clusteringRatio
plain-language theorem explainer
The clustering ratio is defined as the reciprocal of the golden ratio phi. Network theorists comparing recognition-science predictions to Erdős-Rényi baselines would cite this constant when quantifying excess clustering in small-world graphs. The definition is introduced by direct assignment from the imported phi constant.
Claim. The predicted clustering coefficient ratio between the recognition-science model and the Erdős-Rényi baseline equals $1/phi$, where $phi$ is the golden ratio fixed point of the recognition composition law.
background
The NetworkScience module derives power-law degree distributions from the phi-recurrence on the recognition graph. It fixes the exponent gamma at 3 as the unique positive solution of the sigma-conservation equation (gamma-1)(gamma-2)=2 and establishes logarithmic path-length growth in base phi for the small-world property. The clustering ratio supplies the numerical factor by which the RS model exceeds the Erdős-Rényi expectation for the same degree sequence.
proof idea
The declaration is a direct definition that assigns the value 1/phi using the imported constant phi. No lemmas are applied; it serves as the base for subsequent theorems that bound it between 0.617 and 0.622 and prove positivity and being less than one.
why it matters
This definition completes the one-statement summary in networkScience_one_statement, which asserts gamma=3 together with the fixed-point equation and the clustering band. It realizes the module claim that the clustering coefficient ratio is 1/phi and feeds the SmallWorldFromSigmaCert structure. Within the framework it connects the phi fixed point to observable network statistics, supplying a falsifiable prediction for networks whose degree exponents lie near 3.
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