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Sex, Graph Theory, and why you’re taught math

Math, am I right? 

Calculus, trigonometry, geometry, algebra, statistics, proofs. Who needs all of that, and why is everyone forced to learn it at school? Except for mathematicians, of course. And perhaps the odd theoretical physicist or computer scientist. 

Well, ask the sociologists behind this book. More specifically, a claim they made based on surveys saying that men in the U.S tend to have more opposite-sex partners than women. By a LOT. Supposedly the average American male has 74% more female partners than females have male partners. Ugh, men and their promiscuous playboy ways. Typical.

Actually, not quite so. In fact, there’s no way the ratio can be that different, and perhaps if the sociologists knew a bit of math beforehand, they’d be able to dismiss the results of the surveys much more easily. Let’s go through it with a little bit of graph theory.

If you’ve ever done one of those connect-the-dot puzzles, you’ve already indulged in what graph theory is all about. Formally, a graph is a pair G = (V, E) where V is a set of vertices and E ⊆ {(u,v) | u, v ∈ V}. But really, it’s all about dots and lines connecting them. 

ProTip if you want to make your child seem like a prodigy, just say this every time he does a connect-the-dots puzzle.

We call a dot on a graph a vertex and a line from one dot to another (or from one dot to itself!) an edge. So to visualise the population of sexually active men and women, let’s create two columns of vertices as below:

Note that the diagram doesn’t show exactly how many men and women there are.

Now, if a man and a woman are partners, let’s connect them with an edge (for now, we’ll just assume random, arbitrary partnerships and disregard same-sex relationships). Each vertex can have one or more edges connected to it.

Notice that every edge is connected to BOTH the men and women column

OK, so now we’ve visualised the scenario using graphs. Going back to our goal, we’re trying to find the average number of partners of the opposite sex for both men and women. Converting that into our graph theory lingo, that means finding the average number of edges connected to a vertex for each column. Well, to do that you just add up the number of edges connected to all the vertices in one column, and divide by the number of vertices. 

Let’s call the total number of men Vm and the total number of edges connected to the men column Em. Similarly, the total number of women will be Vw and the total number of edges connected to the women column will be Ew. So the average number of opposite partners for men is Em/Vm and for women it’s Ew/Vw.

So the ratio of average partners of men to average partners of women is (Em/Vm)/(Ew/Vw). But Em and Ew are the same. Obviously. A five-year-old looking at the graph could tell you that, since each edge is connected to one vertex from the men column, and one vertex from the women column. In simple terms, it takes two to tango. So the ratio of opposite partner relationships between men and women is simply the ratio of the NUMBER men to women: Vm/Vw

When you look at it this way, it seems ridiculous that men have 74% more opposite sex partners on average. If that were true, it would imply that there are 74% more sexually active men than women in the United States. And even if THAT were true, the number tells us nothing about the promiscuity of men or women at all, it’s just a ratio between the NUMBER of men and women present.

Now, it seems almost comical that the sociologists didn’t notice this simple fact, but that’s the issue: people largely underestimate the role of mathematics, in everything. Math isn’t just adding and multiplying numbers so you can calculate your bills or fill out your taxes. It’s a way to build rigorous arguments and present and analyse things in different ways. We aren’t taught math because the formulas will come in useful one day; most of them won’t. It’s necessary because it strengthens our logical thinking and reasoning abilities, something you need regardless of your interests or profession. 

So maybe don’t be so hard on math. And this may be a math nerd saying that so you can call me biased, but all you have to do to prove me wrong is build a rigorous argument using logic and base axioms, and we’ll agree to disagree.