Inference

Two-graph hypothesis testing

class graspy.inference.LatentPositionTest(embedding='ase', n_components=None, n_bootstraps=500, test_case='rotation')

Two-sample hypothesis test for the problem of determining whether two random dot product graphs have the same latent positions [R3d27477db6c1-1].

This test assumes that the two input graphs are vertex aligned, that is, there is a known mapping between vertices in the two graphs and the input graphs have their vertices sorted in the same order. Currently, the function only supports undirected graphs.

Read more in the tutorials

Parameters:

embedding : string, { 'ase' (default), 'omnibus'}

String describing the embedding method to use:

• 'ase'

  Embed each graph separately using adjacency spectral embedding and use Procrustes to align the embeddings.

• 'omnibus'

  Embed all graphs simultaneously using omnibus embedding.

n_components : None (default), or int

Number of embedding dimensions. If None, the optimal embedding dimensions are found by the Zhu and Godsi algorithm.

test_case : string, {'rotation' (default), 'scalar-rotation', 'diagonal-rotation'}

describes the exact form of the hypothesis to test when using 'ase' or 'lse' as an embedding method. Ignored if using 'omnibus'. Given two latent positions, \(X_1\) and \(X_2\), and an orthogonal rotation matrix \(R\) that minimizes \(||X_1 - X_2 R||_F\):

• 'rotation'

  \[H_o: X_1 = X_2 R\]

• 'scalar-rotation'

  \[H_o: X_1 = c X_2 R\]

  where \(c\) is a scalar, \(c > 0\)

• 'diagonal-rotation'

  \[H_o: X_1 = D X_2 R\]

  where \(D\) is an arbitrary diagonal matrix

n_bootstraps : int, optional (default 500)

Number of bootstrap simulations to run to generate the null distribution

Attributes:

null_distribution_1_, null_distribution_2_ : np.ndarray (n_bootstraps,)

The distribution of T statistics generated under the null, using the first and and second input graph, respectively. The latent positions of each sample graph are used independently to sample random dot product graphs, so two null distributions are generated

sample_T_statistic_ : float

The observed difference between the embedded positions of the two input graphs after an alignment (the type of alignment depends on test_case)

p_value_1_, p_value_2_ : float

The p value estimated from the null distributions from sample 1 and sample 2.

p_ : float

The overall p value from the test; this is the max of p_value_1_ and p_value_2_

See also

graspy.embed.AdjacencySpectralEmbed

graspy.embed.OmnibusEmbed

graspy.embed.selectSVD

References

[R3d27477db6c1-1]

Tang, M., A. Athreya, D. Sussman, V. Lyzinski, Y. Park, Priebe, C.E. "A Semiparametric Two-Sample Hypothesis Testing Problem for Random Graphs" Journal of Computational and Graphical Statistics, Vol. 26(2), 2017

fit(self, A1, A2)

Fits the test to the two input graphs

Parameters:

A1, A2 : nx.Graph, nx.DiGraph, nx.MultiDiGraph, nx.MultiGraph, np.ndarray

The two graphs to run a hypothesis test on. If np.ndarray, shape must be (n_vertices, n_vertices) for both graphs, where n_vertices is the same for both

Returns:

p : float

The p value corresponding to the specified hypothesis test

get_params(self, deep=True)

Get parameters for this estimator.

Parameters:

deep : boolean, optional

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

params : mapping of string to any

Parameter names mapped to their values.

set_params(self, **params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it's possible to update each component of a nested object.

Returns:

self

class graspy.inference.LatentDistributionTest(n_components=None, n_bootstraps=200, bandwidth=None)

Two-sample hypothesis test for the problem of determining whether two random dot product graphs have the same distributions of latent positions [R63152004fa12-1].

This test can operate on two graphs where there is no known matching between the vertices of the two graphs, or even when the number of vertices is different. Currently, testing is only supported for undirected graphs.

Read more in the tutorials

Parameters:

n_components : int or None, optional (default=None)

Number of embedding dimensions. If None, the optimal embedding dimensions are found by the Zhu and Godsi algorithm.

n_bootstraps : int (default=200)

Number of bootstrap iterations.

bandwidth : float, optional (default=0.5)

Bandwidth to use for gaussian kernel. If None, the median heuristic will be used.

Attributes:

sample_T_statistic_ : float

The observed difference between the embedded latent positions of the two input graphs.

p_ : float

The overall p value from the test.

null_distribution_ : ndarray, shape (n_bootstraps, )

The distribution of T statistics generated under the null.

References

[R63152004fa12-1]

Tang, M., Athreya, A., Sussman, D. L., Lyzinski, V., & Priebe, C. E. (2017). "A nonparametric two-sample hypothesis testing problem for random graphs." Bernoulli, 23(3), 1599-1630.

fit(self, A1, A2)

Fits the test to the two input graphs

Parameters:

A1, A2 : nx.Graph, nx.DiGraph, nx.MultiDiGraph, nx.MultiGraph, np.ndarray

The two graphs to run a hypothesis test on.

Returns:

p_ : float

The p value corresponding to the specified hypothesis test

get_params(self, deep=True)

Get parameters for this estimator.

Parameters:

deep : boolean, optional

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

params : mapping of string to any

Parameter names mapped to their values.

set_params(self, **params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it's possible to update each component of a nested object.

Returns:

self