Adjusted spectral test — adj_spec

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

adj_spec_test(
  A,
  K,
  z = NULL,
  DC = T,
  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.