r"""
Localization of Fourier modes.
=============================

The Fourier modes (the eigenvectors of the graph Laplacian) can be localized in
the spatial domain. As a consequence, graph signals can be localized in both
space and frequency (which is impossible for Euclidean domains or manifolds, by
the Heisenberg's uncertainty principle).

This example demonstrates that the more isolated a node is, the more a Fourier
mode will be localized on it.

The mutual coherence between the basis of Kronecker deltas and the basis formed
by the eigenvectors of the Laplacian, :attr:`pygsp2.graphs.Graph.coherence`, is
a measure of the localization of the Fourier modes. The larger the value, the
more localized the eigenvectors can be.

See `Global and Local Uncertainty Principles for Signals on Graphs
<https://arxiv.org/abs/1603.03030>`_ for details.
"""

import matplotlib as mpl
import numpy as np
from matplotlib import pyplot as plt

import pygsp2 as pg

fig, axes = plt.subplots(2, 2, figsize=(8, 8))

for w, ax in zip([10, 1, 0.1, 0.01], axes.flatten()):

    adjacency = [
        [0, w, 0, 0],
        [w, 0, 1, 0],
        [0, 1, 0, 1],
        [0, 0, 1, 0],
    ]
    graph = pg.graphs.Graph(adjacency)
    graph.compute_fourier_basis()

    # Plot eigenvectors.
    ax.plot(graph.U)
    ax.set_ylim(-1, 1)
    ax.set_yticks([-1, 0, 1])
    ax.legend([f'$u_{i}(v)$, $\lambda_{i}={graph.e[i]:.1f}$' for i in range(graph.n_vertices)], loc='upper right')

    ax.text(0, -0.9, f'coherence = {graph.coherence:.2f}'
            f'$\in [{1/np.sqrt(graph.n_vertices)}, 1]$')

    # Plot vertices.
    ax.set_xticks(range(graph.n_vertices))
    ax.set_xticklabels([f'$v_{i}$' for i in range(graph.n_vertices)])

    # Plot graph.
    x, y = np.arange(0, graph.n_vertices), -1.20 * np.ones(graph.n_vertices)
    line = mpl.lines.Line2D(x, y, lw=3, color='k', marker='.', markersize=20)
    line.set_clip_on(False)
    ax.add_line(line)

    # Plot edge weights.
    for i in range(graph.n_vertices - 1):
        j = i + 1
        ax.text(i + 0.5, -1.15, f'$w_{{{i}{j}}} = {adjacency[i][j]}$', horizontalalignment='center')

fig.tight_layout()
plt.show()