adj_spec_test {nett}

Adjusted spectral test

Description

The adjusted spectral goodness-of-fit test based on Poisson DCSBM.

The test is a natural extension on Lei's work of testing goodness-of-fit for SBM. The residual matrix \tilde{A} is computed from the DCSBM estimation expectation of A. To speed up computation, the residual matrix uses Poisson variance instead. Specifically,

\tilde{A}_{ij} = (A_{ij} - \hat P_{ij}) / ( n \hat P_{ij})^{1/2}, \quad \hat P_{ij} = \hat \theta_i \hat \theta_j \hat B_{\hat{z}_i, \hat{z}_j} \cdot 1\{i \neq j\}

where \hat{\theta} and \hat{B} are computed using estim_dcsbm if not provided.

Adjusted spectral test

Usage

adj_spec_test(
  A,
  K,
  z = NULL,
  DC = TRUE,
  theta = NULL,
  B = NULL,
  cluster_fct = spec_clust,
  ...
)

Arguments

A

adjacency matrix.

K

number of communities.

z

label vector for rows of adjacency matrix. If not given, will be calculated by the spectral clustering.

DC

whether or not include degree correction in the parameter estimation.

theta

give the propensity parameter directly.

B

give the connectivity matrix directly.

cluster_fct

community detection function to get z , by default using spec_clust.

...

additional arguments for cluster_fct.

Value

Adjusted spectral test statistics.

References

Details of modification can be seen at Adjusted chi-square test for degree-corrected block models, Linfan Zhang, Arash A. Amini, arXiv preprint arXiv:2012.15047, 2020.

The original spectral test is from A goodness-of-fit test for stochastic block models Lei, Jing, Ann. Statist. 44 (2016), no. 1, 401–424. doi:10.1214/15-AOS1370.