Fourier basis.

The eigenvectors of the graph Laplacian form the Fourier basis. The eigenvalues are a measure of variation of their corresponding eigenvector. The lower the eigenvalue, the smoother the eigenvector. They are hence a measure of “frequency”.

In classical signal processing, Fourier modes are completely delocalized, like on the grid graph. For general graphs however, Fourier modes might be localized. See pygsp2.graphs.Graph.coherence.

import numpy as np
from matplotlib import pyplot as plt

import pygsp2 as pg

n_eigenvectors = 7

fig, axes = plt.subplots(2, 7, figsize=(15, 4))


def plot_eigenvectors(G, axes):
    G.compute_fourier_basis(n_eigenvectors)
    limits = [f(G.U) for f in (np.min, np.max)]
    for i, ax in enumerate(axes):
        G.plot(G.U[:, i], limits=limits, colorbar=False, vertex_size=50, ax=ax)
        energy = abs(G.dirichlet_energy(G.U[:, i]))
        ax.set_title(r'$u_{0}^\top L u_{0} = {1:.2f}$'.format(i + 1, energy))
        ax.set_axis_off()


G = pg.graphs.Grid2d(10, 10)
plot_eigenvectors(G, axes[0])
fig.subplots_adjust(hspace=0.5, right=0.8)
cax = fig.add_axes([0.82, 0.60, 0.01, 0.26])
fig.colorbar(axes[0, -1].collections[1], cax=cax, ticks=[-0.2, 0, 0.2])

G = pg.graphs.Sensor(seed=42)
plot_eigenvectors(G, axes[1])
fig.subplots_adjust(hspace=0.5, right=0.8)
cax = fig.add_axes([0.82, 0.16, 0.01, 0.26])
_ = fig.colorbar(axes[1, -1].collections[1], cax=cax, ticks=[-0.4, 0, 0.4])
plt.show()

Estimated memory usage: 180 MB