# Simulations¶

graspy.simulations.er_np(n, p, directed=False, loops=False, wt=1, wtargs=None, dc=None, dc_kws={})[source]

Samples a Erdos Renyi (n, p) graph with specified edge probability.

Erdos Renyi (n, p) graph is a simple graph with n vertices and a probability p of edges being connected.

Parameters: n: int Number of vertices p: float Probability of an edge existing between two vertices, between 0 and 1. directed: boolean, optional (default=False) If False, output adjacency matrix will be symmetric. Otherwise, output adjacency matrix will be asymmetric. loops: boolean, optional (default=False) If False, no edges will be sampled in the diagonal. Otherwise, edges are sampled in the diagonal. wt: object, optional (default=1) Weight function for each of the edges, taking only a size argument. This weight function will be randomly assigned for selected edges. If 1, graph produced is binary. wtargs: dictionary, optional (default=None) Optional arguments for parameters that can be passed to weight function wt. dc: function or array-like, shape (n_vertices) dc is used to generate a degree-corrected Erdos Renyi Model in which each node in the graph has a parameter to specify its expected degree relative to other nodes. function: should generate a non-negative number to be used as a degree correction to create a heterogenous degree distribution. A weight will be generated for each vertex, normalized so that the sum of weights is 1. array-like of scalars, shape (n_vertices): The weights should sum to 1; otherwise, they will be normalized and a warning will be thrown. The scalar associated with each vertex is the node's relative expected degree. dc_kws: dictionary Ignored if dc is none or array of scalar. If dc is a function, dc_kws corresponds to its named arguments. If not specified, in either case all functions will assume their default parameters. A : ndarray, shape (n, n) Sampled adjacency matrix

Examples

>>> np.random.seed(1)
>>> n = 4
>>> p = 0.25


To sample a binary Erdos Renyi (n, p) graph:

>>> er_np(n, p)
array([[0., 0., 1., 0.],
[0., 0., 1., 0.],
[1., 1., 0., 0.],
[0., 0., 0., 0.]])


To sample a weighted Erdos Renyi (n, p) graph with Uniform(0, 1) distribution:

>>> wt = np.random.uniform
>>> wtargs = dict(low=0, high=1)
>>> er_np(n, p, wt=wt, wtargs=wtargs)
array([[0.        , 0.        , 0.95788953, 0.53316528],
[0.        , 0.        , 0.        , 0.        ],
[0.95788953, 0.        , 0.        , 0.31551563],
[0.53316528, 0.        , 0.31551563, 0.        ]])

graspy.simulations.er_nm(n, m, directed=False, loops=False, wt=1, wtargs=None)[source]

Samples an Erdos Renyi (n, m) graph with specified number of edges.

Erdos Renyi (n, m) graph is a simple graph with n vertices and exactly m number of total edges.

Parameters: n: int Number of vertices m: int Number of edges, a value between 1 and $$n^2$$. directed: boolean, optional (default=False) If False, output adjacency matrix will be symmetric. Otherwise, output adjacency matrix will be asymmetric. loops: boolean, optional (default=False) If False, no edges will be sampled in the diagonal. Otherwise, edges are sampled in the diagonal. wt: object, optional (default=1) Weight function for each of the edges, taking only a size argument. This weight function will be randomly assigned for selected edges. If 1, graph produced is binary. wtargs: dictionary, optional (default=None) Optional arguments for parameters that can be passed to weight function wt. A: ndarray, shape (n, n) Sampled adjacency matrix

Examples

>>> np.random.seed(1)
>>> n = 4
>>> m = 4


To sample a binary Erdos Renyi (n, m) graph:

>>> er_nm(n, m)
array([[0., 1., 1., 1.],
[1., 0., 0., 1.],
[1., 0., 0., 0.],
[1., 1., 0., 0.]])


To sample a weighted Erdos Renyi (n, m) graph with Uniform(0, 1) distribution:

>>> wt = np.random.uniform
>>> wtargs = dict(low=0, high=1)
>>> er_nm(n, m, wt=wt, wtargs=wtargs)
array([[0.        , 0.66974604, 0.        , 0.38791074],
[0.66974604, 0.        , 0.        , 0.39658073],
[0.        , 0.        , 0.        , 0.93553907],
[0.38791074, 0.39658073, 0.93553907, 0.        ]])

graspy.simulations.sbm(n, p, directed=False, loops=False, wt=1, wtargs=None, dc=None, dc_kws={})[source]

Samples a graph from the stochastic block model (SBM).

SBM produces a graph with specified communities, in which each community can have different sizes and edge probabilities.

