The Principle of Dynamic Majorization Equilibrium: A Unified Framework for the Emergence of Scale-Free Structures

Power-law distributions appear across a wide range of physical, biological and socioeconomic systems. Mechanisms such as preferential attachment (PA), self-organized criticality (SOC), and stochastic multiplicative growth can each produce power-law behaviour, yet a unified perspective remains elusive. This paper develops a common language for these mechanisms based on Majorization Theory, specifically the Lorenz order. We classify stochastic dynamics into Class A (concentrating) processes, which increase the Lorenz order (inequality), and Class B (stabilizing) mechanisms, which counteract concentration via equalization. We rigorously prove that multiplicative dynamics are inherently Class A. We analyze additive dynamics, showing they are ambiguous with respect to the Lorenz order when taken in isolation. Crucially, we rigorously prove that in mixed multiplicative--additive processes, additive components act as stabilizing mechanisms by inducing equalization (reduction of relative gaps), in direct analogy with the Pigou--Dalton principle of transfers. The Principle of Dynamic Majorization Equilibrium (DME) states that complex structures emerge from the equilibrium between these opposing forces. We formulate the Principle of Critical DME, stating that when a scale-invariant Class A process asymptotically dominates a stabilizing mechanism, a power law emerges as the stationary tail. We demonstrate this principle through canonical examples: the Kesten process (using the Kesten--Goldie theorem together with Breiman's theorem) and stabilized Geometric Brownian Motion (via the Fokker--Planck formalism). We further interpret PA within this framework by analyzing the structure of its Master Equation. This provides a unified conceptual framework for understanding the emergence of scale-free structures as a universal balance between concentration and diffusion.