.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/eigenvector_localization.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_eigenvector_localization.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_eigenvector_localization.py:


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.

.. GENERATED FROM PYTHON SOURCE LINES 21-67



.. image-sg:: /examples/images/sphx_glr_eigenvector_localization_001.png
   :alt: eigenvector localization
   :srcset: /examples/images/sphx_glr_eigenvector_localization_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    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()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.383 seconds)

**Estimated memory usage:**  176 MB


.. _sphx_glr_download_examples_eigenvector_localization.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: eigenvector_localization.ipynb <eigenvector_localization.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: eigenvector_localization.py <eigenvector_localization.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: eigenvector_localization.zip <eigenvector_localization.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_