The standard advice is a lookup table. One observation of a network, use an exponential random graph model; several observations, use a stochastic actor-oriented model — SIENA, in the software's name. The table is not wrong. It is the answer to a question about data availability, and it gets mistaken for an answer to a question about theory.

My thesis: wave count decides which models are available, not which one is right. The ERGM and the SAOM encode different assumptions about how ties depend on one another, and those assumptions imply different substantive claims. When a design permits both — and designs with two or more waves permit both — the dependence structure your theory asserts should decide, and the choice should be defended rather than inherited from a methods course.

The models disagree about where dependence lives

The formal difference is not longitudinal versus cross-sectional. Block, Stadtfeld and Snijders set it out from first principles: the ERGM is defined on the graph level, while the SAOM is defined on the transition level, which lets the SAOM represent asymmetric or one-sided tie-transition dependence that a graph-level model cannot.[^1] A second difference compounds it. ERGM statistics are global — counted over the whole network — whereas SAOM statistics are nested within actors, evaluated as the change one actor can make at one moment. Their conclusion is uncomfortable for the lookup table: dependence assumptions in the SAOM are generally stronger, the meso- and macro-level properties the two models can represent differ substantively, and analysing the same network with both can lead to distinct results.[^1]

Interpretation, mine: this reframes what a wave is. Additional waves do not merely unlock a longitudinal tool; they let you assert that change is generated by actors making individual decisions under constraint. That is a strong theoretical claim about mechanism. On a dataset where it is false — say ties recorded by an institution rather than chosen by the actors they connect — extra waves do not make the SAOM appropriate. They only make it available.

This is why "SIENA when you have panels, ERGM when you do not" quietly answers the wrong question. It treats an assumption about mechanism as a consequence of sample design.

Where the temporal question is genuinely decisive

The lookup table earns its keep at the boundaries, and those boundaries are real.

With a single observation, the SAOM has nothing to estimate: it models the rate and direction of change between observed states, and with one state there is no change to decompose. The ERGM is the appropriate family here, modelling the probability of the observed graph as a function of its structural statistics.[^8]

With repeated observations and a question about change, the asymmetry runs the other way. Pooling panel waves into one ERGM discards the very variation the design was built to capture. The SAOM instead decomposes change between waves into unobserved micro-steps, and can model network and behaviour dynamics jointly when actor outcomes co-evolve with ties.[^4]

Between those poles sits the case that the table handles badly: several waves, but a question about structural association rather than about change. Here both families are available and the decision is genuinely yours to argue.

Temporal ERGMs, and a caution about how the comparison is run

The obvious reconciliation — extend the ERGM through time — has been examined directly, and the finding is specific. Comparing auto-regressive network models such as TERGM against process-based models such as the SAOM, Block and colleagues report that the TERGM has no consistent interpretation at the tie level and on network change, and that its parameters depend strongly on the interval between measurements; the SAOM does not suffer from either limitation.[^2]

The same paper carries a result that should temper the whole debate: both model families performed poorly at out-of-sample prediction of individual ties.[^2] Interpretation: if neither family predicts individual ties well, predictive accuracy is a weak criterion for choosing between them. Consistency of interpretation is the stronger one, and it is the criterion the lookup table never mentions.

The literature has also had to correct itself on method. A subsequent exchange documents circular specifications and prediction using information from the future in a published SAOM–TERGM comparison.[^3] Interpretation: a model-comparison result is only as good as the protocol that produced it, and in this area the protocols themselves have been contested. Treat any single head-to-head benchmark, including the ones cited here, as evidence rather than settlement.

Specification and diagnostics are where the work actually is

In practice the failure mode is rarely picking the wrong family. It is specifying either one badly, and both families announce that failure loudly enough to be misread as a defect of the tool.

