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networkx.linalg.algebraicconnectivity.algebraic_connectivity

algebraic_connectivity(G, weight='weight', normalized=False, tol=1e-08, method='tracemin_pcg', seed=None)[source]

Return the algebraic connectivity of an undirected graph.

The algebraic connectivity of a connected undirected graph is the second smallest eigenvalue of its Laplacian matrix.

Parameters:

  • G (NetworkX graph) – An undirected graph.

  • weight (object, optional (default: None)) – The data key used to determine the weight of each edge. If None, then each edge has unit weight.

  • normalized (bool, optional (default: False)) – Whether the normalized Laplacian matrix is used.

  • tol (float, optional (default: 1e-8)) – Tolerance of relative residual in eigenvalue computation.

  • method (string, optional (default: ‘tracemin_pcg’)) – Method of eigenvalue computation. It must be one of the tracemin options shown below (TraceMIN), ‘lanczos’ (Lanczos iteration) or ‘lobpcg’ (LOBPCG).

    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

  • seed (integer, random_state, or None (default)) – Indicator of random number generation state. See Randomness.
Returns:

algebraic_connectivity – Algebraic connectivity.
Return type:

float
Raises:
  • NetworkXNotImplemented – If G is directed.
  • NetworkXError – If G has less than two nodes.

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()