SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\Re \mathbf {v} _ {N / 4} = \frac {1}{\sqrt {N}} \left(\cos \left(\frac {\pi n}{2}\right)\right) _ {n = 0} ^ {N - 1}, \Im \mathbf {v} _ {N / 4} = \frac {1}{\sqrt {N}} \left(\sin \left(\frac {\pi n}{2}\right)\right) _ {n = 0} ^ {N - 1}
$$
Their sum is then equivalent to
$$
\begin{array}{l} \Re \mathbf {v} _ {N / 4} + \Im \mathbf {v} _ {N / 4} = \frac {1}{\sqrt {N}} \left(\cos \left(\frac {\pi n}{2}\right) + \sin \left(\frac {\pi n}{2}\right)\right) _ {n = 0} ^ {N - 1} \\ = \frac {\sqrt {2}}{\sqrt {N}} \left(\sin \left(\frac {\pi n}{2} + \frac {\pi}{4}\right)\right) _ {n = 0} ^ {N - 1} \\ = \sqrt {\frac {2}{N}} \left(\sin \left(\frac {\pi}{4} (2 n + 1)\right)\right) _ {n = 0} ^ {N - 1} \\ = \frac {1}{\sqrt {N}} (1, 1, - 1, - 1, \dots) \\ \end{array}
$$
hence, the mid eigenvector of  $\mathbf{L}$  guarantees homophily  $1/2$ . A visual explanation is shown in Figure 4.