The Earth is flat.

Well not really, but the surface of the Earth is approximately a 2-dimensional or “flat” thing. Humans interact with each other on this flat surface, and the pure mathematical properties of 2-dimensionality impose some constraints on how these interactions play out.

Imagine instead that humans interacted with each other on a 1-dimensional thing instead of a 2-dimensional thing. Then everyone’s social circle would be more like a social line segment, and all of these social line segments would clearly divide into two parts, a “northern” half, say, and a “southern” half. Your southern friends would have their own southern friends, and it would be hard for you to meet them because your own southern friends would always be in the way.

Human society would be characterized by this linearity and sequentiality among friendships, and it wouldn’t be possible for your friends of friends of friends in the northern direction to join social line segments with your friends of friends of friends in the southern direction. Instead of “six degrees of separation,” there would be as many degrees of separation as there would be patches of friends between the southernmost person and the northernmost person—a lot of degrees. We’re lucky we don’t live on a 1-dimensional thing, but maybe we’d be luckier if we lived on a 3- or 4-dimensional thing.

It’s easy to notice that geography does not completely determine our relationships. You’re more likely to know your next-door neighbor than your next-next-door neighbor, but you may not know either of them well at all while your best friend actually lives all the way across town or maybe even on another continent. Certain communities are characterized by a higher degree of geographic determination than others. In suburban but especially rural areas, geographic distance is often a good predictor of whether two people know each other. In dense cities, it’s pretty terrible; on the one hand, people often live in skyscrapers, so their immediate environment is actually more 3-dimensional than 2-dimensional, on the other hand, many people don’t know anyone in their skyscraper. Something beyond distance over the 2-dimensional surface of the Earth or even in the small 3-dimensional vertical spread is clearly at play.

If we adopt some more advanced mathematical analyses, we can analyze how we might live on a higher-than-2-dimensional thing. I’ll describe one way to do so, and it involves two mathematical pieces. One piece allows us to define a “social distance” between two people, and the other allows us to take a whole bunch of distances between points and determine the N of the N-dimensional space with the least number of dimensions that allows all of these points with their pairwise distances to fit.

The Jaccard Index and Multidimensional Scaling

The first mathematical piece is the Jaccard similarity coefficient or Jaccard index, which is calculated as the size of the intersection of two sets divided by the size of the union of two sets. The Jaccard index is equal to 1 if two sets contain identical members and 0 and if two sets contain none of the same members. This index can be applied to people in many ways, but a straightforward one is by looking at their set of friends. With social media like Facebook, this is very easy to do. Take two people and count the number of mutual friends they have and count the total number of friends (without double-counting mutual friends) they have. Divide the first number by the second and that is their Jaccard similarity coefficient.

The second mathematical piece is multidimensional scaling. Multidimensional scaling is complicated, but briefly what it does is to take a set of entities with pairwise distances among them all and computes how much distortion must be applied to fit this web into a 1-dimensional space, or into a 2-dimensional space, or into a 3-dimensional space, etc. Imagine I have four points where each is 1 meter distant from all the rest. I can easily fit this into a 3-dimensional space, and the resulting shape that the points and edges would make is a regular tetrahedron, with a total distortion of 0.0 meters. I can somewhat fit this into a 2-dimensional space by making a square-with-diagonals, but then the diagonals are distorted from 1.0 to 1.414 meters (=sqrt(2)), for a total of distortion of 0.828 meters. I can’t really fit this into a 1-dimensional space at all without wildly distorting most of the distances.

The Jaccard similarity coefficient that we computed before for people can be interpreted as a measure of social closeness. Distance is the opposite of closeness, so the Jaccard index can be inverted (e.g. by taking 1 minus the Jaccard index or by taking 1 divided by the Jaccard index), and the resulting distances can be used in multidimensional scaling. You would run multidimensional scaling with your data under a constraint of 1 dimension, 2 dimension, 3 dimensions, etc. and see how many dimensions you need to achieve negligible distortion.

Let’s think about what we might get when we do this in a very rural environment, say the midwestern United States in the agricultural era. Each farm has a family, and each family regularly interacts with neighboring farms, but less and less often the farther away the farm is. The Jaccard similarity between family friends will correspond very closely to how far away farms are situated from each other in geographic space, and multidimensional scaling will show that this ensemble of distances fits fairly accurately into a 2-dimensional space.

Let’s examine a wildly different environment: a university campus, and let’s just use Facebook friend lists as I suggested earlier. People make friends through their dorms, their studies, their groups, their parties, and this structure becomes manifest in who friends who. A multidimensional scaling analysis would clearly not allow this web of relationship to fit into 2 dimensions. You’d have to actually perform the analysis to get the right answer, but I’d imagine you’d get far more than 3 dimensions, corresponding to majors, housing, class, ethnicity, personality, etc. The level of dimensionality could be computed for many different Universities, and this number would be an interest fact about that University. Universities with more targeted focuses and smaller or less diverse student bodies would likely have lower dimensionality.

Entire cities and countries could be analyzed this way, and the analysis could be extended back through history. A clear pattern would emerge: dimensionality would increase steadily over time and with population density. Dimensionality would also be higher with greater diversity. Areas like New York City, the San Francisco Bay, and London would have very high dimensionalities. Rural areas would generally have dimensionalities down to Earth’s surface’s default of 2.