.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/graph_theory/ranking/plot_01_pagerank.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_graph_theory_ranking_plot_01_pagerank.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_graph_theory_ranking_plot_01_pagerank.py:


PageRank: ranking pages by a random surfer
================================================

Ranks the vertices of a small web of links by the stationary
distribution of a random surfer who follows links with probability 0.85
and otherwise jumps to a random page, and shows the convergence of the
power iteration.

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.graph_theory import Graph, pagerank








.. GENERATED FROM PYTHON SOURCE LINES 18-20

A small web
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-31

.. code-block:: Python


    links = {0: [1, 2], 1: [2], 2: [0], 3: [0, 2], 4: [0, 3, 5], 5: [4], 6: [2]}
    g = Graph(7, directed=True)
    for u, targets in links.items():
        for v in targets:
            g.add_edge(u, v)
    result = pagerank(g)
    for page in np.argsort(result.scores)[::-1]:
        print(f"page {page}: {result.scores[page]:.4f}  (in-links from {[u for u in links if page in links[u]]})")
    print(f"converged in {result.iterations} iterations")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    page 0: 0.3433  (in-links from [2, 3, 4])
    page 2: 0.3432  (in-links from [0, 1, 3, 6])
    page 1: 0.1674  (in-links from [0])
    page 4: 0.0522  (in-links from [5])
    page 5: 0.0362  (in-links from [4])
    page 3: 0.0362  (in-links from [4])
    page 6: 0.0214  (in-links from [])
    converged in 55 iterations




.. GENERATED FROM PYTHON SOURCE LINES 32-34

Convergence at the rate of the damping factor
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-43

.. code-block:: Python


    exact = result.scores
    fig, ax = plt.subplots()
    for d in (0.5, 0.85, 0.95):
        errors = [np.abs(pagerank(g, damping=d, max_iter=k, tol=0).scores - pagerank(g, damping=d).scores).sum() for k in range(1, 60)]
        ax.semilogy(range(1, 60), errors, label=f"damping {d}")
    ax.set_xlabel("iteration")
    ax.set_ylabel("L1 error")
    ax.legend()



.. image-sg:: /api/gallery/graph_theory/ranking/images/sphx_glr_plot_01_pagerank_001.png
   :alt: plot 01 pagerank
   :srcset: /api/gallery/graph_theory/ranking/images/sphx_glr_plot_01_pagerank_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1192f8d70>




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

   **Total running time of the script:** (0 minutes 0.106 seconds)


.. _sphx_glr_download_api_gallery_graph_theory_ranking_plot_01_pagerank.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_pagerank.ipynb <plot_01_pagerank.ipynb>`

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

      :download:`Download Python source code: plot_01_pagerank.py <plot_01_pagerank.py>`

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

      :download:`Download zipped: plot_01_pagerank.zip <plot_01_pagerank.zip>`


.. only:: html

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

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