IndisputableMonolith.NetworkScience.SmallWorldFromSigma
SmallWorldFromSigma predicts the power-law degree-distribution exponent gamma for small-world networks from the Recognition Science sigma parameter. Complex-network researchers cite it when deriving topology metrics directly from the J-cost functional. The module structures its content as a sequence of definitions for gamma and supporting network quantities, followed by positivity, fixed-point, and uniqueness lemmas.
claimThe module defines the exponent $gamma$ as the unique positive fixed point induced by the Recognition Science J-cost, together with average path length exhibiting logarithmic growth in network size and clustering ratio $C$ satisfying $0 < C < 1$.
background
Recognition Science derives all structure from the J-cost $J(x) = (x + x^{-1})/2 - 1$ and the Recognition Composition Law. This module imports the base time quantum $tau_0 = 1$ tick from Constants and the cost primitives from the Cost module. It introduces gamma as the power-law exponent in the degree distribution, with gamma_fixed_point and gamma_unique establishing its self-similar character and uniqueness under the cost composition.
proof idea
This is a definition module. Gamma is introduced as the fixed point of the map derived from the J-cost; gamma_pos follows by direct evaluation, gamma_fixed_point by substitution into the composition law, and gamma_unique by monotonicity. The network metrics avgPathLength and clusteringRatio are defined directly, with their growth and bound properties proved by bounding arguments.
why it matters in Recognition Science
The module supplies the RS-native prediction of the power-law exponent that follows from the T6 phi fixed point and the eight-tick octave. It feeds the SmallWorldFromSigmaCert certificate and extends the framework into complex-network topology. The derivation fills the gap between the cost functional and observable network statistics.
scope and limits
- Does not compute a numerical value for gamma without an explicit sigma input.
- Does not simulate network growth or dynamics over time.
- Does not incorporate empirical data fitting or validation against real networks.
- Does not treat directed, weighted, or multilayer networks.
depends on (2)
declarations in this module (14)
-
def
gamma -
theorem
gamma_pos -
theorem
gamma_fixed_point -
theorem
gamma_unique -
def
avgPathLength -
theorem
avgPathLength_pos -
theorem
path_length_log_growth -
def
clusteringRatio -
theorem
clusteringRatio_pos -
theorem
clusteringRatio_lt_one -
theorem
clusteringRatio_band -
structure
SmallWorldFromSigmaCert -
def
smallWorldFromSigmaCert -
theorem
networkScience_one_statement