Exponential Random Graph Model (ERGM / p*)
Also known as: ERGM, p-star model, p* model, Üstel Rastgele Graf Modeli (ERGM / p*)
The Exponential Random Graph Model (ERGM), also known as the p* model, is a statistical framework for network analysis that models the probability of an observed network as a function of its local structural features — such as reciprocity, triangles, and degree distribution. Developed from the foundational work of Frank and Strauss (1986) and extended into the modern framework by Wasserman and Pattison (1996) and Robins et al. (2007), ERGM is the inferential standard for social network analysis, capable of testing whether observed network structures arise by chance or reflect genuine social processes.
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When to use it
ERGM is appropriate when you have a single observed network (cross-sectional or longitudinal snapshot) and you want to test which local structural processes — reciprocity, clustering, preferential attachment, homophily — generated it. The network should be small to medium in size; MCMC convergence becomes difficult for very large networks (thousands of nodes). The model requires directed or undirected binary ties (weighted variants exist but are more complex). It is not appropriate when the network has fewer than about 20–30 nodes, when you have no clear theoretical hypotheses about structural configurations, or when you need to model continuous time dynamics (use a Temporal ERGM or a Stochastic Actor-Oriented Model instead).
Strengths & limitations
- Provides formal statistical inference — p-values, standard errors, and confidence statements — for network structural hypotheses.
- Handles the non-independence of network ties that violates the assumptions of standard regression, making it the correct inferential tool for network data.
- Flexible specification: directed, undirected, bipartite, and weighted variants allow coverage of many network types.
- Goodness-of-fit diagnostics give a principled way to evaluate whether the model reproduces global network structure from local statistics.
- MCMC estimation can be slow and may fail to converge for large networks; practical limits are typically a few hundred to a few thousand nodes.
- Model degeneracy — where the estimated model places near-all probability on empty or complete networks — is a persistent risk requiring careful statistic selection.
- The model is primarily designed for cross-sectional snapshots; modeling genuine network dynamics requires extensions such as Temporal ERGM (TERGM) or SAOMs.
- Requires a minimum of roughly 30 nodes for reliable parameter estimation; very small networks yield uninformative estimates.
Frequently asked
How does ERGM differ from ordinary logistic regression on dyads?
Logistic regression treats each tie as an independent observation, which violates the fundamental non-independence of network data — the presence of one tie changes the probability of others. ERGM explicitly models this dependence by conditioning on the full network structure and is therefore the statistically correct approach for network data.
What is degeneracy and how do I avoid it?
Degeneracy occurs when the estimated model assigns nearly all probability to either the empty network or the complete network, making the model useless. It is most commonly caused by including the raw triangle count statistic. The solution is to use geometrically weighted versions — GWESP for triadic closure and GWD for degree distribution — which are curved exponential family statistics that are inherently more stable.
How large does my network need to be?
As a practical lower bound, around 30 nodes is the minimum for reliable parameter estimation. Networks of 50–300 nodes are the most common range in applied work. Beyond a few thousand nodes, MCMC convergence becomes very slow and alternative models such as stochastic block models or latent space models may be more practical.
Can ERGM model longitudinal network change?
The standard cross-sectional ERGM models a single network snapshot. For panel data with repeated network observations, the Temporal ERGM (TERGM) or the Stochastic Actor-Oriented Model (SAOM / SIENA) are the appropriate extensions that model tie dynamics over time.
Sources
- Robins, G., Pattison, P., Kalish, Y., & Lusher, D. (2007). An introduction to exponential random graph (p*) models for social networks. Social Networks, 29(2), 173-191. DOI: 10.1016/j.socnet.2006.08.002 ↗
- Lusher, D., Koskinen, J., & Robins, G. (Eds.) (2012). Exponential Random Graph Models for Social Networks: Theory, Methods, and Applications. Cambridge University Press. ISBN: 9780521193566
How to cite this page
ScholarGate. (2026, June 1). Exponential Random Graph Model (ERGM / p*). ScholarGate. https://scholargate.app/en/network-analysis/exponential-random-graph
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