hergm  R Documentation 
The function hergm
estimates and simulates three classes of hierarchical exponentialfamily random graph models:
1. The p_1 model of Holland and Leinhardt (1981) in exponentialfamily form and extensions by Vu, Hunter, and Schweinberger (2013) and Schweinberger, PetrescuPrahova, and Vu (2014) to both directed and undirected random graphs with additional model terms, with and without covariates, and with parametric and nonparametric priors (see arcs_i
, arcs_j
, edges_i
, edges_ij
, mutual_i
, mutual_ij
).
2. The stochastic block model of Snijders and Nowicki (1997) and Nowicki and Snijders (2001) in exponentialfamily form and extensions by Vu, Hunter, and Schweinberger (2013) and Schweinberger, PetrescuPrahova, and Vu (2014) with additional model terms, with and without covariates, and with parametric and nonparametric priors (see arcs_i
, arcs_j
, edges_i
, edges_ij
, mutual_i
, mutual_ij
).
3. The exponentialfamily random graph models with local dependence of Schweinberger and Handcock (2015), with and without covariates, and with parametric and nonparametric priors (see arcs_i
, arcs_j
, edges_i
, edges_ij
, mutual_i
, mutual_ij
, twostar_ijk
, triangle_ijk
, ttriple_ijk
, ctriple_ijk
).
The exponentialfamily random graph models with local dependence replace the longrange dependence of conventional exponentialfamily random graph models by shortrange dependence.
Therefore, exponentialfamily random graph models with local dependence replace the strong dependence of conventional exponentialfamily random graph models by weak dependence,
reducing the problem of model degeneracy (Handcock, 2003; Schweinberger, 2011) and improving goodnessoffit (Schweinberger and Handcock, 2015).
In addition, exponentialfamily random graph models with local dependence satisfy a weak form of selfconsistency in the sense that these models are selfconsistent under neighborhood sampling (Schweinberger and Handcock, 2015),
which enables consistent estimation of neighborhooddependent parameters (Schweinberger and Stewart, 2017; Schweinberger, 2017).
hergm(formula, max_number = 2, hierarchical = TRUE, parametric = FALSE, parameterization = "offset", initialize = FALSE, initialization_method = 1, estimate_parameters = TRUE, initial_estimate = NULL, n_em_step_max = 100, max_iter = 4, perturb = FALSE, scaling = NULL, alpha = NULL, alpha_shape = NULL, alpha_rate = NULL, eta = NULL, eta_mean = NULL, eta_sd = NULL, eta_mean_mean = NULL, eta_mean_sd = NULL, eta_precision_shape = NULL, eta_precision_rate = NULL, mean_between = NULL, indicator = NULL, parallel = 1, simulate = FALSE, method = "ml", seeds = NULL, sample_size = NULL, sample_size_multiplier_blocks = 20, NR_max_iter = 200, NR_step_len = NULL, NR_step_len_multiplier = 0.2, interval = 1024, burnin = 16*interval, mh.scale = 0.25, variational = FALSE, temperature = c(1,100), predictions = FALSE, posterior.burnin = 2000, posterior.thinning = 1, relabel = 1, number_runs = 1, verbose = 0, ...)
formula 
formula of the form 
max_number 
maximum number of blocks. 
hierarchical 
hierarchical prior; if 
parametric 
parametric prior; if 
parameterization 
There are three possible parameterizations of withinblock terms when using

initialize 
if 
initialization_method 
if 
estimate_parameters 
if 
initial_estimate 
if 
n_em_step_max 
if 
max_iter 
if 
perturb 
if 
scaling 
if 
alpha 
concentration parameter of truncated Dirichlet process prior of natural parameters of exponentialfamily model. 
alpha_shape, alpha_rate 
shape and rate parameter of Gamma prior of concentration parameter. 
eta 
the parameters of 
eta_mean, eta_sd 
means and standard deviations of Gaussian baseline distribution of Dirichlet process prior of natural parameters. 
eta_mean_mean, eta_mean_sd 
means and standard deviations of Gaussian prior of mean of Gaussian baseline distribution of Dirichlet process prior. 
eta_precision_shape, eta_precision_rate 
shape and rate (inverse scale) parameter of Gamma prior of precision parameter of Gaussian baseline distribution of Dirichlet process prior. 
mean_between 
if 
indicator 
if the indicators of block memberships of nodes are specified as integers between 
parallel 
number of computing nodes; if 
simulate 
if 
method 
if 
seeds 
seed of pseudorandom number generator; if 
sample_size 
if 
sample_size_multiplier_blocks 
if 
NR_max_iter 
if 
NR_step_len 
if 
NR_step_len_multiplier 
if 
interval 
if 
burnin 
if 
mh.scale 
if 
variational 
if 
temperature 
if 
predictions 
if 
posterior.burnin 
number of posterior burnin iterations; if computing is parallel, 
posterior.thinning 
if 
relabel 
if 
number_runs 
if 
verbose 
if Progress: 50.00% of 1000000 ... means of block parameters: 0.2838 1.3323 precisions of block parameters: 0.9234 1.4682 block parameters: 0.2544 0.2560 0.1176 0.0310 0.1915 1.9626 0.4022 1.8887 1.9719 0.6499 1.7265 0.0000 block indicators: 1 3 1 1 1 1 3 1 1 2 2 2 2 2 1 1 1 block sizes: 10 5 2 0 0 block probabilities: 0.5396 0.2742 0.1419 0.0423 0.0020 block probabilities prior parameter: 0.4256 posterior prediction of statistics: 66 123 where ... indicates additional information about the Markov chain Monte Carlo algorithm that is omitted here. The console output corresponds to:  "means of block parameters" correspond to the mean parameters of the Gaussian base distribution of parameters of  "precisions of block parameters" correspond to the precision parameters of the Gaussian base distribution of parameters of  "block parameters" correspond to the parameters of  "block indicators" correspond to the indicators of block memberships of nodes.  "block sizes" correspond to the block sizes.  "block probabilities" correspond to the prior probabilities of block memberships of nodes.  "block probabilities prior parameter" corresponds to the concentration parameter of truncated Dirichlet process prior of parameters of  if 
... 
additional arguments, to be passed to lowerlevel functions in the future. 
