Information Network Topology From Sigma Degree Exponent Eq Two Plus Inv

The degree exponent identity

In the study of complex networks, the degree exponent γ describes how the number of connections, or degree, is distributed among nodes. A scale-free network follows a power law, P(k) ∝ k^(-γ), meaning the probability P(k) that a node has k connections decays as a power of k. The exponent γ is a key structural parameter. For many real-world networks like the internet or social graphs, measured values of γ typically fall between 2.1 and 2.3.

The framework of Recognition Science (RS) models network growth as a series of recognition cost decisions. Within this framework, a theorem proves that for a network growing by preferential attachment while conserving a quantity called σ, the degree exponent is forced to be γ = 1 + φ, where φ is the golden ratio. Since 1/φ = φ - 1, this is equivalently written as γ = 2 + 1/φ. The numerical value is approximately 2.618. This is a precise, formal result: the machine-checked library of formal theorems contains a proof, with no gaps and no extra axioms, that the defined exponent equals 2 + 1/φ.

This identity is notable because the predicted value of 2.618 is higher than the commonly observed range of 2.1 to 2.3 for many real networks. The framework's theorem establishes a logical consequence of its model, not a measurement of the internet. The value is a prediction of the RS model, not an empirical fact. The theorem itself is a statement about the definition of the exponent within the framework, not a claim about any specific network's measured properties.

The key consequence is a sharp, testable prediction: if the RS model of network growth is correct, the degree exponent should be near 2.618, not 2.2. This provides a clear falsifier for the model. A network that clearly exhibits a different exponent, such as 2.2, would contradict this specific prediction. The formal theorem is a piece of mathematics; the physical claim is a hypothesis that can be tested against real-world data.

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

/-- 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

/-- 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 theorem is not a measurement of any real network's degree exponent. The theorem does not claim that all scale-free networks have an exponent of 2.618. The derivation of the exponent from the cost function is not formalized in the machine-checked library.

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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