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