The function hergm
returns an object of class hergm
with components:
network 

formula 
formula of the form 
n 
number of nodes. 
hyper_prior 
indicator of whether hyper prior has been specified, i.e., whether the parameters 
alpha 
concentration parameter of truncated Dirichlet process prior of parameters of 
ergm_theta 
parameters of 
eta_mean 
mean parameters of Gaussian base distribution of parameters of 
eta_precision 
precision parameters of Gaussian base distribution of parameters of 
d1 
total number of parameters of 
d2 
total number of parameters of 
hergm_theta 
parameters of 
relabeled.hergm_theta 
relabeled parameters of 
number_fixed 
number of fixed indicators of block memberships of nodes. 
indicator 
indicators of block memberships of nodes. 
relabel 
if 
relabeled.indicator 
relabeled indicators of block memberships of nodes by using 
size 
the size of the blocks, i.e., the number of nodes of blocks. 
parallel 
number of computing nodes; if 
p_i_k 
posterior probabilities of block membership of nodes. 
p_k 
probabilities of block memberships of nodes. 
predictions 
if 
simulate 
if 
prediction 
posterior predictions of statistics. 
edgelist 
edge list of simulated network. 
sample_size 
if 
extract 
indicator of whether function 
verbose 
if 
Babkin, S., Stewart, J., Long, X., and M. Schweinberger (2020). Largescale estimation of random graph models with local dependence. Computational Statistics and Data Analysis, 152, 1–19.
Cao, M., Chen, Y., Fujimoto, K., and M. Schweinberger (2018). A twostage working model strategy for network analysis under hierarchical exponential random graph models. Proceedings of the 2018 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, 290–298.
Handcock, M. S. (2003). Assessing degeneracy in statistical models of social networks. Technical report, Center for Statistics and the Social Sciences, University of Washington, Seattle. http://www.csss.washington.edu/Papers.
Holland, P. W. and S. Leinhardt (1981). An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association, Theory & Methods, 76, 33–65.
Krivitsky, P. N., Handcock, M. S., & Morris, M. (2011). Adjusting for network size and composition effects in exponentialfamily random graph models. Statistical Methodology, 8(4), 319339.
Krivitsky, P.N, and Kolaczyk, E. D. (2015). On the question of effective sample size in network modeling: An asymptotic inquiry. Statistical science: a review journal of the Institute of Mathematical Statistics, 30(2), 184.
Nowicki, K. and T. A. B. Snijders (2001). Estimation and prediction for stochastic blockstructures. Journal of the American Statistical Association, Theory & Methods, 96, 1077–1087.
Peng, L. and L. Carvalho (2016). Bayesian degreecorrected stochastic block models for community detection. Electronic Journal of Statistics 10, 2746–2779.
Schweinberger, M. (2011). Instability, sensitivity, and degeneracy of discrete exponential families. Journal of the American Statistical Association, Theory & Methods, 106, 1361–1370.
Schweinberger, M. (2020). Consistent structure estimation of exponentialfamily random graph models with block structure. Bernoulli, 26, 1205–1233.
Schweinberger, M. and M. S. Handcock (2015). Local dependence in random graph models: characterization, properties, and statistical inference. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 7, 647676.
Schweinberger, M., Krivitsky, P. N., Butts, C.T. and J. Stewart (2020). Exponentialfamily models of random graphs: Inference in finite, super, and infinite population scenarios. Statistical Science, 35, 627662.
Schweinberger, M. and P. Luna (2018). HERGM: Hierarchical exponentialfamily random graph models. Journal of Statistical Software, 85, 1–39.
Schweinberger, M., PetrescuPrahova, M. and D. Q. Vu (2014). Disaster response on September 11, 2001 through the lens of statistical network analysis. Social Networks, 37, 42–55.
Schweinberger, M. and J. Stewart (2020). Concentration and consistency results for canonical and curved exponentialfamily random graphs. The Annals of Statistics, 48, 374–396.
Snijders, T. A. B. and K. Nowicki (1997). Estimation and prediction for stochastic blockmodels for graphs with latent block structure. Journal of Classification, 14, 75–100.
Stewart, J., Schweinberger, M., Bojanowski, M., and M. Morris (2019). Multilevel network data facilitate statistical inference for curved ERGMs with geometrically weighted terms. Social Networks, 59, 98–119.
Vu, D. Q., Hunter, D. R. and M. Schweinberger (2013). Modelbased clustering of large networks. Annals of Applied Statistics, 7, 1010–1039.
network, ergm.terms, hergm.terms, hergm.postprocess, summary, print, plot, gof, simulate
data(example) m < summary(d ~ edges)
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