graspologic/embed/ase.py [12:124]: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 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.select_svd`). Read more in the `Adjacency Spectral Embedding Tutorial `_ 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`` is 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` Does not support ``graph`` input of type scipy.sparse.csr_matrix - '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. svd_seed : int or None (default ``None``) Only applicable for ``algorithm="randomized"``; allows you to seed the randomized svd solver for deterministic, albeit pseudo-randomized behavior. Attributes ---------- n_features_in_: int Number of features passed to the :func:`~graspologic.embed.AdjacencySpectralEmbed.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.select_svd 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 .. [2] Levin, K., Roosta-Khorasani, F., Mahoney, M. W., & Priebe, C. E. (2018). Out-of-sample extension of graph adjacency spectral embedding. PMLR: Proceedings of Machine Learning Research, 80, 2975-2984. """ def __init__( self, n_components: Optional[int] = None, n_elbows: Optional[int] = 2, algorithm: SvdAlgorithmType = "randomized", n_iter: int = 5, check_lcc: bool = True, diag_aug: bool = True, concat: bool = False, svd_seed: Optional[int] = None, - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - graspologic/embed/omni.py [121:226]: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - r""" Omnibus embedding of arbitrary number of input graphs with matched vertex sets. Given :math:`A_1, A_2, ..., A_m` a collection of (possibly weighted) adjacency matrices of a collection :math:`m` undirected graphs with matched vertices. Then the :math:`(mn \times mn)` omnibus matrix, :math:`M`, has the subgraph where :math:`M_{ij} = \frac{1}{2}(A_i + A_j)`. The omnibus matrix is then embedded using adjacency spectral embedding. Read more in the `Omnibus Embedding for Multiple Graphs Tutorial `_ 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`` is 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 (defult = True) Whether to check if the average of all input graphs are connected. May result in non-optimal results if the average graph is unconnected. If True and average graph is unconnected, a UserWarning is thrown. diag_aug : bool, optional (default = True) Whether to replace the main diagonal of each adjacency matrices with a vector corresponding to the degree (or sum of edge weights for a weighted network) before embedding. concat : bool, optional (default = False) If graph(s) are directed, whether to concatenate each graph's left and right (out and in) latent positions along axis 1. svd_seed : int or None (default = ``None``) Only applicable for ``algorithm="randomized"``; allows you to seed the randomized svd solver for deterministic, albeit pseudo-randomized behavior. lse : bool, optional (default = False) Whether to construct the Omni matrix use the laplacian matrices of the graphs and embed the Omni matrix with LSE Attributes ---------- n_graphs_ : int Number of graphs n_vertices_ : int Number of vertices in each graph latent_left_ : array, shape (n_graphs, n_vertices, n_components) Estimated left latent positions of the graph. latent_right_ : array, shape (n_graphs, n_vertices, n_components), or None Only computed when the graph is directed, or adjacency matrix is asymmetric. 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.select_svd graspologic.embed.select_dimension References ---------- .. [1] Levin, K., Athreya, A., Tang, M., Lyzinski, V., & Priebe, C. E. (2017, November). A central limit theorem for an omnibus embedding of multiple random dot product graphs. In Data Mining Workshops (ICDMW), 2017 IEEE International Conference on (pp. 964-967). IEEE. """ def __init__( self, n_components: Optional[int] = None, n_elbows: Optional[int] = 2, algorithm: SvdAlgorithmType = "randomized", n_iter: int = 5, check_lcc: bool = True, diag_aug: bool = True, concat: bool = False, svd_seed: Optional[int] = None, - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -