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

.. only:: html

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

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

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

.. _sphx_glr_auto_examples_dowker_plot_dowker_homology.py:


.. _dowker_homology_gallery:

Homology
=========================================

In this example we compute the homology of a Dowker complex.

.. GENERATED FROM PYTHON SOURCE LINES 9-12

.. code-block:: Python


    import oat_python as oat








.. GENERATED FROM PYTHON SOURCE LINES 13-18

Define a dowker complex
------------------------
Dowker complexes are represented as a list of sorted-lists of integers.
Each sorted list represents a simplex, and the Dowker complex consists of all
subsets of these simplices. 

.. GENERATED FROM PYTHON SOURCE LINES 18-30

.. code-block:: Python


    dowker_simplices        =   [
                                    [0,1,2,6],   # simplex 0
                                    [0,1,2],     # simplex 1
                                    [0,1,3],     # simplex 2
                                    [0,2,3],     # simplex 3
                                    [1,2,3],     # simplex 4
                                    [3,4],       # simplex 5
                                    [4,5],       # simplex 6
                                    [3,5],       # simplex 7                                                                                                                         
                                ]








.. GENERATED FROM PYTHON SOURCE LINES 31-35

Plot the Dowker complex
------------------------

This step isn't necessary, but the complex is small and it will help to visualize it.

.. GENERATED FROM PYTHON SOURCE LINES 35-53

.. code-block:: Python


    points                  =   oat.plot.vertex_embedding_for_simplices(
                                    dowker_simplices,
                                    dimension       =   3,      # 3D embedding   
                                )
    fig                     =   oat.plot.fig_3d_for_simplices(
                                    simplices       =   dowker_simplices,
                                    points          =   points,
                                    kwargs_points   =   dict(
                                                            mode="markers+text",  # indicates we want to plot text labels on points
                                                            text=[str(i) for i in range(len(points))] # text labels for each vertex
                                                        ),
                                    kwargs_triangles=   dict(color="white") # color the triangles (transparent) black
                                )
    fig.update_layout(template="plotly_dark") # dark theme
    fig







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.. GENERATED FROM PYTHON SOURCE LINES 56-62

Compute homology by decomposing the boundary matrix
-----------------------------------------------------

Each row of the homology dataframe represents a homology class.
Taken together, these homology classes form a basis for the homology groups of the Dowker complex
in dimensions 0 .. max_homology_dimension.

.. GENERATED FROM PYTHON SOURCE LINES 62-70

.. code-block:: Python


    decomposition           =   oat.core.dowker.BoundaryMatrixDecompositionDowker(
                                    dowker_simplices        =   dowker_simplices,
                                    max_homology_dimension  =   2  
                                )
    homology                =   decomposition.homology()
    homology






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>dimension</th>
          <th>cycle_representative</th>
          <th>cycle_representative_nonzero_coefficient_count</th>
          <th>unique_simplex_id</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>0</td>
          <td>simplex coefficient
    0    (0,)           1</td>
          <td>1</td>
          <td>(0,)</td>
        </tr>
        <tr>
          <th>1</th>
          <td>1</td>
          <td>simplex coefficient
    0  (3, 4)           1
    1 ...</td>
          <td>3</td>
          <td>(4, 5)</td>
        </tr>
        <tr>
          <th>2</th>
          <td>2</td>
          <td>simplex coefficient
    0  (0, 1, 2)         ...</td>
          <td>4</td>
          <td>(1, 2, 3)</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 71-77

Cycle representatives
-----------------------------------------------------

The "cycle representative" column in the homology dataframe contains a
cycle representative for each homology class.


.. GENERATED FROM PYTHON SOURCE LINES 77-80

.. code-block:: Python

    homology["cycle_representative"][1]







.. raw:: html

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    <div>
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          <td>1</td>
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      </tbody>
    </table>
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    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 81-92

Plotting
-----------------------------------------------------

Let's plot some cycle representatives from the homology dataframe.

See also
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

- :ref:`vertex_embedding_gallery`
- :ref:`cycle_representative_strategies_gallery`
- :ref:`styling_3d_gallery`

.. GENERATED FROM PYTHON SOURCE LINES 94-96

Plot a 1-cycle
^^^^^^^^^^^^^^^^^^^^

.. GENERATED FROM PYTHON SOURCE LINES 98-99

Pull the cycle from the homology dataframe.

