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Simple Maths Explains Why Rich Get Richer - Power Law - FutureIQ

10,693 views Wait, is this logic right? • Sep 02, 2023
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Slog Reference: Understanding Power Laws

Description

Power law distribution is one of the more interesting distributions that can be often seen in ordinary situations. The prominent examples of power law in real life are returns on different stocks, social networks and the number of followers on these sites, populations of countries, and the number of lies told by people.

All these things can be described by a power law. But what is a power law? How will it help you to learn how power law works and occurs in different situations? Let's find out all these things and everything else you need to know about power laws.

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More videos for you:
Benford's law: https://youtu.be/JHA2QpJKGN4
Karmanye Vadhikaraste: https://youtu.be/95Zi_4OthbY

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Chapters:
00:00 Power law
02:02 Example 1
04:18 Theory
06:23 Real Life Examples
13:34 What follows power law & what doesn't
16:20 How to spot?
20:50 Problems
24:40 How to stay safe?
27:22 The power law of effort

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Understanding Power Laws

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Transcript

Did you know that a quarter of all the lies are said by just 1% of the population? Okay. Uh so you're thinking exactly what I'm thinking. Right. Right. Uh why can't we do anything about them? Yes. Why do we have to keep doing that thing every 5 years for them? Not fair. Okay. I have no idea what he's talking about. But so this explains uh why there are so many lies. But it also gives us an idea of how this can be a quarter of all the lies.
Yeah, that seems like a fairly large number. Yes. In fact, this concept is going to come back again and again, right? That a very large fraction of something is contributed by a very small number of uh the population, right? Okay. Um I mean you must have heard of uh the parto law right parto principle that 80% of the output is generated by 20% of uh the people which kind of is seen in in a lot of social networks because the greatest number of followers are typically concentrated in those few accounts like Barack Obama or Justin Bieber or I don't even know PewDiePie on YouTube.
We need to get to the 10 million number subscribe. I hate saying that. Uh this concept is called a power law or a power law distribution. And today's episode we are going to talk about what are all the different places in normal life that power laws start showing up and how if you're not careful about understanding power laws, you can make mistakes. Conversely, how if you understand power laws, you know how to use it to your advantage. Right. Okay. Right. Power loss. Yes. So let's take an example of a social network.
The only power loss we know is the electricity going off in our house. Power loss. Sorry, bad joke. Continue. Example. All right. So um to give you an idea of where power loss come from. Okay. Uh we're going to take another toy example. Right. Imagine I'm creating a social network. Okay. Start with 100 of my friends and they know each other. Navin is on a very weird power trip and a toy trip. Recently we did an episode on the power of compounding. Now we are doing power laws and now he's giving toys as no toy examples not toys as examples.
So toy examples are very important in understanding the world. Okay. Yes. Thought experiments we call them in physics. Go on. So I start with I start a social network. It starts with say 100ish people. Okay. And there each person connects to their friends. Correct? Now every new person that is added to this social network. Okay. Let's say they immediately connect to up to say 50 people, right? Okay. Some of them connect to 10, some to 20, some to 40, whatever. But each person whom they connect to is proportional to how many followers or friends connections that person already has.
Seems quite reasonable, right? As in if somebody is very popular, more people are likely to connect to them. and somebody who's not very friendly, fewer people are likely to connect with you. Or somebody who is introvert, not necessarily not friendly, that also. Yeah. If you simulate this and then you grow it to a million people and then if you count how many connections each person has and sort them uh in decreasing order of number of connections, right? What you will notice? What do you think the curve is going to be like? Right?
We're sorting by decreasing order of number of connections. Do you think it'll be like a linear straight line going like this? I have a feeling it won't be a straight line. It is an exponentially decreasing curve. Right. There you go. The first person will have a ridiculously high number of followers like you know probably 500,000. The next one will also have high but it'll be a sharp fall. It will not be like just one or two less than the top right the 1% right? The next one will also be another sharp fall, right? And there'll be like a initially it'll be a very steep fall and then it starts flattening out and then by the time you come to the end it
just seems to be going almost flat. This is the reverse power of compounding. In the power of compounding, we looked at the other direction. We started slow and grew fast. This is top and then slide down. In fact, graph wise, huh? Yeah. No, it's the same concept, right? It's just we're looking at it in reverse like you uh said it occurs because of the same basic principles. Accidental knowledge happened. Yeah. So in power of compounding we talked about how compounding happens when the growth at any stage is proportional to the size at that stage.
