Earlier this week I spoke at Maths and Computer Science in the Pub. The event was hosted by Phil Dooley and sponsored by the Mathematical Sciences Institute. I had a great time talking and hearing from the other presenters. Below is a photo of me presenting and a transcript of my talk.

Right now the number of people with an odd number of Facebook friends is an even number. In fact at any point in time the number of people with an odd number of Facebook friends is an even number. Why is this true? We could check this claim by counting exactly how many people have an odd number of friends but this is much too complicated. An easy answer can be founded by using a common mathematical approach called counting in different ways.

For each Facebook user, imagine looking at how many Facebook friends they have and add up all those numbers. We can write this quantity as an intimidating sum:

$\sum\limits_{\text{Facebook users}} \text{number of friends}.$

Another way of counting the same thing is to look at the number of friendships and multiply that number by 2. This is because each friendship is between two people. When we counted the first number we double counted every friendship. We can write this in an equation:

$\sum\limits_{\text{Facebook users}} \text{number of friends} = 2\cdot(\text{number of friendships}).$

The important thing that this tells us is that the sum on the left is an even number. We can split this sum into two sums. We could first add up how many Facebook friends all of the people who have an even number of friends. Then we can add up how many Facebook friends all of the people with an odd number of friends have. These two sums added together gives us the original sum which is an even number.

$\sum\limits_{\text{even Facebook users}} \text{number of friends}+\sum\limits_{\text{odd Facebook users}} \text{number of friends} = 2\cdot(\text{number of friendships}).$

In the first sum, all of the people have an even number of Facebook friends so all of the numbers are even. Thus the first sum is an even number. If we subtract this first sum from both sides, then we can see that the second sum is equal to an even number minus an even number. Hence the second sum in an even number.

$\sum\limits_{\text{odd Facebook users}} \text{number of friends} = 2\cdot(\text{number of friendships})-\sum\limits_{\text{even Facebook users}} \text{number of friends}.$

In the second sum, all of the people have an odd number of friends. Thus all of the numbers must be odd. So we are adding up a bunch of odd numbers and getting an even number. The only way this is possible is if we have an even number of terms. That is if we have an even number of people with an odd number of Facebook friends.

So there you have it, the number of people with an odd number of Facebook friends is an even number. You can be sure of this fact thanks to a mathematical proof but why should we stop there? One thing mathematicians love to do is generalise their results. The only things we needed to know about Facebook was that there are a bunch of people with Facebook accounts and there are some pairs of people whose Facebook accounts are linked by being friends. If we take this abstract view of Facebook we arrive at the definition of a graph. A graph consists of a set of points called vertices and a set of connections between some pairs of vertices called edges.

Facebook is an example of a graph but there are many other examples. For each graph we get a fact analogous to the number of people with an odd number of Facebook friends being even. We learn that the number of vertices with an odd number of neighbours is an even number. The exact same proof works if at every point we replace “person” with “vertex”, “Facebook friend” with “neighbour” and “friendship” with “edge”.

For example if there’s a big party, we can make a graph where the vertices are people who attended the party and there is an edge between two people if they hugged at the party. Then the theorem tells us that the number of people who hugged an odd number of people is an even number.

Unfortunately not everything is a graph. In particular Instagram and Twitter are not graphs. On these social media platforms, you can follow someone without them following you back. This breaks the proof I gave before and it’s not always true that the number of people with an odd number of Instagram followers is always even. In fact it fluctuates between even and odd whenever someone follows or unfollows someone else.

So if someone asks you what the difference between Facebook and Instagram is, now you know. On Facebook the number of people with an odd number of Facebook friends is always even.