.. GENERATED FROM PYTHON SOURCE LINES 101-104

.. code-block:: Python

    one_cycle              =   homology["cycle_representative"][1]
    one_cycle






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    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 105-118

.. code-block:: Python

    edges               =   one_cycle.simplex
    trace_1cycle        =   oat.plot.trace_3d_for_edges(
                                edges           =   edges,
                                points          =   points,
                                line            =   dict( width=10, color="red" ),
                                opacity         =   1.0,
                                showlegend      =   True,
                                name            =   "1-cycle"
                            )
    fig.add_trace(trace_1cycle)
    fig







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    <br />

.. GENERATED FROM PYTHON SOURCE LINES 119-122

Plot a 2-cycle
^^^^^^^^^^^^^^^^^^^^^^^^


.. GENERATED FROM PYTHON SOURCE LINES 124-125

Pull the cycle from the homology dataframe.

.. GENERATED FROM PYTHON SOURCE LINES 127-130

.. code-block:: Python

    two_cycle              =   homology["cycle_representative"][2]
    two_cycle






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>simplex</th>
          <th>coefficient</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>(0, 1, 2)</td>
          <td>-1</td>
        </tr>
        <tr>
          <th>1</th>
          <td>(0, 1, 3)</td>
          <td>1</td>
        </tr>
        <tr>
          <th>2</th>
          <td>(0, 2, 3)</td>
          <td>-1</td>
        </tr>
        <tr>
          <th>3</th>
          <td>(1, 2, 3)</td>
          <td>1</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 131-132

Toggle off the one-cycle (but don't remove it from the figure).

.. GENERATED FROM PYTHON SOURCE LINES 134-136

.. code-block:: Python

    fig.data[3].visible = "legendonly"








.. GENERATED FROM PYTHON SOURCE LINES 137-138

Add a trace for the 2-cycle, colored red.

.. GENERATED FROM PYTHON SOURCE LINES 140-156

.. code-block:: Python


    triangles           =   two_cycle.simplex
    trace_2cycle        =   oat.plot.trace_3d_for_triangles(
                                triangles       =   triangles,
                                points          =   points,
                                color           =   "red",
                                opacity         =   0.5,
                                showlegend      =   True,
                                name            =   "2-cycle"
                            )
    fig.add_trace(trace_2cycle)
    fig









.. raw:: html

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    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 157-162

Fundamental subspaces
-----------------------------------------------------

The ``.fundamental_subspace_dimensions()`` method returns a data frame with the dimensions of
the spaces of chains, cycles, and boundaries.

.. GENERATED FROM PYTHON SOURCE LINES 162-165

.. code-block:: Python


    decomposition.fundamental_subspace_dimensions()






.. raw:: html

    <div class="output_subarea output_html rendered_html output_result">
    <div>
    <style scoped>
        .dataframe tbody tr th:only-of-type {
            vertical-align: middle;
        }

        .dataframe tbody tr th {
            vertical-align: top;
        }

        .dataframe thead th {
            text-align: right;
        }
    </style>
    <table border="1" class="dataframe">
      <thead>
        <tr style="text-align: right;">
          <th></th>
          <th>homology</th>
          <th>space_of_chains</th>
          <th>space_of_cycles</th>
          <th>space_of_boundaries</th>
        </tr>
        <tr>
          <th>dimension</th>
          <th></th>
          <th></th>
          <th></th>
          <th></th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <th>0</th>
          <td>1</td>
          <td>7</td>
          <td>7</td>
          <td>6</td>
        </tr>
        <tr>
          <th>1</th>
          <td>1</td>
          <td>12</td>
          <td>6</td>
          <td>5</td>
        </tr>
        <tr>
          <th>2</th>
          <td>1</td>
          <td>7</td>
          <td>2</td>
          <td>1</td>
        </tr>
      </tbody>
    </table>
    </div>
    </div>
    <br />
    <br />

.. GENERATED FROM PYTHON SOURCE LINES 166-171

Betti numbers
-----------------------------------------------------

The ``.betti_numbers()`` method returns a list [b_0, b_1, ..., b_max_homology_dimension],
where b_i is the dimension of the i-th homology group.

.. GENERATED FROM PYTHON SOURCE LINES 171-173

.. code-block:: Python


    decomposition.betti_numbers()




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

 .. code-block:: none


    [1, 1, 1]




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

   **Total running time of the script:** (0 minutes 0.174 seconds)


.. _sphx_glr_download_auto_examples_dowker_plot_dowker_homology.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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