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Hybrid Network histogram estimation

hnethist.Rd

Estimating hybrid network histogram for single-layer networks and returning the indices of partitions.

Usage

hnethist(A, h = NA, method = "LSE", control = nethist_control(), ...)

Arguments

A

An adjacency matrix or an igraph object. It must be an undirected and simple graph.

h

A bandwidth parameter. If NA, the bandwidth is selected by Olhede and Wolfe (2014). If specified, the user-supplied value is used.

method

Type of loss function for network histogram. Must be one of LSE (default) or PLL for single-layer hybrid network histogram. See Details.

control

A control object from nethist_control. Governs max_itr, greedy_swap_rule, greedy_stop_threshold, and verbose.

...

Currently unused.

Value

A list of class c("hnethist", "nethist") with the following fields:

  • cluster an integer vector of length n with vertex-level block assignments from the initial nethist fit.

  • thetahat a k-by-k probability matrix of the selected hybrid model, where k is the number of nethist blocks.

  • rho_hat estimated sparsity parameter from the initial nethist fit.

  • normalized_LL normalized log-likelihood from the initial nethist fit.

  • MSE mean squared error of the selected model.

  • method loss function used ("LSE" or "PLL").

  • h bandwidth used for the initial nethist fit.

  • blockcluster a kmeans object describing how the k-by-k blocks were merged into s shapes.

  • BIC BIC value of the selected model.

  • s number of distinct shapes in the selected model.

  • details list of all candidate models, one entry per number of shapes from 1 up to the maximum considered, each containing s, blockcluster, thetahat, MSE, normalized_LL, and BIC.

  • initial the nethist object used as the starting point for block clustering.

Details

Among the outputs, the best model is selected based on the BIC criterion.The original reference provided theoretical guarantees for LSE, but we also allow PLL for the initial nethist fit. The block clustering and model selection steps are performed on the LSE loss regardless of the initial method, as the theoretical results pertain to LSE.

References

Verdeyme, A. & Olhede, S. C. (2024). Hybrid of Node and Link Communities for Graphon Estimation. arXiv:2401.05088