6.2.2. networkx.linalg.laplacianmatrix.normalized_laplacian_matrix¶

networkx.linalg.laplacianmatrix.normalized_laplacian_matrix(G, nodelist=None, weight='weight')[source]¶

Return 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:	G : graph A NetworkX graph nodelist : list, 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(). weight : string 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:	N : NumPy matrix The normalized Laplacian matrix of G.

See also

laplacian_matrix

Notes

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

If the Graph contains selfloops, D is defined as diag(sum(A,1)), where A is the adjacency matrix [R1181].

References

[R1180]

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

[R1181]

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