How many people can be famous?

Many people look to influencers/content creators/celebrities and wish to emulate their success in gaining a large following and making a living off of their content, but how many people can actually achieve fame? In order to answer this question, you first have to define what it means to be famous. I think a good and flexible definition is that a large number of people know you. Unfortunately this raises the followup questions of "How large of a number?" and "What counts as knowing someone?". Let's call people \(n\)-famous if they are known by at least \(n\)% of the people in a given population. Let's also put an upper-limit, \(k\), on the number of people each person can know (by whatever definition you choose for "knowing" someone). Interestingly, \(k\) and a population size is all it takes to put an upper-bound on how many people can be \(n\)-famous.

The number of people in a population of size \(p\) that can be \(n\)-famous I hereby with as the following formula: $$ fame(n,k,p)=\dfrac{kp}{np/100} = \dfrac{k}{n/100} = \dfrac{100k}{n}$$ Interestingly the population size drops out of the formula, so the max number of \(n\)-famous people is independent of the population size. This can be thought of as each person having \(k\) "slots" in their brain for people they can know. In total, a population of size \(p\) then has \(kp\) slots altogether. Each person who is \(n\)-famous takes up at least \(\frac{np}{100}\) of these slots. The maximum number of \(n\)-famous people is achieved by assuming that each \(n\)-famous person takes up exactly \(\frac{np}{100}\) slots. It doesn't matter if the audiences of two \(n\)-famous people are disjoint or not, because the total number of slots occupied is the same regardless.

Here's a calculator you can use to play around with the formula: In the sliders below, \(1\leq n\leq 100\), \(1\leq k\leq 50000\), and \(1\leq p\leq 8\; billion\).

Number of \(n\)-famous people: