scipy.sparse.csgraph.laplacian#

scipy.sparse.csgraph.laplacian(csgraph, normed=False, return_diag=False, use_out_degree=False, *, copy=True, form='array', dtype=None, symmetrized=False)[source]#

Return the Laplacian of a directed graph.

Parameters:

csgrapharray_like or sparse matrix, 2 dimensions

compressed-sparse graph, with shape (N, N).

normedbool, optional

If True, then compute symmetrically normalized Laplacian. Default: False.

return_diagbool, optional

If True, then also return an array related to vertex degrees. Default: False.

use_out_degreebool, optional

If True, then use out-degree instead of in-degree. This distinction matters only if the graph is asymmetric. Default: False.

copy: bool, optional

If False, then change csgraph in place if possible, avoiding doubling the memory use. Default: True, for backward compatibility.

form: ‘array’, or ‘function’, or ‘lo’

Determines the format of the output Laplacian:

‘array’ is a numpy array;
‘function’ is a pointer to evaluating the Laplacian-vector or Laplacian-matrix product;
‘lo’ results in the format of the LinearOperator.

Choosing ‘function’ or ‘lo’ always avoids doubling the memory use, ignoring copy value. Default: ‘array’, for backward compatibility.

dtype: None or one of numeric numpy dtypes, optional

The dtype of the output. If dtype=None, the dtype of the output matches the dtype of the input csgraph, except for the case normed=True and integer-like csgraph, where the output dtype is ‘float’ allowing accurate normalization, but dramatically increasing the memory use. Default: None, for backward compatibility.

symmetrized: bool, optional

If True, then the output Laplacian is symmetric/Hermitian. The symmetrization is done by csgraph + csgraph.T.conj without dividing by 2 to preserve integer dtypes if possible prior to the construction of the Laplacian. The symmetrization will increase the memory footprint of sparse matrices unless the sparsity pattern is symmetric or form is ‘function’ or ‘lo’. Default: False, for backward compatibility.

Returns:

lapndarray, or sparse matrix, or LinearOperator: The N x N Laplacian of csgraph. It will be a NumPy array (dense) if the input was dense, or a sparse matrix otherwise, or the format of a function or LinearOperator if form equals ‘function’ or ‘lo’, respectively.
diagndarray, optional: The length-N main diagonal of the Laplacian matrix. For the normalized Laplacian, this is the array of square roots of vertex degrees or 1 if the degree is zero.

Notes

The Laplacian matrix of a graph is sometimes referred to as the “Kirchhoff matrix” or just the “Laplacian”, and is useful in many parts of spectral graph theory. In particular, the eigen-decomposition of the Laplacian can give insight into many properties of the graph, e.g., is commonly used for spectral data embedding and clustering.

The constructed Laplacian doubles the memory use if copy=True and form="array" which is the default. Choosing copy=False has no effect unless form="array" or the matrix is sparse in the coo format, or dense array, except for the integer input with normed=True that forces the float output.

Sparse input is reformatted into coo if form="array", which is the default.

If the input adjacency matrix is not symmetic, the Laplacian is also non-symmetric unless symmetrized=True is used.

Diagonal entries of the input adjacency matrix are ignored and replaced with zeros for the purpose of normalization where normed=True. The normalization uses the inverse square roots of row-sums of the input adjacency matrix, and thus may fail if the row-sums contain negative or complex with a non-zero imaginary part values.

The normalization is symmetric, making the normalized Laplacian also symmetric if the input csgraph was symmetric.

References

[1]

Laplacian matrix. https://en.wikipedia.org/wiki/Laplacian_matrix

Examples

>>> import numpy as np
>>> from scipy.sparse import csgraph

Our first illustration is the symmetric graph

>>> G = np.arange(4) * np.arange(4)[:, np.newaxis]
>>> G
array([[0, 0, 0, 0],
       [0, 1, 2, 3],
       [0, 2, 4, 6],
       [0, 3, 6, 9]])

and its symmetric Laplacian matrix

>>> csgraph.laplacian(G)
array([[ 0,  0,  0,  0],
       [ 0,  5, -2, -3],
       [ 0, -2,  8, -6],
       [ 0, -3, -6,  9]])

The non-symmetric graph

>>> G = np.arange(9).reshape(3, 3)
>>> G
array([[0, 1, 2],
       [3, 4, 5],
       [6, 7, 8]])

has different row- and column sums, resulting in two varieties of the Laplacian matrix, using an in-degree, which is the default

>>> L_in_degree = csgraph.laplacian(G)
>>> L_in_degree
array([[ 9, -1, -2],
       [-3,  8, -5],
       [-6, -7,  7]])

or alternatively an out-degree

>>> L_out_degree = csgraph.laplacian(G, use_out_degree=True)
>>> L_out_degree
array([[ 3, -1, -2],
       [-3,  8, -5],
       [-6, -7, 13]])

Constructing a symmetric Laplacian matrix, one can add the two as

>>> L_in_degree + L_out_degree.T
array([[ 12,  -4,  -8],
        [ -4,  16, -12],
        [ -8, -12,  20]])

or use the symmetrized=True option

>>> csgraph.laplacian(G, symmetrized=True)
array([[ 12,  -4,  -8],
       [ -4,  16, -12],
       [ -8, -12,  20]])

that is equivalent to symmetrizing the original graph

>>> csgraph.laplacian(G + G.T)
array([[ 12,  -4,  -8],
       [ -4,  16, -12],
       [ -8, -12,  20]])

