ergm-references {ergm.count} R Documentation

## Reference Measures for Exponential-Family Random Graph Models for Counts

### Description

This page describes the possible reference measures (baseline distributions) for modeling count data found in the ergm.count package.

Each of these is specified on the RHS of a one-sided formula passed as the reference argument to ergm. See the ergm documentation for a complete description of how reference measures are specified.

### Known issues

#### Parameter space constraints

Poisson- and geometric-reference ERGMs have an unbouded sample space. This means that the parameter space may be constrained in complex ways that depend on the terms used in the model. At this time ergm has no way to detect when a parameter configuration had strayed outside of the parameter space, but it may be noticeable on a runtime trace plot (activated via MCMC.runtime.traceplot control parameter), when the simulated values keep climbing upwards. (See Krivitsky (2012) for a further discussion.)

A possible remedy if this appears to occur is to try lowering the control parameter MCMLE.steplength.

#### MCMLE.trustregion

Because Monte Carlo MLE's approximation to the likelihood becomes less accurate as the estimate moves away from the one used for the sample, ergm limits how far the optimization can move the estimate for every iteration: the log-likelihood may not change by more than MCMLE.trustregion control parameter, which defaults to 20. This is an adequate value for binary ERGMs, but because each dyad in a valued ERGM contains more information, this number may be too small, resulting in unnecessarily many iterations needed to find the MLE.

Automatically setting MCMLE.trustregion is work in progress, but, in the meantime, you may want to set it to a high number (e.g., 1000).

### Possible reference measures to represent baseline distributions

Reference measures currently available are:

Poisson

Poisson-reference ERGM: Specifies each dyad's baseline distribution to be Poisson with mean 1: h(y)=∏_{i,j} 1/y_{i,j}!, with the support of y_{i,j} being natural numbers (and 0). Using valued ERGM terms that are “generalized” from their binary counterparts, with form "sum" (see previous link for the list) produces Poisson regression. Using CMP induces a Conway-Maxwell-Poisson distribution that is Poisson when its coefficient is 0 and geometric when its coefficient is 1.

Three proposal functions are currently implemented, two of them designed to improve mixing for sparse networks. They can can be selected via the MCMC.prop.weights= control parameter. The sparse proposals work by proposing a jump to 0. Both of them take an optional proposal argument p0 (i.e., MCMC.prop.args=list(p0=...)) specifying the probability of such a jump. However, the way in which they implement it are different:

"random"

Select a dyad (i,j)at random, and draw the proposal y_{i,j}^\star \sim \mathrm{Poisson}_{\ne y_{i,j}}(y_{i,j}+0.5) (a Poisson distribution with mean slightly higher than the current value and conditional on not proposing the current value).

"0inflated"

As "random" but, with probability p0, propose a jump to 0 instead of a Poisson jump (if not already at 0). If p0 is not given, defaults to the "surplus" of 0s in the observed network, relative to Poisson.

"TNT" (the default)

As "0inflated" but instead of selecting a dyad at random, select a tie with probability p0, and a random dyad otherwise, as with the binary TNT. Currently, p0 defaults to 0.2.

Geometric

Geometric-reference ERGM: Specifies each dyad's baseline distribution to be uniform on the natural numbers (and 0): h(y)=1. In itself, this "distribution" is improper, but in the presence of sum, a geometric distribution is induced. Using CMP (in addition to sum) induces a Conway-Maxwell-Poisson distribution that is geometric when its coefficient is 0 and Poisson when its coefficient is -1.

Binomial(trials)

Binomial-reference ERGM: Specifies each dyad's baseline distribution to be binomial with trials trials and success probability of 0.5: h(y)=∏_{i,j}{{\code{trials}}\choose{y_{i,j}}}. Using valued ERGM terms that are “generalized” from their binary counterparts, with form "sum" (see previous link for the list) produces logistic regression.

### References

Krivitsky PN (2012). Exponential-Family Random Graph Models for Valued Networks. Electronic Journal of Statistics, 2012, 6, 1100-1128. doi: 10.1214/12-EJS696

Shmueli G, Minka TP, Kadane JB, Borle S, and Boatwright P (2005). A Useful Distribution for Fitting Discrete Data: Revival of the Conway–Maxwell–Poisson Distribution. Journal of the Royal Statistical Society: Series C, 54(1): 127-142.