A set of terms and functions implementing Tapered exponential-family random graph models (ERGMs). Tapered ERGMs are a modification of ERGMs that reduce the effects of phase transitions, and with properly chosen hyper-parameters, provably removes all multiphase behavior.
Each ERGM has a corresponding Tapered ERGM. Indeed, the ergm.tapered package fits any ergm as it is based on ergm itself.
To install the latest development version from github, you can also use:
# If pak is not installed:
# install.packages("pak")
pak::pak("statnet/ergm.tapered")
Load package and example data
library(ergm.tapered)
data(sampson)
Sampson (1969) recorded the social interactions among a group of monks while he was a resident as an experimenter at the cloister. Of particular interest are the data on positive affect relations ("liking," in which each monk was asked if he had positive relations to each of the other monks. Each monk ranked only his top three choices (or four, in the case of ties) on "liking". Here, we consider a directed edge from monk A to monk B to exist if A nominated B among these top choices at any one of three time points during the year. For details see:
help(sampson)
We can make a quick visualization of the network
plot(sampson)
A natural model is one that includes a term measuring the transitivity of triples in the network, defined as a set of edges {(i,j), (j,k), (i,k)}.
fit <- ergm(samplike ~ edges + ttriple)Starting maximum pseudolikelihood estimation (MPLE):
...
Optimizing with step length 0.0054.
The log-likelihood improved by 27.9793.
Estimating equations are not within tolerance region.
Iteration 3 of at most 60:
Error in ergm.MCMLE(init, nw, model, initialfit = (initialfit <- NULL), :
Unconstrained MCMC sampling did not mix at all. Optimization cannot continue.
Calls: ergm -> ergm.MCMLE
In addition: Warning message:
In ergm_MCMC_sample(s, control, theta = mcmc.init, verbose = max(verbose - :
Unable to reach target effective size in iterations alotted.
Calls: ergm -> ergm.MCMLE -> ergm_MCMC_sample
This fit fails to converge computationally as the model is near
degenerate. We could try to get it to fit by working on the
computational algorithm. However, ergm.tapered considers a variant of
the ERGM that reflects our prior belief that the true generating process
is non-degenerate:
fit <- ergm.tapered(samplike ~ edges + ttriple)Starting maximum pseudolikelihood estimation (MPLE):
...
Iteration 2 of at most 60:
Optimizing with step length 1.0000.
The log-likelihood improved by 0.0047.
Precision adequate twice. Stopping.
Finished MCMLE.
Evaluating log-likelihood at the estimate. Fitting the dyad-independent submodel...
Bridging between the dyad-independent submodel and the full model...
Setting up bridge sampling...
Using 16 bridges: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .
Bridging finished.
This model was fit using MCMC. To examine model diagnostics and check
for degeneracy, use the mcmc.diagnostics() function.
summary(fit) Results:
Estimate Std. Error MCMC % z value Pr(>|z|)
edges -1.83992 0.27737 0 -6.634 < 1e-04 ***
ttriple 0.21810 0.05816 0 3.750 0.000177 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The estimated tapering scaling factor is 2.
Null Deviance: 424.2 on 306 degrees of freedom
Residual Deviance: 358.0 on 304 degrees of freedom
AIC: 362 BIC: 369.4 (Smaller is better. MC Std. Err. = 0)
This tapered ERGM fits. The summary indicates that the coefficient on the transitive triple term is positive (about 0.22) and statistically above zero. This coefficient has the same interpretation as those in a standard ERGM.
This model fixes the tapering parameter at 2 units. Let’s try to taper
less by increasing the tapering parameter to 3 (r=3):
fit <- ergm.tapered(samplike ~ edges + ttriple, r=3)Starting maximum pseudolikelihood estimation (MPLE):
...
Iteration 2 of at most 60:
This model was fit using MCMC. To examine model diagnostics and check
for degeneracy, use the mcmc.diagnostics() function.
summary(fit) Results:
Estimate Std. Error MCMC % z value Pr(>|z|)
edges -1.82890 0.27246 0 -6.712 <1e-04 ***
ttriple 0.21174 0.05255 0 4.029 <1e-04 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The estimated tapering scaling factor is 3.
Null Deviance: 424.2 on 306 degrees of freedom
Residual Deviance: 358.6 on 304 degrees of freedom
AIC: 362.6 BIC: 370.1 (Smaller is better. MC Std. Err. = 0)
It does not effect it much.
The software allows the tapering to be estimated based on the shape of the distributions of the model statistics. Let’s try that:
fit <- ergm.tapered(samplike ~ edges + ttriple, fixed=FALSE)Starting maximum pseudolikelihood estimation (MPLE):
Iteration 4 of at most 60:
Optimizing with step length 1.0000.
The log-likelihood improved by 0.0003.
Precision adequate twice. Stopping.
Finished MCMLE.
summary(fit) Results:
Estimate Std. Error MCMC % z value Pr(>|z|)
edges -1.83372 0.28426 0 -6.451 <1e-04 ***
ttriple 0.21236 0.09938 0 2.137 0.0326 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The estimated tapering scaling factor is 2.813.
Null Deviance: 424.2 on 306 degrees of freedom
Residual Deviance: 359.0 on 304 degrees of freedom
AIC: 365 BIC: 376.2 (Smaller is better. MC Std. Err. = 0)
The estimated tapering parameter is about 2.8 (It is printed under the
coefficient table). This is between the default value and the second
guess. The coefficients of the ERGM terms are about the same as before.
Enjoy trying ergm.tapering!
See the following papers for more information and examples:
- Fellows, Ian E. and Handcock, Mark S. (2017) Removing Phase Transitions from Gibbs Measures, Proceedings of the 20th International Conference on Artificial Intelligence and Statistics (AISTATS)s, Volume 54.
- Blackburn, Bart and Handcock, Mark S. (2022) Practical Network Modeling via Tapered Exponential-family Random Graph Models, Journal of Computational and Graphical Statistics.
To facilitate open development of the package while giving the core developers an opportunity to publish on their developments before opening them up for general use, this project comprises two repositories:
- A public repository
statnet/ergm.tapered - A private repository
statnet/ergm.tapered-private
The intention is that all developments in statnet/ergm.tapered-private will eventually make
their way into statnet/ergm.tapered and onto CRAN.
Developers and Contributing Users to the Statnet Project should read https://statnet.org/private/ for information about the relationship between the public and the private repository and the workflows involved.
R-Universe builds a set of binaries after every commit to the main
branch of the repository. We strongly encourage testing against them before filing a bug
report, as they may contain fixes that have not yet been sent to CRAN. To obtain the binaries
from r-universe, navigate to the package page at https://statnet.r-universe.dev/ergm.tapered
.
You may also want to install the corresponding latest binaries for packages on which
ergm.tapered depends, in particular
statnet.common,
network, and
ergm.