terrorist

25:23

attacks and

25:25

deaths complex systems are built with

25:29

networks a tree an insect a human a city

25:33

all are networks and the mathematics of

25:36

networks is clear that's the answer

25:40

that's why power laws work so widely

25:44

networks Nature's deep

25:47

secret the implications are awesome

25:51

power laws reveal regularities at the

25:54

foundations of reality that's closer to

26:01

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26:08

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26:13

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