Parameters: n: list of int, shape (n_communities) Number of vertices in each community. Communities are assigned n[0], n[1], ... p: array-like, shape (n_communities, n_communities) Probability of an edge between each of the communities, where p[i, j] indicates the probability of a connection between edges in communities [i, j]. 0 < p[i, j] < 1 for all i, j. directed: boolean, optional (default=False) If False, output adjacency matrix will be symmetric. Otherwise, output adjacency matrix will be asymmetric. loops: boolean, optional (default=False) If False, no edges will be sampled in the diagonal. Otherwise, edges are sampled in the diagonal. wt: object or array-like, shape (n_communities, n_communities) if Wt is an object, a weight function to use globally over the sbm for assigning weights. 1 indicates to produce a binary graph. If Wt is an array-like, a weight function for each of the edge communities. Wt[i, j] corresponds to the weight function between communities i and j. If the entry is a function, should accept an argument for size. An entry of Wt[i, j] = 1 will produce a binary subgraph over the i, j community. wtargs: dictionary or array-like, shape (n_communities, n_communities) if Wt is an object, Wtargs corresponds to the trailing arguments to pass to the weight function. If Wt is an array-like, Wtargs[i, j] corresponds to trailing arguments to pass to Wt[i, j]. dc: function or array-like, shape (n_vertices) or (n_communities), optional dc is used to generate a degree-corrected stochastic block model [1] in which each node in the graph has a parameter to specify its expected degree relative to other nodes within its community. function: should generate a non-negative number to be used as a degree correction to create a heterogenous degree distribution. A weight will be generated for each vertex, normalized so that the sum of weights in each block is 1. array-like of functions, shape (n_communities): Each function will generate the degree distribution for its respective community. array-like of scalars, shape (n_vertices): The weights in each block should sum to 1; otherwise, they will be normalized and a warning will be thrown. The scalar associated with each vertex is the node's relative expected degree within its community. dc_kws: dictionary or array-like, shape (n_communities), optional Ignored if dc is none or array of scalar. If dc is a function, dc_kws corresponds to its named arguments. If dc is an array-like of functions, dc_kws should be an array-like, shape (n_communities), of dictionary. Each dictionary is the named arguments for the corresponding function for that community. If not specified, in either case all functions will assume their default parameters. A: ndarray, shape (sum(n), sum(n)) Sampled adjacency matrix

References

 [1] Tai Qin and Karl Rohe. "Regularized spectral clustering under the Degree-Corrected Stochastic Blockmodel," Advances in Neural Information Processing Systems 26, 2013

Examples

>>> np.random.seed(1)
>>> n = [3, 3]
>>> p = [[0.5, 0.1], [0.1, 0.5]]


To sample a binary 2-block SBM graph:

>>> sbm(n, p)
array([[0., 0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0., 1.],
[1., 1., 0., 0., 0., 0.],
[0., 0., 0., 0., 1., 0.],
[0., 0., 0., 1., 0., 0.],
[0., 1., 0., 0., 0., 0.]])


To sample a weighted 2-block SBM graph with Poisson(2) distribution:

>>> wt = np.random.poisson
>>> wtargs = dict(lam=2)
>>> sbm(n, p, wt=wt, wtargs=wtargs)
array([[0., 4., 0., 1., 0., 0.],
[4., 0., 0., 0., 0., 2.],
[0., 0., 0., 0., 0., 0.],
[1., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0.],
[0., 2., 0., 0., 0., 0.]])

graspy.simulations.rdpg(X, Y=None, rescale=True, directed=False, loops=True, wt=1, wtargs=None)[source]

Samples a random graph based on the latent positions in X (and optionally in Y)

If only X $$\in\mathbb{R}^{n\times d}$$ is given, the P matrix is calculated as $$P = XX^T$$. If X, Y $$\in\mathbb{R}^{n\times d}$$ is given, then $$P = XY^T$$. These operations correspond to the dot products between a set of latent positions, so each row in X or Y represents the latent positions in $$\mathbb{R}^{d}$$ for a single vertex in the random graph Note that this function may also rescale or clip the resulting P matrix to get probabilities between 0 and 1, or remove loops. A binary random graph is then sampled from the P matrix described by X (and possibly Y).

Parameters: X: np.ndarray, shape (n_vertices, n_dimensions) latent position from which to generate a P matrix if Y is given, interpreted as the left latent position Y: np.ndarray, shape (n_vertices, n_dimensions) or None, optional right latent position from which to generate a P matrix rescale: boolean, optional (default=True) when rescale is True, will subtract the minimum value in P (if it is below 0) and divide by the maximum (if it is above 1) to ensure that P has entries between 0 and 1. If False, elements of P outside of [0, 1] will be clipped directed: boolean, optional (default=False) If False, output adjacency matrix will be symmetric. Otherwise, output adjacency matrix will be asymmetric. loops: boolean, optional (default=True) If False, no edges will be sampled in the diagonal. Diagonal elements in P matrix are removed prior to rescaling (see above) which may affect behavior. Otherwise, edges are sampled in the diagonal. wt: object, optional (default=1) Weight function for each of the edges, taking only a size argument. This weight function will be randomly assigned for selected edges. If 1, graph produced is binary. wtargs: dictionary, optional (default=None) Optional arguments for parameters that can be passed to weight function wt. A: ndarray (n_vertices, n_vertices) A matrix representing the probabilities of connections between vertices in a random graph based on their latent positions

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

Examples

>>> np.random.seed(1)


Generate random latent positions using 2-dimensional Dirichlet distribution.

>>> X = np.random.dirichlet([1, 1], size=5)


Sample a binary RDPG using sampled latent positions.

>>> rdpg(X, loops=False)
array([[0., 1., 0., 0., 1.],
[1., 0., 0., 1., 1.],
[0., 0., 0., 1., 1.],
[0., 1., 1., 0., 0.],
[1., 1., 1., 0., 0.]])


Sample a weighted RDPG with Poisson(2) weight distribution

>>> wt = np.random.poisson
>>> wtargs = dict(lam=2)
>>> rdpg(X, loops=False, wt=wt, wtargs=wtargs)
array([[0., 4., 0., 2., 0.],
[1., 0., 0., 0., 0.],
[0., 0., 0., 0., 2.],
[1., 0., 0., 0., 1.],
[0., 2., 2., 0., 0.]])