Intro to ERGM in Stata

Every tutorial so far has treated a network’s structure as a fixed thing to describe — count ties, compute centrality, extract a path. An exponential-family random graph model (ERGM) turns that around: instead of describing one observed network, it asks what statistical process could plausibly have generated it, and estimates that process from the data.

The idea

An ERGM models the probability of the entire observed network as a function of a handful of network statistics — how many ties there are, whether ties tend to cluster into triangles, whether nodes with the same attribute tend to tie to each other, and so on. Each statistic gets its own coefficient, estimated so that the statistics of the observed network are about as likely as possible under the fitted model. A coefficient’s sign and size then say whether that particular structural or covariate effect makes a tie more or less likely, holding the others fixed — the same logic as a logistic regression coefficient, because a dyad-independent ERGM is one.

A minimal example

Every nwergm model needs an edges term — it plays the role of the intercept, and by itself gives a model where every potential tie is equally likely:

. nwwebuse florentine, nwclear

. nwergm flomarriage, edges

Exponential-family random graph model

Network:               =  flomarriage
Nodes:                 =  16
Ties:                  =  20
Directed:              =  No
Estimation:            =  MPLE

------------------------------------------------------------------------------
 flomarriage | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
       edges |  -1.609438    .244949    -6.57   0.000    -2.089529   -1.129347
------------------------------------------------------------------------------

Maximum pseudolikelihood estimate (not full ERGM maximum likelihood unless the model is dyad-independent).

That coefficient isn’t arbitrary — for an edges-only model it’s exactly the log-odds of the network’s own density: flomarriage has 20 ties out of 120 possible dyads (density ≈ 0.167), and logit(0.167) ≈ -1.609, matching the output above.

Adding a covariate term asks a real question: are wealthier families more likely to have married into each other?

. nwergm flomarriage, edges nodecov(wealth)

Exponential-family random graph model

Network:               =  flomarriage
Nodes:                 =  16
Ties:                  =  20
Directed:              =  No
Estimation:            =  MPLE

--------------------------------------------------------------------------------
   flomarriage | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
---------------+----------------------------------------------------------------
         edges |  -2.594929   .5360561    -4.84   0.000     -3.64558   -1.544278
nodecov_wealth |   .0105459   .0046743     2.26   0.024     .0013845    .0197073
--------------------------------------------------------------------------------

Maximum pseudolikelihood estimate (not full ERGM maximum likelihood unless the model is dyad-independent).

nodecov_wealth is positive and significant (p = 0.024): combined family wealth is associated with a higher chance of a marriage tie, controlling for the baseline density captured by edges.

Where to go from here

nwergm supports a large family of additional terms beyond nodecov() — homophily (nodematch()), triadic closure (gwesp()), degree-distribution effects, directed and two-mode variants, and both MPLE and full MCMC maximum-likelihood (method(mcmle)) estimation — see the command reference for the complete term list. A forthcoming Stata Press book covers ERGM (and the longitudinal models in the next tutorial) in full depth: model selection, goodness-of-fit diagnostics, convergence checking for MCMC estimation, and worked applications beyond this orientation.