# Copyright (c) Microsoft Corporation and contributors.
# Licensed under the MIT License.
import warnings
from .base import BaseSpectralEmbed
from ..utils import (
import_graph,
is_fully_connected,
augment_diagonal,
pass_to_ranks,
is_unweighted,
)
[docs]class AdjacencySpectralEmbed(BaseSpectralEmbed):
r"""
Class for computing the adjacency spectral embedding of a graph.
The adjacency spectral embedding (ASE) is a k-dimensional Euclidean representation
of the graph based on its adjacency matrix. It relies on an SVD to reduce
the dimensionality to the specified k, or if k is unspecified, can find a number of
dimensions automatically (see :class:`~graspologic.embed.selectSVD`).
Read more in the :ref:`tutorials <embed_tutorials>`
Parameters
----------
n_components : int or None, default = None
Desired dimensionality of output data. If "full",
n_components must be <= min(X.shape). Otherwise, n_components must be
< min(X.shape). If None, then optimal dimensions will be chosen by
:func:`~graspologic.embed.select_dimension` using ``n_elbows`` argument.
n_elbows : int, optional, default: 2
If ``n_components=None``, then compute the optimal embedding dimension using
:func:`~graspologic.embed.select_dimension`. Otherwise, ignored.
algorithm : {'randomized' (default), 'full', 'truncated'}, optional
SVD solver to use:
- 'randomized'
Computes randomized svd using
:func:`sklearn.utils.extmath.randomized_svd`
- 'full'
Computes full svd using :func:`scipy.linalg.svd`
- 'truncated'
Computes truncated svd using :func:`scipy.sparse.linalg.svds`
n_iter : int, optional (default = 5)
Number of iterations for randomized SVD solver. Not used by 'full' or
'truncated'. The default is larger than the default in randomized_svd
to handle sparse matrices that may have large slowly decaying spectrum.
check_lcc : bool , optional (default = True)
Whether to check if input graph is connected. May result in non-optimal
results if the graph is unconnected. If True and input is unconnected,
a UserWarning is thrown. Not checking for connectedness may result in
faster computation.
diag_aug : bool, optional (default = True)
Whether to replace the main diagonal of the adjacency matrix with a vector
corresponding to the degree (or sum of edge weights for a weighted network)
before embedding. Empirically, this produces latent position estimates closer
to the ground truth.
concat : bool, optional (default False)
If graph is directed, whether to concatenate left and right (out and in) latent positions along axis 1.
Attributes
----------
n_features_in_: int
Number of features passed to the fit method.
latent_left_ : array, shape (n_samples, n_components)
Estimated left latent positions of the graph.
latent_right_ : array, shape (n_samples, n_components), or None
Only computed when the graph is directed, or adjacency matrix is assymetric.
Estimated right latent positions of the graph. Otherwise, None.
singular_values_ : array, shape (n_components)
Singular values associated with the latent position matrices.
See Also
--------
graspologic.embed.selectSVD
graspologic.embed.select_dimension
Notes
-----
The singular value decomposition:
.. math:: A = U \Sigma V^T
is used to find an orthonormal basis for a matrix, which in our case is the
adjacency matrix of the graph. These basis vectors (in the matrices U or V) are
ordered according to the amount of variance they explain in the original matrix.
By selecting a subset of these basis vectors (through our choice of dimensionality
reduction) we can find a lower dimensional space in which to represent the graph.
References
----------
.. [1] Sussman, D.L., Tang, M., Fishkind, D.E., Priebe, C.E. "A
Consistent Adjacency Spectral Embedding for Stochastic Blockmodel Graphs,"
Journal of the American Statistical Association, Vol. 107(499), 2012
"""
def __init__(
self,
n_components=None,
n_elbows=2,
algorithm="randomized",
n_iter=5,
check_lcc=True,
diag_aug=True,
concat=False,
):
super().__init__(
n_components=n_components,
n_elbows=n_elbows,
algorithm=algorithm,
n_iter=n_iter,
check_lcc=check_lcc,
concat=concat,
)
if not isinstance(diag_aug, bool):
raise TypeError("`diag_aug` must be of type bool")
self.diag_aug = diag_aug
[docs] def fit(self, graph, y=None):
"""
Fit ASE model to input graph
Parameters
----------
graph : array_like or networkx.Graph
Input graph to embed.
Returns
-------
self : object
Returns an instance of self.
"""
A = import_graph(graph)
if self.check_lcc:
if not is_fully_connected(A):
msg = (
"Input graph is not fully connected. Results may not"
+ "be optimal. You can compute the largest connected component by"
+ "using ``graspologic.utils.get_lcc``."
)
warnings.warn(msg, UserWarning)
if self.diag_aug:
A = augment_diagonal(A)
self.n_features_in_ = len(A)
self._reduce_dim(A)
return self