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

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, $x_{\pi(1)} < x_{\pi(2)} < \dots < x_{\pi(n)}$,webuildagraphon$\left{1, \dots, n\right}$ 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.