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fiedler_vector

fiedler_vector(G, weight='weight', normalized=False, tol=1e-08, method='tracemin')

Return the Fiedler vector of a connected undirected graph.

The Fiedler vector of a connected undirected graph is the eigenvector corresponding to the second smallest eigenvalue of the Laplacian matrix of of the graph.

Parameters:

• G (NetworkX graph) – An undirected graph.
• weight (object, optional) – The data key used to determine the weight of each edge. If None, then each edge has unit weight. Default value: None.
• normalized (bool, optional) – Whether the normalized Laplacian matrix is used. Default value: False.
• tol (float, optional) – Tolerance of relative residual in eigenvalue computation. Default value: 1e-8.
• method (string, optional) –

  Method of eigenvalue computation. It should be one of ‘tracemin’ (TraceMIN), ‘lanczos’ (Lanczos iteration) and ‘lobpcg’ (LOBPCG). Default value: ‘tracemin’.

  The TraceMIN algorithm uses a linear system solver. The following values allow specifying the solver to be used.

  Value	Solver
  ’tracemin_pcg’	Preconditioned conjugate gradient method
  ’tracemin_chol’	Cholesky factorization
  ’tracemin_lu’	LU factorization

Returns:

fiedler_vector – Fiedler vector.

Return type:

NumPy array of floats.

Raises:

• NetworkXNotImplemented – If G is directed.
• NetworkXError – If G has less than two nodes or is not connected.

Notes

Edge weights are interpreted by their absolute values. For MultiGraph’s, weights of parallel edges are summed. Zero-weighted edges are ignored.

To use Cholesky factorization in the TraceMIN algorithm, the scikits.sparse package must be installed.

See also

laplacian_matrix()