Correct? Or the growth is a percentage. So where the interest is a percentage of the principle itself. Right? Um same thing is happening in our social network. Right? There is a person growth in how many connections a person gets is proportional to how many connections that person already has. Right? There is another name for this dynamic and the name that I don't really like very much. The name that I've heard is the rich get richer. Yeah. I mean you might not like it but everything in the world works on that principle.
That's why I don't unfortunately or fortunately or whatever. Right. I mean in some cases have a fight on this very well. In some cases it is very bad but because whenever there is a rich get richer dynamic in how a complex system grows you will end up with a situation where some metric that you are measuring. If you add it all up and then you sort by decreasing order of that metric you will find that there is a power law. is a very distinct power law and by power law what it simply means is that the curve that forms is an exponential decay. Uh mathematically it is a ra - x etc. That's why it is called a power
law. Okay. Right. But we are not going to talk about the mathematics. Okay. Today we are going to talk about what it means in uh I mean you know to you in your daily life and what are the implications of a power law. But before I wanted to impress on you that it occurs very naturally as a result of very reasonable well it's not always all natural there are some artificial situations also where the power law is kind of created as a result of situations so just to give examples right uh wealth distribution is a power law okay if you look at the richest person secondary HS person, third rich HS person, it will be a power law. It it actually
is true. So uh income distribution is a different power law. Yeah. If you look at just populations, right? Populations of different countries, China, India, uh US, so on, right? That is a power law. If you look at populations within a country like populations of cities and towns and villages, you will notice that there is a power law. There will be a few cities like with ridiculously high populations and then after that it falls sharply Delhi Mumbai within a city within a city if you look at populations of different uh areas of the pockets of the city you will notice there's a power law there also right so that is in fact one of the important
properties of anything depending on a power law and now that you mention it the concept of the rich getting richer applies to each one of these examples that you mentioned because uh cities keep growing exponentially in size because there are already lots of people then more people come and then it keeps growing whereas a smaller town or a village might not get as many people coming and settling there for multiple reasons that right and yeah and is a hot area that's why more people want to stay in and because there are lots of people in there there are lots of services there lots of shops there everything is there so then more people want to stay. Whereas if you go
like outside to an empty area, very few people stay there. So nobody wants to stay there, right? So it's a power law up there. Eventually there will be people staying there but it's a slow growth there. This is where you have to look at the other episode which is the power of compounding on how but anyway we are talking about the power law here. So the other important thing here that you noticed was that the whole thing looks very similar at different levels. Correct? world, country, city etc. So uh in fact this might remind you of fractals right self- similar and that is one of the important characteristics of anything that is uh described by a power
law right if you look at a social network the entire social network uh will look very similar to if you look at a smaller pocket of it just that social network within say India or within Pune or if you just take a subset of only the uh maths geeks in a social network that will have its own smaller power law. Right? So, uh fair it appears in other places also. If you look at the number of say if you look at a WhatsApp large WhatsApp group, right?
And you look at how many people are posting bad posts, right? Like offensive, it's usually one or two. It's literally usually one, two, or maybe like a small group of idiots. In fact, if you just look at count the number of bad posts by person and then you sort it in decreasing order, you will see a power law uh coming. That makes me wonder why admins don't kick off. If you look at traffic offenders in a city, huh, you'll notice there will be a power law there, right? If you get wait traffic offenders, power law, how does that work? Like all of or a major chunk of the traffic offenses are caused by one person or one
a small number a much smaller number like you know again 80/20 rule you call right 80% of the offenses will probably be by 20% of the people. So 50% will be by 1%. Why not take away their driving licenses and make sure that they never drive on the roads again. In fact, that is one of the ways in which you use a power law which is that whenever something is power law distributed it is not and you want to stop that behavior. It is not necessary to catch and punish everybody. You just take a small fraction of the top guys and you shut them down.
That results in two things. One is that it gets rid of a lot of the volume entirely and it serves as an example for the others. So can we just clip this little part, this little explanation by Naven and send it to everywhere where the power law is harmful to humanity like everywhere. Traffic offenders, liars, rich getting richer. Yes. So for example, if you are running a large WhatsApp group, right? All you have to do is that those few people who are doing most of the uh say rude posts, right? you scold them one or two times, maybe kick out one or two of them and after that the group is going to be very
