An implementation of the Dowker complex originally introduced in Homology Groups of Relations and adapted to the setting of persistent homology in A functorial Dowker theorem and persistent homology of asymmetric networks. The complex is implemented as a class named DowkerComplex that largely follows the API conventions from scikit-learn.

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Example of running DowkerComplex

The following is an example of computing persistent homology of the filtered complex $\left\{\mathrm{D}_{\varepsilon}(X,Y)\right\}_{\varepsilon\in\mathbb{R}^{+}}$, that is, of the Dowker complex with relations $R_{\varepsilon}\subseteq X\times Y$ defined by $(x,y)\in R_{\varepsilon}$ iff $d(x,y)\leq\varepsilon$ for $\varepsilon\geq 0$, and where $X$ and $Y$ are subsets of $\mathbb{R}^{n}$ equipped with the Euclidean norm. In the following example, we refer to $X$ and $Y$ as vertices and witnesses, respectively.

>>> from dowker_complex import DowkerComplex
>>> from sklearn.datasets import make_blobs
>>> X, y = make_blobs(
        n_samples=200,
        centers=[[-1, 0], [1, 0]],
        cluster_std=0.75,
        random_state=42,
    )
>>> vertices, witnesses = X[y == 0], X[y == 1]
>>> dc = DowkerComplex()  # use default parameters
>>> persistence = dc.fit_transform([vertices, witnesses])
>>> persistence
[array([[0.39632083, 0.4189592 ],
        [0.17218397, 0.24239225],
        [0.07438909, 0.1733489 ],
        [0.13146844, 0.25247844],
        [0.16269607, 0.29266369],
        [0.0815455 , 0.24042536],
        [0.10576964, 0.32222553],
        [0.1382231 , 0.358332  ],
        [0.07358198, 0.37408252],
        [0.24082383, 0.57726198],
        [0.02419385,        inf]]),
 array([[0.5035793 , 0.63405836]])]

The output above is a list of arrays, where the $i$-th array contains (birth, death)-times of homological generators in dimension $i-1$. Validity of Dowker duality can be verified by swapping the roles of vertices as witnesses as follows.

>>> import numpy as np
>>> persistence_swapped = DowkerComplex().fit_transform([witnesses, vertices])
>>> all(
        np.allclose(homology, homology_swapped)
        for homology, homology_swapped
        in zip(persistence, persistence_swapped)
    )
True

Any DowkerComplex object accepts further parameters during instantiation. A full description of these can be displayed by calling help(DowkerComplex). These parameters, among other things, allow the user to specify persistence-related parameters such as the maximal homological dimension to compute or which metric to use.

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Installation and requirements

The package can be installed via pip by running pip install -U dowker-complex.

Required Python dependencies are specified in pyproject.toml. Provided that uv is installed, these dependencies can be installed by running uv pip install -r pyproject.toml. The environment specified in uv.lock can be recreated by running uv sync.

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Installing from PyPI for uv users

$ uv init
$ uv add dowker-complex
$ uv run python
>>> from dowker-complex import DowkerComplex
>>> ...