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Normalizers

Contains a collection of matrix normalizers, i.e., functions that take a matrix as input and return a normalized/stochastic matrix version of it.

normalization_same_eigenvec_centr(A)

Add a dummy node to the matrix and normalize it so that the row sums are equal to one. The dummy node is added in such a way that the first eigenvector of the normalized matrix is the same as the first eigenvector of the original matrix.

Parameters:

  • A (ndarray) –

    Matrix to be normalized.

Returns:

  • ndarray –

    Normalized matrix.

Source code in pykda\normalizers.py 
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def normalization_same_eigenvec_centr(A: np.ndarray) -> np.ndarray:
    """
    Add a dummy node to the matrix and normalize it so that the row sums are
    equal to one. The dummy node is added in such a way that the first
    eigenvector of the normalized matrix is the same as the first eigenvector
    of the original matrix.

    Parameters
    ----------
    A : np.ndarray
        Matrix to be normalized.

    Returns
    -------
    np.ndarray
        Normalized matrix.

    """

    rows_sums = A.sum(axis=1, keepdims=True)
    max_row_sum = np.max(rows_sums)

    A_with_dummy = expand_matrix_with_row_and_column(A)  # added at beginning
    eigvec_centr, eigval = eigenvec_centrality(A)
    A_with_dummy[[0], 1:] = eigvec_centr.T
    A_with_dummy[1:, [0]] = max_row_sum - rows_sums

    return standard_row_normalization(A_with_dummy)

normalization_with_self_loops(A)

Add self-loops to each node so that the row sums of A are equal. Afterward apply standard row normalization.

Parameters:

  • A (ndarray) –

    Non-negative matrix to be normalized.

Returns:

  • ndarray –

    Normalized matrix.

Source code in pykda\normalizers.py 
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def normalization_with_self_loops(A: np.ndarray) -> np.ndarray:
    """
    Add self-loops to each node so that the row sums of A are equal. Afterward
    apply standard row normalization.

    Parameters
    ----------
    A : np.ndarray
        Non-negative matrix to be normalized.

    Returns
    -------
    np.ndarray
        Normalized matrix.

    """

    assert is_nonnegative_matrix(A), "Ensure the matrix elements are >= 0."

    rows_sums = A.sum(axis=1, keepdims=True)
    max_row_sum = np.max(rows_sums)
    self_loops_matrix = np.diag((max_row_sum - rows_sums).flatten())

    return standard_row_normalization(A + self_loops_matrix)

standard_row_normalization(A)

Normalize the rows of a given matrix so that they sum up to one.

Parameters:

  • A (ndarray) –

    Non-negative matrix with positive row sums to be normalized.

Returns:

  • ndarray –

    Normalized matrix.

Source code in pykda\normalizers.py 
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def standard_row_normalization(A: np.ndarray) -> np.ndarray:
    """
    Normalize the rows of a given matrix so that they sum up to one.

    Parameters
    ----------
    A : np.ndarray
        Non-negative matrix with positive row sums to be normalized.

    Returns
    -------
    np.ndarray
        Normalized matrix.

    """

    assert is_nonnegative_matrix(A), "Ensure the matrix elements are >= 0."
    assert has_positive_row_sums(A), "Ensure all row sums are > 0."

    return A / A.sum(axis=1, keepdims=True)