networkx.linalg.laplacianmatrix.normalized_laplacian_matrix¶

normalized_laplacian_matrix(G, nodelist=None, weight='weight')[source]

Returns the normalized Laplacian matrix of G.

The normalized graph Laplacian is the matrix

$N = D^{-1/2} L D^{-1/2}$

where L is the graph Laplacian 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
NScipy sparse matrix

The normalized Laplacian matrix of G.

laplacian_matrix
normalized_laplacian_spectrum

Notes

For MultiGraph/MultiDiGraph, the edges weights are summed. See to_numpy_array for other options.

If the Graph contains selfloops, D is defined as diag(sum(A,1)), where A is the adjacency matrix [2].

References

1

Fan Chung-Graham, Spectral Graph Theory, CBMS Regional Conference Series in Mathematics, Number 92, 1997.

2

Steve Butler, Interlacing For Weighted Graphs Using The Normalized Laplacian, Electronic Journal of Linear Algebra, Volume 16, pp. 90-98, March 2007.