gdim

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gdim estimates graph dimension using cross-validated eigenvalues, via the graph-splitting technique developed in https://arxiv.org/abs/2108.03336. Theoretically, the method works by computing a special type of cross-validated eigenvalue which follows a simple central limit theorem. This allows users to perform hypothesis tests on the rank of the graph.

Installation

You can install gdim from CRAN with:

install.packages("gdim")

# to get the development version from GitHub:
install.packages("pak")
pak::pak("RoheLab/gdim")

Example

eigcv() is the main function in gdim. The single required parameter for the function is the maximum possible dimension, k_max.

In the following example, we generate a random graph from the stochastic block model (SBM) with 1000 nodes and 5 blocks (as such, we would expect the estimated graph dimension to be 5).

library(fastRG)
#> Loading required package: Matrix

B <- matrix(0.1, 5, 5)
diag(B) <- 0.3

model <- sbm(
  n = 1000,
  B = B,
  expected_degree = 40,
  poisson_edges = FALSE,
  allow_self_loops = FALSE
)

A <- sample_sparse(model)

Here, A is the adjacency matrix.

Now, we call the eigcv() function with k_max=10 to estimate graph dimension.

library(gdim)

eigcv_result <- eigcv(A, k_max = 10)
#> 'as(<dsCMatrix>, "dgCMatrix")' is deprecated.
#> Use 'as(., "generalMatrix")' instead.
#> See help("Deprecated") and help("Matrix-deprecated").
eigcv_result
#> Estimated graph dimension:    5
#> 
#> Number of bootstraps:         10
#> Edge splitting probabaility:  0.1
#> Significance level:       0.05
#> 
#>  ------------ Summary of Tests ------------
#>   k          z        pvals         padj
#>   1 41.1972023 0.000000e+00 0.000000e+00
#>   2  6.5483842 2.908147e-11 2.908147e-11
#>   3  6.2885741 1.601976e-10 1.601976e-10
#>   4  6.9601015 1.700138e-12 1.700138e-12
#>   5  7.1673010 3.824537e-13 3.824537e-13
#>   6 -0.3594110 6.403562e-01 6.403562e-01
#>   7 -0.2062852 5.817159e-01 5.817159e-01
#>   8 -0.6096004 7.289367e-01 7.289367e-01
#>   9 -0.7202233 7.643062e-01 7.643062e-01
#>  10 -0.6707828 7.488205e-01 7.488205e-01

In this example, eigcv() suggests k=5.

To visualize the result, use plot() which returns a ggplot object. The function displays the test statistic (z score) for each hypothesized graph dimension.

plot(eigcv_result)

Reference

Chen, Fan, Sebastien Roch, Karl Rohe, and Shuqi Yu. “Estimating Graph Dimension with Cross-Validated Eigenvalues.” ArXiv:2108.03336 [Cs, Math, Stat], August 6, 2021. https://arxiv.org/abs/2108.03336.