A complete directed triad census
Thirteen structures.
One external test.
Classify every connected three-node digraph, count induced occurrences in a published network, then ask which structures survive a degree-preserving null comparison.
Primary result030T
Feed-forward loop ranks first
Published comparison / Milo et al. 2002
Feed-forward loop
A regulator reaches the same target directly and through an intermediate. In the frozen ColiNet 1.0 network it appears far more often than in the randomized ensemble.
Observed40
vsNull mean7.531
Z-score+10.588
Onto2D reading
The ranking hides a structural transition.
The same frozen result can be read as identity, support and admissibility evidence. These views do not add a biological history that the dataset does not contain.
Closure contrast
Three open classes converge on 030T.
Each two-edge precursor is below its degree-preserving null mean, while the closed feed-forward loop is above it. The nearly matched differences are a structural signature in the induced census, not evidence that one observed triad literally changed into another.
- Census share
- Null enrichment
- 5.311x
- Above null
Finite support
Zero is not one state.
Onto2D keeps empirical absence separate from structural exclusion by the declared null-model invariants.
- Observed
- 4 of 13 classes
- Allowed, absent
- 1 030C
- Null-fixed
- 8 mutual-edge classes
- Identity
- 40 -> 1 FFL occurrences to canonical ID
Why eight are fixed: the measured network has zero mutual directed edges, and the null ensemble preserves every node's mutual degree. Their undefined Z-scores are therefore correct, not missing data.
Admissibility probe
A declared path, not a recovered past.
Start with three declared nodes, add one final 030T edge per step, and apply the filter once the partial triad becomes weakly connected.
Interpretation boundaryEmpirical counts select admissible structural intermediates inside a disclosed edge-addition model. The result is not an evolutionary reconstruction or a historical-load measurement of the organism.
Complete catalogue
Inspect the 13 classes
Reading ruleArrows encode directed edges. Observed is the real induced count; mu and Z come from the null ensemble. A dash is meaningful: mutual-edge motifs cannot occur when mutual degree is fixed at zero.
Reproduction contract
What was frozen before the run?
The case changes no rule after seeing the ranking. Raw source files are fetched and hash-verified rather than redistributed without a stated license.
- 01Graph convention
Simple directed graph; 424 dictionary nodes; `target, source` columns reversed on import; unsigned edges; no self-loops.
- 02Occurrence rule
Complete induced subgraphs on unordered triples of distinct nodes; weak connectivity; each triple counted once.
- 03Null model
Per-node in-, out- and mutual-degree-preserving target swaps; 1,000 deterministic trials; mFinder-compatible sample deviation.
- 04Ranking test
Descending finite Z-score. Primary agreement required 030T first and alone at empirical upper-tail P < 0.01.