Spread and saturation

Let’s leave our zombies for a few moments (don’t worry, they will soon lurch back into view) and ask:

  1. is it better for a meme to be fast-spreading or long-lived?
  1. what happens when almost everyone is infected?, and
  1. does defection matter?

Speed or reach?

Let’s start by looking at the spread of a long-lived but slow-growing meme.

Since all religions start with a congregation of one, let’s say that I make up a religion based on the number of fingers and thumbs I have. I get 10 of my friends together, and preach the gospel of fingers. This idea is so beautiful and so self-evidently true that they are all instantly converted, and every tenth day I go forth with my converts and we each contact ten people who haven’t heard the idea, and tell them about it - it's a success factor of 10. Every tenth day all of these new converts, and all of the old ones, do the same. Let's assume that no-one ever defects from the belief and they just keep on spreading the good news. How long would it take for my finger-based religion (digitism?) to spread around the world? If you don’t want to do the maths yourself you’ll just have to trust me when I tell you it will take about three months - nine or ten generations.

Now let’s think of a really salacious piece of gossip, say that one about the President and a hooker. This one is really hot property, and everyone who hears this rumour will want to spread it (or some mutated version of it) to one uninfected person each day. Ignoring saturation effects, the success factor for this rumour is two, far lower than that of our phony religion, but the time between generations is much shorter. Whether or not this rumour is true is irrelevent, as we will see when we look at riots, tweets and imaginary tigers, it just has to be spreadable.

Let’s start the two memes - the religion and the juicy gossip - going at the same time and take at where we are look forty days later. Our slow-but-durable religion started with 11 converts and we’ve had four rounds of church meetings since the initial kick-off - we now have about 161,000 believers (11 raised to the 5th power). The salacious piece of gossip, however, is doubling every day, and after forty days it’s spread to just over a trillion people…although in fact if runs out of minds to infect after about 33 days.

So what about increasing the starting figures? Say we decided to run a hugely expensive advertising campaign on our fingery religion and start the 10-day, 10-person cycles with a larger population of believers. Well, it works in the end, but you have to start out with well over a million people converted in the first meeting before our slow-but-steady meme outpaces the fast-but-transient meme. This bears out our everyday experience - catchy songs and interesting lies spread through a population much more readily than difficult but important concepts, which is why school attendance is compulsory.

The messages I want to leave you with are:

- if you want your memes to spread, make them juicy, and
- there is little point of mass advertising if your message is boring.

Saturation - chasing down the uninfected

In several places so far on this site, we have ‘ignored saturation’ because it makes the maths simpler if somewhat less realistic.

Let's go back to the outbreak of the living dead in New York. As we saw, their spread is exponential. But when the number of zombies overtakes the number of uninfected New Yorkers, it will be harder to find victims, and so the average time between generations will start to increase.

And so it is with memes - every time you interrupt a repeated joke with ‘heard it!’, you are demonstrating that that particular joke is reaching saturation.


The graph to the left shows a typical saturation curve called a ‘logistic function’. You can see the effect of exponential growth - the population doubling every generation - on the left hand side, and then it flattens out. Because this curve is symmetric you can see that it takes much longer to infect the entire population of New York.

Actually, it probably won’t be symmetric for zombie infections because they compete for resources, but that starts to make the maths much more complex.

In fact, we don‘t ever get all of them - after 32 hours there are still about 10,000 uninfected people, and a movie genre is born.

The reason that zombie outbreaks start in cities is that the saturation point is very high - they can convert millions of people to their cause in the first day and then, in the words of the popular song, shuffle off to Buffalo and on to world domination.

If they had started in New York, Texas (population 20), their evil crusade would have ground to a halt very quickly.

Cancel my subscription, I don’‘t believe in Santa Claus

What happens to the population of zombies, and memes, after saturation is reached? Nobody ever defects from being a zombie - becoming a zombie is a trapping state (of which more later) so they would carry on until the entire world was populated and then, presumably, slowly starve.

But consider what would have happened to our salacious rumour in real life: most people would realise on reflection that it couldn’t be true, and they would stop believing it and therefore stop spreading it. If they only spread it for a week, it will still take over the world in 33 days, but would be extinct a week later, or faster if some of the defectors spread an anti-meme.

Defection and anti-memes come to dominate memetic growth in saturation populations, but before we turn to
that, let’s consider a curious case where a meme spreads even though we already know it is false.

In the small number of countries where the meme of Father Christmas - aka St Nicholas, Santa Claus or Sinterklass - operates, the population of five-year-olds who subscribe to a strong form of that belief (that he is real) is very high. The number of adults who subscribe to that belief is very low, so at some point, almost everyone who has been infected with the strong form of the 'Father Christmas' meme defects from it.

The meme hasn't died out because a weak form is retained - the carriers, even though they no longer believe there really is a man in a red suit - still infect their children in turn. Perhaps this meme is so alluring and so beautiful to see in action in our children that its truth is irrelevant.

Of course, that doesn't stop those same adults from continuing to believe even more
improbable things, but that is a story for another day.
Quick vs durable