avgPathLength
plain-language theorem explainer
The definition supplies the average path length L(N) in a recognition-derived network as the logarithm of N to base phi. Network theorists examining small-world scaling from phi-recurrence would cite this relation when certifying logarithmic growth. It is introduced as a direct transcription of the predicted L(N) = log_phi N formula.
Claim. The average path length in a network of size $N$ is $L(N) = (log N) / (log phi)$.
background
The NetworkScience.SmallWorldFromSigma module derives network properties from the phi-recurrence on the recognition graph, establishing gamma = 3 as the unique positive solution to the sigma-conservation fixed-point equation (gamma - 1) * (gamma - 2) = 2. It introduces the average path length scaling as log_phi N to capture the small-world property, alongside a clustering ratio of 1/phi. The local setting follows the module's status as a theorem-level derivation of power-law degree distributions and path-length scaling from the self-similar fixed point.
proof idea
The declaration is a direct definition expressing the average path length as the ratio of the natural logarithm of N to the natural logarithm of phi.
why it matters
This definition supports the SmallWorldFromSigmaCert structure, which certifies the small-world property through positive path length for N > 1, logarithmic growth, and clustering ratio 1/phi. It fills the path-length scaling claim in the module's derivation of network properties from phi-recurrence, aligning with the self-similar fixed point and eight-tick octave in the Recognition Science framework. It touches the open question of matching real network power-law exponents within the falsifier band [2.5, 3.5].
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