Six degrees of separation, or on Facebook 3.57. Why are we attracted to Kevin Bacon? Why are there some people with a bunch of friends, while others of us have few? Why do social networks look like brain neurons or street maps or the cosmic microwave background radiation? Why does preferential attachment in scale free networks act like gravity? Gravity? What if? Let’s do a little back of the envelope exploration of the gravity in our social networks.
On Facebook, it now takes only 4.57 hops to get from you to any other person. Instead of Kevin Bacon, let’s call her That Girl. So, my connection to her would go from me, to you, to your friend, to their friend, and then to That Girl. You count me as a hop, but the degrees of separation is 3.57 (there are three people between me and That Girl). We will assume that the half person is accounted for by your friend, as he has an extra large beard, and his friend, as she is tall and plays volleyball. Me, to you, to Beard, to Stretch, to That Girl.
If you divided the hops into equidistant units, you would get 1.1425 units per hop. The type of unit doesn’t matter, so let’s just call them iluminaries. Next, we can bring gravity into our discussion by looking at masses and acceleration and all of that, or we can look at effects. At the micro scale we are talking about (human interactions), gravity is impossible to measure. But remember, gravity may get weaker, but gravity never quits. Gravity also knows no bounds. It has been working on everything, and I mean everything for all but the first second of our 13.8 billion year old universe. Every piece of matter and every micro-joule of energy has at least one, but more than likely trillions of gravitational signatures embedded. So, let’s keep this simple and say in this case all gravity in our social network is of the same force. Let’s look at what happens to gravity’s effect as we take our hops to That Girl.
So what is the gravitational attraction between me and you, or any one hop? Well, gravity gets weaker by the inverse of the distance squared. Thus, for traveling just over one iluminari, we could say that our generic everything is equal gravity is now weaker than 0.766104, oh, uh, we need another unit, OK, let’s call them hugs (1/1.1425^2=0.766104). So on our first hop of 1.1425 iluminaries, gravity had a force of something or other (another new unit), but by the time I get to you, it is now 0.766104 hugs of it’s former force. Between me and That Girl, gravity is a shadow force at 0.047881 hugs of something or other. Let’s get rid of something or other by dividing everything by one something or other, and we end up with our hugs as the “force of gravity” at each hop. This is called normalization.
Let’s see what the loss of force does to our social network and how I am connected to That Girl. The average Facebook user has 338 friends. Some have more, some have less, but the person in the middle of Facebook has 338 friends. We could play around with the more friends the more gravity, but that is another post. I’ll just post myself at the center of Facebook with all of my 338 friends. If just gravity accounted for how connections are winnowed down to That Girl, we would expect at each hop, some number of connections would drop off and we would get to some non zero result at the end, hopefully near one.
Let’s explore that last statement just a tad more. While in reality the path to get to That Girl will be Me, You, Beard, Stretch, That Girl, yet there could be more than one path. So my 338 connections have some number of common connections with your connections, and you have some number of common connections with Beard’s connections, and so on. In the end, however, many connections are ruled out as we hop our way to That Girl. That what binds me to That Girl goes through each of our connection repositories (me, you, Beard, and Stretch). At our four hops, I still have 0.047881 hugs binding me to That Girl. So, let’s just use gravity to reduce the number of possibilities to get to That Girl.
If I have 338 friends when we start, the weakening of gravity will reduce the number of common connections with you to 259 (338*0.766104). At Beard, we are down to 50 (259*0.19153), and at Stretch we are at 4 (50*0.08512). When we get to That Girl, we are at the non-zero number of 0.2 (4*0.04788). Nifty. Our use of gravity seems to agree with the Kevin Bacon algorithm. However, it would be a stronger story if we ended up closer to one, as well, That Girl is a one. We fudged the 4.57 “distance” with Beard and Stretch, so it seems a stronger argument if we get closer to one.
Well, if this is about gravity, density matters. If you are a certain type of nerd you chuckled just then. Crazy thing, in network analysis today, we use a term called density to describe the total number of connections that could exist versus what actually does. Let’s say the planet Jupiter could have 100 moons, but the density of planets, moons, comets, dwarf planets, asteroids and other planetary material in our solar system limits it to 67 moons or a system density of .67. For Facebook, density is .12. Density is calculated by the actual number of connections divided by the total number of potential connections. Remember, Jupiter due to its size could have many more moons, it just hadn’t had the opportunity to capture those other 33. So, even though I only have 338 connections, I could have as many as 6834!
In this scenario, our 338 connections would spawn 6834 potential connections at you (338*((338-1)/2)*.12). From there, we have 4008 at Beard, 145 at Stretch and .81 at That Girl! So much closer to one.
We have a distance of 4.57 iluminaries. My gravitational attraction has captured 338 friends into my orbit, but because of the mass of my friends coupled with the potential density of friends around me, I am pretty assured of being connected to anyone on Facebook that comes within 4.57 iluminaries from me. Once I get to 375 friends, I am a lock to capture anyone on Facebook into my gravity well.
Why did the numbers come out so close using gravity parameters? Actually, I was somewhat surprised. Remember, I also took a bunch of liberties. I used how the force of gravity decreases as a stand in for the actual force of gravity. I also used network density to the advantage of my argument, yet this is all pretty well within the bounds of reasonable. In both of our cases, we had non zero/non negative numbers less than one. If you build a spreadsheet with these numbers, you can see how having just 375 friends gets you to one. Big data is giving us the tools to look at patterns. I argue many of our patterns are the result of gravity’s universal reach that never rests.