Triad Census
Also known as: triad count, triadic census, 16-type triad census, MAN triad census
The triad census counts how many of a directed network's three-actor subgroups fall into each of the 16 possible types of triad, providing a compact fingerprint of the network's local structure. Introduced by Paul Holland and Samuel Leinhardt in 1970, it is the standard way to test structural theories — balance, clustering, transitivity, ranked clusters — by comparing the observed distribution of triad types against what a random network would produce.
Key highlights
- Compactly fingerprints a directed network's complete local (triadic) structure in 16 numbers.
- Directly operationalizes classical structural theories — balance, clustering, ranked clusters, transitivity.
- Comes with derived expectations and covariances for principled testing against random-graph nulls.
- Serves as a standard goodness-of-fit check for exponential random graph models.
Intuition
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How it works
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When to use it
Use the triad census to characterize and test the local structure of a directed network — to assess transitivity, reciprocity, balance, clustering, or hierarchy against a random baseline, and as a goodness-of-fit diagnostic for ERGMs. It is the canonical descriptive tool for triadic structure and connects directly to classical structural theories. It applies to directed networks (an undirected version with four triad types exists), requires reasonably complete data, and always demands a null model for interpretation — raw counts are dominated by the network's size and density. It does not by itself fit a generative model; it summarizes structure that models like the ERGM then explain.
Strengths & limitations
- Compactly fingerprints a directed network's complete local (triadic) structure in 16 numbers.
- Directly operationalizes classical structural theories — balance, clustering, ranked clusters, transitivity.
- Comes with derived expectations and covariances for principled testing against random-graph nulls.
- Serves as a standard goodness-of-fit check for exponential random graph models.
- Raw counts are dominated by network size and density, so a null model is essential for interpretation.
- Captures only local three-node structure, missing larger-scale organization.
- Computationally heavy for very large networks, though efficient matrix algorithms exist.
- Sensitive to missing ties, which alter many triads at once.
Common pitfalls
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Applications
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Frequently asked
Why are there exactly 16 triad types?
In a directed network each of the three dyads in a triple can be mutual, asymmetric, or null, and asymmetric ties have a direction. Enumerating all configurations of three directed dyads and collapsing those that are equivalent under relabeling the three actors (isomorphism) yields exactly 16 distinct triad types, conventionally written with MAN codes such as 003, 030T, and 300. Undirected networks reduce to four triad types.
Why do I need a null model to interpret the census?
The raw number of any triad type depends heavily on how many nodes and ties the network has — a dense network has many complete triads simply by volume. To learn whether a triad type is structurally over- or under-represented, you compare the observed count to its expectation under a null model (such as a random graph conditioning on the dyad census), using the derived covariances to standardize. Only the deviation from expectation is substantively meaningful.
How is the triad census used with ERGMs?
The triad census is a standard goodness-of-fit target for exponential random graph models: after fitting an ERGM, one simulates networks from it and compares their triad censuses to the observed network's. If the model reproduces the triad distribution it was not explicitly fitted to, that is evidence of good fit; large discrepancies signal that important triadic dependence (e.g., transitivity) is unmodeled.
Sources
- 1.Holland, P. W., & Leinhardt, S. (1970). A method for detecting structure in sociometric data. American Journal of Sociology, 76(3), 492–513.
- 2.Davis, J. A. (1967). Clustering and structural balance in graphs. Human Relations, 20(2), 181–187.
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Cite this page
ScholarGate. (2026, June 22). Triad Census. ScholarGate. https://scholargate.app/sociology/triad-census