well behaved, right? Uh there is actual studies of this on things like Reddit, right? Where you see that a lot of the offensive behavior comes from a very tiny fraction of the posters and moderators of groups where they uh clamp down on this behavior quickly. very soon the whole thing settles down into good behavior. Right? So the amount the good thing about the power law is that the amount of effort needed to improve things is much smaller than what you would think just by looking at the size right conversely power law right you said this is a power law it goes in reverse also right so if you don't shut down that 1% they can quickly overwhelm everything
and they can also infect other people and they can end up creating a culture of bad behavior. Right? So it has to be done early and it has to be done consistently. So this reminds me of a lot of things from uh a time when I was not doing this but let's not go there. Yeah. Uh uh but this this example that you gave is applicable in so many scenarios. So what you're essentially saying is if you let the people who are being nasty on WhatsApp continue being nasty then that will infect other people. So it's the one bad apple in a bunch example that applies here right? Oh all these behaviors are infectious and
the number of new people who start behaving badly will be proportional to the number of people behaving badly currently and that is a power of compounding it can take over everything so be careful I know yeah but here's what I'm thinking how how do I know when a power law is applicable and when probably a different kind of distribution is applicable because not everything will be power yeah in a power distribution. So in fact this is a very important point uh that you brought up right. Okay there are two distributions in nature which are very important. One is power law but even more important than a power law is the normal distribution otherwise known as
the bell curve. The bell curve right if you look at IQ's in the general population they are distributed as a bell curve. Correct. 100 IQ 100 is average and that is the maximum. Correct. And then as IQ goes up or goes down the number of people reduces right under that yeah for if you look at say heights of people that's also a normal uh distribution right so whenever there is something in nature where there is sort of a mean behavior and variation from the mean is not common right that is a normal distribution basically if there is no rich get richer uh dynamic going on right because I can't I can't get height from anywhere. It is basically something
that is inherent in me or I can't get IQ from anywhere. In fact, let us get into that a little bit more. Right? A lot of people are under the impression that if uh two parents are very high IQ then the child is going to be even higher IQ. Right? That's not at all true. Right? If you look at some of the smartest people in history and try to think of their children, most of them you have never heard of, right? That's because IQ's do not have a rich get-richer uh dynamic.
So what your family is an outlier, [Music] [Laughter] throw him for a loop. That means entire family is intelligent in case you didn't know. You should listen to the other members of the family and you know get a sense of their smartness. the entire family like this is a proper outlier in his expla in his example. Yeah. So the point is that in most of these cases there is reversion to the mean right if you take a high IQ person and go down say three generations from there you will end up with normal IQ's right so that's why IQ's are normally distributed but when there is a rich get richer dynamic then you will get a power law
distribution so you have to know uh either you have to know or you have to just study it and do some measurements to understand if this is a power law or uh normally distributed thing. Right? So uh how do I spot a power law if uh if I can spot a power law? Is there a way to in general try to think of does looking at average make sense here or not? Okay, let me I mean if I look at a class and I ask what is the average what are the average marks in this class that makes sense very likely the marks are going to be bell curve distributed right if I or performance in an office again a bell
curve distribution there will be uh excellent people there will be people who are slackers and most of the people are average but the income the salaries being paid out if I take a look at say a bar mm M right and the incomes in that bar the average doesn't really make sense because if Bill Gates walks into that bar then the average person in the bar is now a billionaire average right so their average doesn't make sense because the upper end doesn't really have a limit correct right it could be million it could be 10 million it could be 100 million it could be a billion it could be 100 billion even 100 billion doesn't seem to be the limit
right so that's the so their wealth and income is power laws right in fact if you uh plot uh I mean that's where we get this problem right which is that I mean let me talk in terms of the 80/20 rule where we said that you know 80% of revenues come from 20% of the customer correct similarly what you will see is that 80% of the income in a country will come from 20%. Now, of course, this is not a hard and fast rule. It varies a little bit. But because wealth and income are power laws, what you will find is that in every country in the world, the top 1% will have quite a large fraction of
the income. They do. They do. They do. And just that number by itself like if I just say that uh the top 1% in India get about uh 20% of the income. Right? That number by itself doesn't tell you anything. You have to look at it in comparison. Is it good or bad compared to other countries? Correct. It's an indicator. It's a measure. more important. You have to see has it been increasing in the last few years or decreasing. Right? That is more important because that number by itself we are humans are not good at intuitively understanding things related to power loss.