The goal of normalization is to make the non-zero diagonal entries of the Laplacian matrix to be all unit, also scaling off-diagonal entries correspondingly. The normalization can be done manually, e.g.,

>>> G = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]])
>>> L, d = csgraph.laplacian(G, return_diag=True)
>>> L
array([[ 2, -1, -1],
       [-1,  2, -1],
       [-1, -1,  2]])
>>> d
array([2, 2, 2])
>>> scaling = np.sqrt(d)
>>> scaling
array([1.41421356, 1.41421356, 1.41421356])
>>> (1/scaling)*L*(1/scaling)
array([[ 1. , -0.5, -0.5],
       [-0.5,  1. , -0.5],
       [-0.5, -0.5,  1. ]])

Or using normed=True option

>>> L, d = csgraph.laplacian(G, return_diag=True, normed=True)
>>> L
array([[ 1. , -0.5, -0.5],
       [-0.5,  1. , -0.5],
       [-0.5, -0.5,  1. ]])

which now instead of the diagonal returns the scaling coefficients

>>> d
array([1.41421356, 1.41421356, 1.41421356])

Zero scaling coefficients are substituted with 1s, where scaling has thus no effect, e.g.,

>>> G = np.array([[0, 0, 0], [0, 0, 1], [0, 1, 0]])
>>> G
array([[0, 0, 0],
       [0, 0, 1],
       [0, 1, 0]])
>>> L, d = csgraph.laplacian(G, return_diag=True, normed=True)
>>> L
array([[ 0., -0., -0.],
       [-0.,  1., -1.],
       [-0., -1.,  1.]])
>>> d
array([1., 1., 1.])

Only the symmetric normalization is implemented, resulting in a symmetric Laplacian matrix if and only if its graph is symmetric and has all non-negative degrees, like in the examples above.

The output Laplacian matrix is by default a dense array or a sparse matrix inferring its shape, format, and dtype from the input graph matrix:

>>> G = np.array([[0, 1, 1], [1, 0, 1], [1, 1, 0]]).astype(np.float32)
>>> G
array([[0., 1., 1.],
       [1., 0., 1.],
       [1., 1., 0.]], dtype=float32)
>>> csgraph.laplacian(G)
array([[ 2., -1., -1.],
       [-1.,  2., -1.],
       [-1., -1.,  2.]], dtype=float32)

but can alternatively be generated matrix-free as a LinearOperator:

>>> L = csgraph.laplacian(G, form="lo")
>>> L
<3x3 _CustomLinearOperator with dtype=float32>
>>> L(np.eye(3))
array([[ 2., -1., -1.],
       [-1.,  2., -1.],
       [-1., -1.,  2.]])

or as a lambda-function:

>>> L = csgraph.laplacian(G, form="function")
>>> L
<function _laplace.<locals>.<lambda> at 0x0000012AE6F5A598>
>>> L(np.eye(3))
array([[ 2., -1., -1.],
       [-1.,  2., -1.],
       [-1., -1.,  2.]])

The Laplacian matrix is used for spectral data clustering and embedding as well as for spectral graph partitioning. Our final example illustrates the latter for a noisy directed linear graph.

>>> from scipy.sparse import diags, random
>>> from scipy.sparse.linalg import lobpcg

Create a directed linear graph with N=35 vertices using a sparse adjacency matrix G:

>>> N = 35
>>> G = diags(np.ones(N-1), 1, format="csr")

Fix a random seed rng and add a random sparse noise to the graph G:

>>> rng = np.random.default_rng()
>>> G += 1e-2 * random(N, N, density=0.1, random_state=rng)

Set initial approximations for eigenvectors:

>>> X = rng.random((N, 2))

The constant vector of ones is always a trivial eigenvector of the non-normalized Laplacian to be filtered out:

>>> Y = np.ones((N, 1))

Alternating (1) the sign of the graph weights allows determining labels for spectral max- and min- cuts in a single loop. Since the graph is undirected, the option symmetrized=True must be used in the construction of the Laplacian. The option normed=True cannot be used in (2) for the negative weights here as the symmetric normalization evaluates square roots. The option form="lo" in (2) is matrix-free, i.e., guarantees a fixed memory footprint and read-only access to the graph. Calling the eigenvalue solver lobpcg (3) computes the Fiedler vector that determines the labels as the signs of its components in (5). Since the sign in an eigenvector is not deterministic and can flip, we fix the sign of the first component to be always +1 in (4).

>>> for cut in ["max", "min"]:
...     G = -G  # 1.
...     L = csgraph.laplacian(G, symmetrized=True, form="lo")  # 2.
...     _, eves = lobpcg(L, X, Y=Y, largest=False, tol=1e-3)  # 3.
...     eves *= np.sign(eves[0, 0])  # 4.
...     print(cut + "-cut labels:\n", 1 * (eves[:, 0]>0))  # 5.
max-cut labels:
[1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1]
min-cut labels:
[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

As anticipated for a (slightly noisy) linear graph, the max-cut strips all the edges of the graph coloring all odd vertices into one color and all even vertices into another one, while the balanced min-cut partitions the graph in the middle by deleting a single edge. Both determined partitions are optimal.