Network histogram estimation for single-layer networks

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

Usage

nethist(A, h = NA, method = "PLL", control = nethist_control(), ...)

Arguments

A: An adjacency matrix or graph object. Accepted formats: a matrix, sparse dgCMatrix, igraph object, or network object. Must be an undirected 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. One of "PLL" (default, profile log-likelihood) or "LSE" (least squares). See Details.
control: A control object from nethist_control. Governs max_itr, greedy_swap_rule, greedy_stop_threshold, and verbose.
...: Pass max_itr, greedy_swap_rule, greedy_stop_threshold, or verbose via control = nethist_control(...) instead.

Value

An object of class c("nethist", "network_histogram") (see network_histogram-class) with the following fields:

cluster an integer vector of length \(n\) with block assignments.
thetahat a \(k \times k\) probability matrix ordered by group labels.
rho_hat estimated sparsity parameter.
normalized_LL normalized log-likelihood.
MSE mean squared error.
method loss function used ("PLL" or "LSE").
h bandwidth used for estimation.

Details

method = "PLL" is for Olhede and Wolfe (2014). method = "LSE" is for Gao et al. (2015).

Note that cluster labels are not ordered: vertices in cluster 1 are not necessarily more similar to cluster 2 than to cluster 10. Users may specify a custom display order in plot.nethist.

The greedy search is stochastic. At each iteration it draws a pair of vertices uniformly at random and swaps their block labels if the objective improves, so it converges to a local optimum that depends on the state of the random number stream. Call set.seed immediately before the fit to make results reproducible, and expect different seeds to give somewhat different partitions.

References

Olhede, S. C. & Wolfe, P. J. (2014). Network Histograms and Universality of Blockmodel Approximation. PNAS, 111(41), 14722-14727. doi:10.1073/pnas.1400374111

Gao, C., Lu, Y., & Zhou, H. H. (2015). Rate-Optimal Graphon Estimation. The Annals of Statistics, 43(6), 2624-2652. doi:10.1214/15-AOS1354

Examples

# \donttest{
set.seed(42)
data(polblog)
fit <- nethist(polblog)
fit
#> 
#> Network histogram fit (nethist)
#> 
#> Nodes: 1224   Blocks: 16   Bandwidth: 73
#> Method: Profile Likelihood
#> thetahat: 16 x 16 probability matrix
#> Normalized log-likelihood: -2.969964
#> 
#> Use summary() for details.
plot(fit)


fit_h <- nethist(polblog, h = 72)
# }