fiedler_vector¶

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

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.