I’ve seen many “spatial” models in social science. Such as models where voters and politicians sit at points in a space of policies. Or where customers and firms sit at points in a space of products. But I’ve never seen a discussion of how one should expect such models to change in high dimensions, such as when there are more dimensions than points.

In small dimensional spaces, the distances between points vary greatly; neighboring points are much closer to each other than are distant points. However, in high dimensional spaces, distances between points vary much less; all points are about the same distance from all other points. When points are distributed randomly, however, these distances do vary somewhat, allowing us to define the few points closest to each point as that point’s “neighbors”. “Hubs” are closest neighbors to many more points than average, while “anti-hubs” are closest neighbors to many fewer points than average. It turns out that in higher dimensions a larger fraction of points are hubs and anti-hubs (Zimek et al. 2012).

If we think of people or organizations as such points, is being a hub or anti-hub associated with any distinct social behavior? Does it contribute substantially to being popular or unpopular? Or does the fact that real people and organizations are in fact distributed in real space overwhelm such things, which only only happen in a truly high dimensional social world?

How is popularity being defined? By being closer to some neighbors more than average than others from subspace selection?

I could see that some hubs have particular, similar or different social behaviors than other hubs (or antihubs).

How important would being popular/unpopular be along those dimensions if most of those neighbors around a particular hub don't really drive the outcome of the overall system?

How can we begin to look at how the models will change?

Maybe if we could construct a vector for every person (org) with every entry representing all possible aspects that could describe such, and a matrix representing of the interactions between such groups, we could look at the decomposition of such a matrix (like a projecting such onto a low dimension complex space with the number of dimensions equal to that of the "social" vector) and then multiplying it by the vector, and doing something like cosine similarity under various transformations (like: values in the matrix that represent what the interactions look like at one period of time over another and even shuffling around the order at random before decomposition) could give us insight into how people/orgs will change together.

Robin,

Great post! I think you are certainly right that relationships get strange in high-dimensional societies.

This post from ribbonfarm.com explores a similar idea in a few directions that seem complementary: https://www.ribbonfarm.com/...

Thanks for sharing,Kevin