Layered culture-dimension scaling conjecture in the hierarchical Axelrod model

Establish whether, in the Axelrod model with hierarchically organized initial conditions on a λ×λ grid with F binary attributes (and inactive attributes representing taboos), the border–entropy scaling derived for prevailing cultures extends to successive layers surrounding a prevailing null-taboo culture. Specifically, prove that the most probable planar fractal dimension of the set of sites belonging to culture c in the i-th layer equals 2 − 2(i+1)ΔD for i = 1, 2, 3, …, where ΔD denotes the approximate constant spacing between successive critical dimensions D_k (zeros of derivatives of the entropy S(d)) and numerically ΔD ≈ 0.25.

Background

The paper studies an Axelrod model of culture dynamics on a λ×λ lattice with F binary attributes and structured (fractal) initial conditions. Cultures c are vectors in {−1,0,1}^F, with 0 encoding inactive attributes (taboos). The authors analyze spatial and volumetric fractal dimensions of culture patterns at equilibrium and relate them to an entropic formalism S(d) that exhibits regimes of uncontrollability characterized by zeros D_k of successive derivatives.

For prevailing cultures, the border dimension D_∂(c) is argued to satisfy D_∂(c) ≲ 2 − ΔD and, at maximum entropy, ⟨D_∂(c)⟩ ≃ 2 − 2ΔD. Building on this, the authors introduce the notion of successive “layers” of sites around a prevailing culture and conjecture that the same entropic-scaling principle governs the planar dimension of each layer, yielding the formula 2 − 2(i+1)ΔD for the i-th layer. Establishing this would formalize the observed quantization-like peaks in culture dimensions.