Ever since writing about the mathematics of love for Valentine’s Day last year, I have thought about writing an article that covers the mathematics of social connection in general. Being a broader topic, this article will only scratch the surface, but I hope it will provide a variety of subtopics for the reader to look more into as interests them.
The starting point for seeking to model social connections with mathematics is the concept of a graph. I’ve covered graphs in a few different contexts before, but as a refresher, their two main components are vertices and edges. In the social connection context, the vertices represent people, and the edges that connect two vertices represent whether two people know each other. More complex graphs may also assign weights to the edges, which can specify how well two people know each other (if they have a really strong bond, the weight of their connecting edge will be large).
So, in general, we can use graphs to model connections in a neighborhood, city, country, or the entire world; one can also imagine using a graph to model the accounts on a social media platform, or a group of patients who may be connected by certain medical conditions or genetic traits. A goal now in modeling these social networks is to figure out whether different parts of the graph are highly or sparsely connected based on the aspects of the vertices.
Mathematically, highly connected subsets of a graph have an apt name: cliques. Cliques are a set of vertices where each one has an edge connected to all the others. Cliques all know each other, and if we’re using weighted graphs, the weights on the edges are especially high. But, as you might expect, when we have large graphs—let’s say a graph that models every profile on a social media platform—it can be really hard to find cliques by a brute-force approach of checking all the vertices and edges.
This is the intuitive reason why the problem of deciding whether a graph has a clique of a certain size is NP-complete; it would take another article to go into the technicalities of this term, but for now, it means that the problem poses a significant computational challenge. An alternative approach to deterministically figuring out where cliques are is to randomly generate huge graphs and see if they manifest similar behaviors to real-life social networks.
Of course, the generated graphs in this scheme aren’t completely random: they are created with some baseline rules in place. For instance, if we’re trying to model the connections of a neighborhood, it’s very likely that two next-door neighbors know each other, but less likely that two people living on opposite sides of town do — or so it may seem. One of the most fascinating results to come out of social network modeling is that some models, based on their set of rules, allow for “small-world” phenomena, in which two people with vastly different backgrounds still have a decent likelihood of knowing each other through someone else. The phenomenon gets its name from people saying something like, “Wow, it’s a small world,” when they experience it.
This brings me to my last point, that there are multiple ways to construct social networks, and social media platforms have, in large part, been the constructors in recent years. Their model has been very siloing — users must have a profile for each platform, and the platforms rarely communicate with each other. Certain people in the tech world have pushed for better interaction across social media platforms, leading to rules such as the AT Protocol of BlueSky and others like ActivityPub and Nostr.
As is the case every week, last week’s Valentine’s Day issue had some great articles covering the importance of love for oneself as well as love towards and from others. Human connection, more generally, is what makes us distinctly human, and it is great to see mathematics lending a hand in describing how this connection arises across scales and facets of society.
