We'll start off by looking at the most famous experiment that really started all of the theory and thinking about the small world or the six degree of separation phenomenon that occurs today. And so, in 1967 Stanley Milgram did an experiment. And Stanley Milgram was a social psychologist. And what the experiment consisted of was, started with 296 people who were living in Omaha, Nebraska. And what they were given was, they were given a letter that was destined to some stockbroker in Boston, Massachusetts. So they're given a letter. Basically. And what it has on the letter is some information which includes the person's name, their occupation, and their address. So the goal of this is to get this letter. To the recipient. Using the minimum number of hops possible. So you want to get it to the stockbroker in Boston, Massachusetts, by going through the smallest amount of people. So, the rule of the game though is that you can only send it to someone who constitutes your friend. So the person you send it to has to be your friend. And a friend. Is someone in this case, who you knew by first name. Who you know on a first name basis. So, the idea is that you're starting here maybe and you have to get it to somewhere, someone that you don't know but whose in Boston, Massachusetts. So you could send it to. One of your friends, than that person can then send it to one of their friends. And so forth, then they can send it to one of their friends. Maybe they'll send it back to one of their friends over here. And then we will get lucky and this person happens to know the recipient, right. So you want to get to this point, and eventually the letter has to get to the destination. And then he looked at all of the different implications of this. And so to get to the end, really, the end is really determined once you get to someone who knows the recipient by first name. So once you get to the node here, the game really has ended. So, now, how would you forward? So supposed you were someone participating in this experiment. How would you forward it? And keep in mind that, you know, there's no internet or anything. You see, can't look up to see who would possible know the recipient by first name. You have to just go based upon what you know locally. So, one way you might forward is based upon location, right. So do you know someone near Boston? Do you know someone who lives. Either in Boston or near Boston who has a higher chance since they live in Boston of knowing the recipient by first name than you do. So you may think okay well if I saw someone who lives in Boston, they probably have a higher chance of knowing this person than I do. So I could forward to them. First way to do is based on location. Another way is based upon the occupation of the person. Which you may not think would be that important but really it turned out to be quite important in this experiment. And that's really just because different occupations have different social circles and people make connections amongst others in similar occupations. So the other one is do you know stockbrokers? So if you know a stockbroker, you might forward it to that person because you think okay, well this person has a higher chance of knowing the recipient that I do. And be even better if you knew someone who was a stockbroker in Boston. Because then you'd say, wow, they probably have a really high chance of knowing the person. And so, if you try to think of what the links would look like in this scenario from source to destination, you'd think, well there's probably going to be a few short range links, right? Maybe one or two, long range links, but also some short range links too. So, the long range links are really going to go from, you know, from where you are to closer to where the person is. Kind of like your. You're traveling, right? Usually you jump on a highway, and that highway's going to take you really the majority of the distance from the source to the destination. And then along the way, you have some of these smaller intermediate paths that you may smaller roads that you may go for. So, this long range rink, link here could be okay. Well, if you're the source, and if. Distance here really can be quantified in a number of ways but supposes is just geographic distance. Alright so this could say okay well you're the source and you forward it to someone you know who is in your area who happens to be a stockbroker. And they know someone whose a stockbroker in the Boston. Massachusetts area stockbroker over this one long range link and so forth. So, probably going to be one or two long range links but also some short range links. So, let's take a look at some of the results of Milgram's experiment. So, out of the roughly 300 letters that were assigned to be sent About 217 of them, or exactly 217 of them were actually sent out. Which means that the the remainder, the 298 minus 217 just ended up not participating. Or maybe felt that there was no way they could get to the destination. And out of those 217 that were sent, that actually went. From the, the source to at least one other hop. 64 of them actually arrived and made it to the destination. So this has seemingly small arrival rate of about 29.5%, or about 30% if you actually take 64 and divide it by 217. But a replica via email done years later actually only had 0.15% arrival rate. Which is pretty remarkable, because you'd probably think that the one that was happening via email would've had higher arrival rate than the one that was happening just via normal mail. But, turns out that, the opposite is the case here. That 30% of the letters that were actually sent. Did end up getting to the destination in Boston, to that, stockbroker. So, from the 64 that actually arrived, Milgram basically looked and he wanted to see what the number of hops was to get from source to destination. So. We can plot, these results here. This is the number of hops. And this is the frequency of the amount of time that number of hops came up when you consider all of the 217 that were sent. So, we can see for instance that two people actually just knew the destination already. And just send it out that way. and, on the other hand three people took ten hops. To get from the source to the destination. Then the majority of them though were obviously in the mid point between one which is the minimum and. Ten which is the maximum. And it turns out that the median number was six hops. Right, so this, this right here six is actually the median if you take the median of this. And the mean which is, slightly different than the median, in case you don't, if don't know the difference. Really for. In order to calculate the mean of this distribution we do just like we have done with ratings, calculating mean ratings for Amazon for instance. You just take the total frequency times the number of hops. So in this case you do, okay 1 hop came up twice so that's 1 times 2. 2 hops came up 3 times, so that's 2 times 3. 3 hops came up 8 times as 3 times 8 you add all those up and you just divide by the total number in this case to get the mean. And the mean is actually lightly lower here of 5. sorry. Of 5.2 so the mean is 5.2 but their roughly. The same. They're on the same order in this case. This is where the concept of the six degrees of separation originally came from, from Milgram's experiment of what he saw. So researchers have long expected that the social distance, which is really the average number of hops it takes to reach anyone. In a population. So if you consider all different variations, in this case empirically, because we're just, basically taking a number of samples and using that to compute the average number of hops from that case. That the average number of hops to reach anyone in a population is, is going to grow small as the size of the network increases. So as the network gets bigger and bigger, the total number of hops it takes to reach someone is still going to be, which is going to grow roughly very small. Relative to the total number of nodes that are in the network and, so as we said, these average number of hops which defines a social distance seems to grow very slowly with the size of the network and Milgram's experiment is just one example of this concept of small world. In which we see that, one, that there is a small number of hops, and two, that, even in a very large network like this, and as the network size increases that the number of hops that it's going to take is going to grow very slowly. There's also a number of other examples that have. Emerged that illustrates small wealth phenomenon in, in society today that we'll take a look at next.