Using Expander Graphs to test whether samples are i.i.d

Abstract

The purpose of this note is to point out that the theory of expander graphs leads to an interesting test whether $n$ real numbers $x_1, \dots, x_n$ could be $n$ independent samples of a random variable. To any distinct, real numbers $x_1, \dots, x_n$, we associate a 4-regular graph $G$ as follows: using $\pi$ to denote the permutation ordering the elements, $x_{\pi(1)} < x_{\pi(2)} < \dots < x_{\pi(n)}$, we build a graph on $\left\{1, \dots, n\right\}$ by connecting $i$ and $i+1$ (cyclically) and $\pi(i)$ and $\pi(i+1)$ (cyclically). If the numbers are i.i.d. samples, then a result of Friedman implies that $G$ is close to Ramanujan. This suggests a test for whether these numbers are i.i.d: compute the second largest (in absolute value) eigenvalue of the adjacency matrix. The larger $\lambda - 2\sqrt{3}$, the less likely it is for the numbers to be i.i.d. We explain why this is a reasonable test and give many examples.