ERGM degeneracy. Certain specifications place nearly all probability mass on near-empty or near-complete graphs. Schweinberger characterises this formally through instability, showing that unstable discrete exponential-family models exhibit excessive sensitivity and near-degeneracy, and that instability can be used to detect and penalise such models.[^5] The practical response is curved exponential-family terms: geometrically weighted statistics such as GWESP, which Hunter treats as curved exponential families and which were introduced in part to address the degeneracy and fitting problems of the earlier Markov specifications.[^6] Terms and their computational behaviour are documented in the ergm specification reference.[^7]

SAOM convergence. The analogous signal is the convergence check: overall maximum convergence ratios and per-parameter t-ratios that indicate whether the simulated moments have actually matched the observed ones. A non-converged SAOM is not a result with wide error bars — it is not a result.

Interpretation: both diagnostics carry the same message. A model that will not fit is usually telling you that the dependence you specified is not the dependence in the data. That is information about your theory, and treating it as a nuisance to be tuned away is how a specification search becomes a fishing expedition.

There is also a design constraint that precedes both: whether the study can detect the effect at all. Statistical power in longitudinal network studies depends on the number of waves, network size and effect size in ways that are not intuitive.[^9] A well-chosen model on an underpowered design still yields an inconclusive answer.

What I do, and what I am not certain about

My own practice, stated as practice rather than as a rule: I use the design to eliminate, then choose on mechanism. One wave and a structural-association question, ERGM. Panel waves and a question about how ties form and dissolve through actor decisions, SAOM. Panel waves where I cannot defend the actor-decision mechanism — institutionally recorded or externally imposed ties — I state that limitation explicitly and treat a cross-sectional model on a defensible snapshot as the more honest analysis. In the venture-capital imprinting work this cut both ways: snapshots for the critical-period question, panel estimation for the evolution question, and the two reported as different questions rather than as a robustness check on one another.

What I am not certain about:

  • Whether the two families should be reconciled at all. The dependence structures differ at the level of first principles,[^1] so a study reporting both may be reporting two findings rather than one confirmed finding. I do not have a general rule for when agreement between them is evidence and when it is coincidence.
  • How much the published comparisons generalise. The benchmark results above rest on a small number of datasets, and the protocols have been disputed in print.[^3] I would not extrapolate from them to a network unlike the ones tested.
  • Where the boundary of the actor-decision assumption sits. I can name clear cases at both ends and cannot specify the threshold in between.

The claim I will defend is narrower than the one the lookup table makes, and I think more useful: when you can justify either model, say which dependence assumption you are asserting and why. When you cannot justify either, more waves will not rescue the design.

[^1]: Block, P., Stadtfeld, C., & Snijders, T. A. B. (2019). Forms of Dependence: Comparing SAOMs and ERGMs From Basic Principles. Sociological Methods & Research, 48(1), 202–239. [^2]: Block, P., Koskinen, J., Hollway, J., Steglich, C., & Stadtfeld, C. (2018). Change we can believe in: Comparing longitudinal network models on consistency, interpretability and predictive power. Social Networks, 52, 180–191. [^3]: Block, P., et al. Circular specifications and "predicting" with information from the future: Errors in the empirical SAOM–TERGM comparison of Leifeld & Cranmer. Network Science. [^4]: Snijders, T. A. B., et al. Statistical Analysis of Network Dynamics. Annual Review of Statistics and Its Application. [^5]: Schweinberger, M. (2011). Instability, Sensitivity, and Degeneracy of Discrete Exponential Families. Journal of the American Statistical Association. [^6]: Hunter, D. R. (2007). Curved Exponential Family Models for Social Networks. Social Networks. [^7]: Hunter, D. R., Handcock, M. S., Butts, C. T., Goodreau, S. M., & Morris, M. Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects. Journal of Statistical Software. [^8]: Handcock, M. S. An Introduction to Exponential Random Graph (p*) Models for Social Networks. [^9]: Stadtfeld, C., Snijders, T. A. B., Steglich, C., & van Duijn, M. (2020). Statistical Power in Longitudinal Network Studies. Sociological Methods & Research.

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