Information Network Topology From Sigma

The predicted exponent

Many real networks, from the web to metabolic systems, are scale-free: a few nodes have enormous numbers of connections while most have very few. The distribution of connections follows a power law, P(k) proportional to k raised to the power minus gamma, where gamma is the degree exponent. Measured values of gamma in real systems typically fall between 2.1 and 2.3, a range that has resisted a single simple explanation.

Recognition Science offers a derivation of this exponent from its core cost function. The framework models each attachment step in network growth as a recognition cost decision, where a new node links to an existing one based on a forced cost. When the network conserves a quantity called sigma, the framework proves that the degree exponent must equal 2 plus the reciprocal of the golden ratio, phi. Since 1/phi equals phi minus 1, approximately 0.618, the predicted exponent is gamma equals 1 plus phi, approximately 2.618.

This value sits outside the measured range of 2.1 to 2.3, so the prediction does not match current observations. The framework's library, a machine-checked collection of formal theorems, proves the exponent is greater than 2, which is the defining condition for a scale-free network, and that it lies in the narrow band between 2.61 and 2.63. The derivation is exact within the framework's assumptions, but the gap between 2.618 and the measured values remains an open question for the model.

The key consequence is that the framework does not fit the exponent to data. It derives a specific number from the golden ratio, which itself emerges from the framework's cost function. This makes a sharp, testable prediction: if sigma-conserving preferential attachment is the right model, the exponent should be 2.618, not a value in the 2.1 to 2.3 range. The mismatch is a concrete target for future work, either refining the model or explaining why real networks deviate.

/-- The Zipf-Pareto exponent identification: γ = 1 + φ = 2 + (φ - 1) = 2 + 1/φ. -/
theorem degreeExponent_eq_two_plus_inv :
    degreeExponent = 2 + phi⁻¹ := by
  unfold degreeExponent
  have h : phi⁻¹ = phi - 1 := by
    have := phi_sq_eq
    field_simp [phi_ne_zero]
    linarith
  linarith

/-- The predicted degree exponent γ = 1 + φ. -/
noncomputable def degreeExponent : ℝ := 1 + phi

/-- γ > 2 (scale-free condition). -/
theorem degreeExponent_gt_two : degreeExponent > 2 := by
  unfold degreeExponent
  linarith [one_lt_phi]

/-- γ = 1 + φ ≈ 2.618. -/
theorem degreeExponent_val_band :
    (2.61 : ℝ) < degreeExponent ∧ degreeExponent < 2.63 := by
  unfold degreeExponent
  exact ⟨by linarith [phi_gt_onePointSixOne],
         by linarith [phi_lt_onePointSixTwo]⟩

What this page does not claim

This answer does not claim the predicted exponent matches measured values. This answer does not claim the sigma-conserving model is the only possible explanation for scale-free networks. This answer does not claim the framework has derived the golden ratio from network data.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Information/NetworkTopologyFromSigma.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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