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

TL;DR

This paper introduces a novel test for independence of samples based on expander graph theory, linking eigenvalues of a constructed graph to the likelihood of samples being i.i.d., with practical examples and theoretical justification.

Contribution

It proposes a new eigenvalue-based test for i.i.d. samples using expander graph properties, connecting graph spectra to statistical independence testing.

Findings

Eigenvalues close to 2√3 indicate likely i.i.d. samples.

The constructed graph is nearly Ramanujan if samples are i.i.d.

The test is supported by theoretical and example-based evidence.

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.