laplacian_matrix

laplacian_matrix(G, nodelist=None, weight='weight')

Returns the Laplacian matrix of G.

The graph Laplacian is the matrix L = D - A, where A is the adjacency matrix and D is the diagonal matrix of node degrees.

Parameters:

Ggraph

A NetworkX graph

nodelistlist, optional

The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes().

weightstring or None, optional (default=’weight’)

The edge data key used to compute each value in the matrix. If None, then each edge has weight 1.

Returns:

LSciPy sparse array

The Laplacian matrix of G.

See also

to_numpy_array()

normalized_laplacian_matrix

directed_laplacian_matrix

directed_combinatorial_laplacian_matrix

laplacian_spectrum()

Notes

For MultiGraph, the edges weights are summed.

This returns an unnormalized matrix. For a normalized output, use normalized_laplacian_matrix, directed_laplacian_matrix, or directed_combinatorial_laplacian_matrix.

This calculation uses the out-degree of the graph G. To use the in-degree for calculations instead, use G.reverse(copy=False) and take the transpose.

References

[1]

Langville, Amy N., and Carl D. Meyer. Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press, 2006.

Examples

For graphs with multiple connected components, L is permutation-similar to a block diagonal matrix where each block is the respective Laplacian matrix for each component.

>>> G = nx.Graph([(1, 2), (2, 3), (4, 5)])
>>> print(nx.laplacian_matrix(G).toarray())
[[ 1 -1  0  0  0]
 [-1  2 -1  0  0]
 [ 0 -1  1  0  0]
 [ 0  0  0  1 -1]
 [ 0  0  0 -1  1]]

>>> edges = [
...     (1, 2),
...     (2, 1),
...     (2, 4),
...     (4, 3),
...     (3, 4),
... ]
>>> DiG = nx.DiGraph(edges)
>>> print(nx.laplacian_matrix(DiG).toarray())
[[ 1 -1  0  0]
 [-1  2 -1  0]
 [ 0  0  1 -1]
 [ 0  0 -1  1]]

Notice that node 4 is represented by the third column and row. This is because by default the row/column order is the order of G.nodes (i.e. the node added order – in the edgelist, 4 first appears in (2, 4), before node 3 in edge (4, 3).) To control the node order of the matrix, use the nodelist argument.

>>> print(nx.laplacian_matrix(DiG, nodelist=[1, 2, 3, 4]).toarray())
[[ 1 -1  0  0]
 [-1  2  0 -1]
 [ 0  0  1 -1]
 [ 0  0 -1  1]]

This calculation uses the out-degree of the graph G. To use the in-degree for calculations instead, use G.reverse(copy=False) and take the transpose.

>>> print(nx.laplacian_matrix(DiG.reverse(copy=False)).toarray().T)
[[ 1 -1  0  0]
 [-1  1 -1  0]
 [ 0  0  2 -1]
 [ 0  0 -1  1]]

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Additional backends implement this function

graphblas : OpenMP-enabled sparse linear algebra backend.