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normalized_laplacian_matrix

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

Return the normalized Laplacian matrix of G.

The normalized graph Laplacian is the matrix

\[NL = D^{-1/2} L D^{-1/2}\]

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 :

L : 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 adjencency matrix [R321].

References

[R320]

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

[R321]

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