Suppose the average person p0 has n acquaintances. Then a naive approach would say that each of p0’s acquaintances (call one of them p1) also has n acquaintances, leading p0 with n^2^ acquaintances of the second degree.

However, in a social network, many of p1’s acquaintances are shared between p0 and p1. Let’s say that r⋅n (1/n≤r≤1) of p1’s acquaintances are actually first-order acquaintances of p0. The lower limit for r is 1/n because naturally one of p1’s acquaintances is p0 themselves.

This gives us n⋅(1−p)⋅n = n^2^⋅(1−p) as the number of second-degree acquaintances, if my math is mathing. Increase n for more extraverted people in the network, and increase p for more closely-knit networks.

To model the headline X % know someone who knows, we solve 1 / [n^2^⋅(1−p)] ≥ x where x is X% expressed as a fraction. Plugging in n=100 and p = 1/10 (I pulled these numbers out of my ass) and X=20% we get 1 / [100^2^ ⋅ (1−.1))] = 1 / [ 10^4 ⋅ 0.9 ] = 1 / 900; again, if my math is mathing.

So this headline is true if about 1 in 900 people are in a relationship with AI.