There is a threshold and this is a discussion that Naven and I keep having off camera also. There is my contention is that there is a number at which or there is a specific threshold at which top x have y% starts beginning to matter. Naven's contention is that it matters when you take into account other factors related uh to that particular statistic or to that particular measure or indicator. What I'm trying to say is that just that number by itself doesn't really give you a deep understanding of how bad uh this is, right?
It doesn't it it but we can't ignore that number either. Yeah. No, no, I'm not saying we should ignore that number. I'm just saying that that number has to be put in context. You have to put given much more data for that number to make sense. Right. Because whenever there is a power law, you are going to get ridiculous numbers like this. Right? 1% of the people are responsible for 25% of the lies, right? That just seems right. But then again, you know, 1% of people are getting 20% of the income, the same number because it came from the same power law.
Correct. No, I I think a lot of these times, a lot of the times when you read say headlines like this or reports like this, they are basically meant to attract attention in a somewhat clickbaity way. For example, we might title this video 1% of the people will watch 80% of this video, 99% won't watch beyond the first 3 minutes and it will get your attention. Yeah, it works. What the eventual result of it is that you reach this point and you heard all about the power law and now you're thinking where power law applies in your life. So it works, right? So I think you know you would argue that when it is used to point out inequality
it's a good thing. Let me give examples where it's a bad thing right where um let's let's look at stocks. Okay. Okay. If there is a person who has say 30 or 50 stocks that they have bought over the last 10 years. Mhm. Right. Now if you look at the return on each of these stocks and then you sort it by decreasing order of returns. This is going to be a power lawn, right? Yeah. There is a rich gets richer dynamic here. There is an unlimited upside here.
So, it will be a power law. Correct. Now, what this person can do is no matter what that portfolio is, the top through three uh stocks will have performed like crazy. And if this person talks about just those stocks, they will look like a genius of investing. And most likely that person actually believes that himself. It's usually a he when he will use this. And this person believes he's a genius in investing because he knows there are three stocks which ridiculously high returns compared to the stock market because it's a power law. It's always easy to come up with examples like this.
Very close to being misandress. Naven be very very careful. Well, I don't know women who go around boasting like this, but there are men who do. Okay. So, fine, fine, fine. Please take it as a joke. All of this is meant as a joke. Nothing serious. No misandry, no misogyny here. This was purely a joke. But, you know, so for example, whenever people are talking about stocks, always they talk about these examples without really talking about the full portfolio. In fact, most of these people don't even actually do that analysis themselves as to how their entire portfolio did. M right and that's where a power law trips you up. I'll give another fun example related to stocks.
Pick one of those top stocks, right? Like performed like crazy, right? Say uh 20 years ago you bought TCS stock or 20 months ago you bought GameStop. Yeah. So again there are studies which show that if you take just the daily returns of uh the stock right and you plot the daily returns in decreasing order of return again you will see a power law most days the stock didn't do anything interesting and there were a few days where there like these big uh jumps and isn't that what day traders are essentially trying to do time those jumps so they don't have to remain invested for the other days when it doesn't but it's a power law which means that
this is really really small number so the smarter thing to do is to stay invested yes power of compounding works best when you are there for a long time consistent but keep in mind that for example if in any important stock like this if you missed the top 10 or 15 biggest daily jumps you would have not made any reasonable able amount of money at all, right? You would have probably underperformed the stock market and this is for a stock which did like amazing returns. So, power laws trip you up in funny ways uh and you have to be careful, very very careful.
To put it differently, it is very easy to come up with clickbay headlines and uh you know genius sounding numbers when there is a power law involved. What you have to do is dig deeper and look at the overall picture not just the top. The digging deeper is something we recommend all the time. There is a reason this is future IQ. Yeah. Uh but yes. So let's let's take a look at or let's try and talk about how not to get tripped by power law. one example you've already given in terms of you know curtailing the nastiness of the most powerful users top users in a WhatsApp community if they are becoming typically in bad behaviors in various
situations like group situations uh is a power law and what that means is that a small fraction of users are responsible for most of the bad behavior shut them down before they infect everybody. That's that's one way of you know kind of combating the power law. What are the other ways? Another one is to keep in mind that if there is a power law average is not a great measure right we gave the example of Bill and a bar. Yeah. Uh so one of the thing averages make most sense when there is a bell curve a normal distribution there the average in the middle is also the same as the most common number right whereas with the
power law an average is just going to throw you. So remember uh there you probably want to ask for a median uh right I'll give you an example from right this week there was a report going around where they analyzed the number of books uh that people read in an year and there the average was 12 right and all of you are like sitting there and thinking oh [ __ ] right I am three books behind I didn't read 12 books and I'm now very below average right the problem is that the there are some people who are so crazy they read a ridiculously high number of books. So the number of books read is a power law.
Oh okay. And that's why if you look at the average there it's a wrong measure. Yeah. The median is four books. That makes you feel much better. Much better because that means I can read a book every 3 months and I still have about I don't know how many days to finish this book. So not just average anytime you see a statistic where it's saying you know the top x% have or control or something such a huge fraction of everything. Remember there's a power law going on. You need to dig deeper. Look at better uh numbers. Better numbers better.
Don't base your decisions or thought process on just that top x% number. Right. Yeah. I personally believe that there is something there for you to consider. But what that number is telling you to directly consider is not exactly the thing you should be considering. Most importantly, something that applies to you in your daily life. I would call it the power law of effort. Okay, that let's take the example uh that Tim Ferrris gives in his book the 4-hour chef. Okay, he says you can become a really high quality chef in 4 hours, right? And to to do do that what he does is that he goes talks to a bunch of chefs okay and tries to understand from them which are
the most important things to learn to be a master chef. Master Chef, right? not the series but the person and what he has realized at is that there is a power law uh there right okay to give a simpler example u you're trying to learn a language right if you look at all the words in English and you order them by frequency of how often they're used that's a power law okay uh you some of you might have heard it as a zip fian distribution because zip was a mathematician who found this or linguist maybe. But that's just one example of a power law, right? Which means that if you take the top n words
or top n phrases and learn those, you will have learned quite a large fraction of the actual language being spoken, right? Time to learn a few new languages. I guess in fact, Tim Ferrris does that uh and he's described it in that book. Similar concept applies to quality chef. This requires that somebody do that initial work. Absolutely. Yeah. Yeah. Yeah. So you you don't you can't just randomly go and learn random things, right? You have to get that distribution. So Tim Ferrris has described it in his book. Take a look. But it involves talking to the experts and finding out.
What's the book called again? The 4hour chef, right? Check it out. more generally more generally power law works for any kind of uh learning right um I mean if you talk in terms of the 80/20 distribution uh learning 80% of something is going to take just 20% of the time correct right correct uh it's only that you want to go from 80 to 90 90 to 95 95 to 99 that takes an increasing amount of time so if that is not your aim then you can learn something very quickly if you pick the right things to learn at the beginning based on the power law. Right?
So, I'll have to go and look at language learning apps that will uh do this 80% uh extraction for me and then probably learn a couple of languages this year. I'm going to try I'm definitely going to try. I wonder if there are any apps like that. If you know of any of those apps, please tell us in the comments. Another place where power law of effort would apply is say in your company, some business, right? uh 80% of the revenue comes from 20% of the customers. So you want to focus on that. 80% of the complaints come from 24% of the customers. So you want to try to get rid of those customers and you know don't
fire those customers but use pricing tricks to get rid of them. We have done an episode on pricing. Take a look at that. Everything you know about pricing is wrong. I think we called it. Yes. And uh it was a fun episode that you should definitely check out. Yeah. Okay. Uh I guess that basically sums up uh everything that we have to discuss about power law today. If you have any questions, any doubts, any additional thoughts, please feel free to send them to us either via comments or uh on our Twitter. But uh this is uh wait, didn't we do an episode on uh something similar with numbers and digits?
Whenever there is a power law involving numbers. So for example uh the number of people who voted for the first candidate versus second candidate versus third candidate I remember this exactly that would be a power law uh number of people who applied or insurance payouts will be a power law and what that means because humans are not good at understanding power laws. It means that when somebody tries to commit fraud, ah, they end up doing a normal distribution or a uniform distribution when it should have been a power law and then the numbers look wrong. We did it in episode called Benford's law which is gives you a very simple method of finding fraud.
Uh we didn't explain it at that time but now is the right time that comes from the power law. Keep in mind that human abilities are normally distributed but the outcomes based on those abilities are power law distributed right which means that effort you put in is always going to be linear right I mean in one month you can only put in maximum one month of effort right correct but what you get out of it can be a power law distribution right so remember karma There we go again. That's like the mantra to Naven's life. Oh, absolutely.
So, I think it would be a good time to go to that episode, check that out and then, you know, combine both these learnings and take away an additional learning with you and while you do that, we will go and take some water to drink. Sri Kant Naven, Future IQ. Thank you for watching till the end. If you like this episode, check out these others. You might like them also. And please share with your friends.