Branching Cascades in Hierarchical Coordination Systems: Criticality Conditions, Finite-Depth Effects, and Tests of Geometric Scaling
Year: 2026
Type: article
Abstract: Cascade size distributions following the power law P(s) ~ s^{−τ} with exponent τ ≈ 3/2 appear across organizational failures, neuronal avalanches, financial contagion, and distributed AI systems. Classical branching process theory recovers τ = 3/2 at critical branching ratio σ = 1 via the Dwass-Otter theorem — a mathematically robust result — but leaves two structural questions open: why does σ ≈ 1 arise naturally in hierarchical coordination systems, and what geometric ingredient of the hierarchy is consistent with the appearance of τ ≈ 3/2? This paper provides a structural consistency explanation for both questions, conditional on three explicitly declared assumptions: (A-B*) optimal hierarchy branching B* ≈ e, (A-dc) coordination dimension d_c = 1/ln B*, and (A3-Df) mean-field cascade cluster geometry D_f ≈ 2. Framing: This paper does not derive the classical τ = 3/2 exponent anew. The contribution is the structural explanation of why hierarchical coordination systems operating near self-tuned criticality (Paper I) naturally exhibit near-critical branching (σ ≈ 1), and the proposal of a hierarchical cascade universality hypothesis linking coordination geometry to cascade statistics through τ = 1 + d_c/D_f. Lemma 1 (Branching Correspondence): At self-tuned criticality (Γ = Γ_c, from Paper I), coordination cascades are approximated by a critical Galton-Watson process with σ ≈ 1, via the mean-field load-splitting approximation p ≈ 1/B. The critical branching condition is derived from the coordination dynamics, not externally imposed. Theorem 1 (Conditional Structural Cascade Universality): Under the three structural assumptions, hierarchical coordination systems operating near self-tuned criticality produce P(s) ~ s^{−τ} with τ = 3/2 as a structural consistency result. The exponent is determined by coordination geometry, not microscopic interaction details. Central proposal — two universality classes: This paper proposes that cascade statistics in coordination systems split into (i) the classical structureless branching class (Route 1, Galton-Watson, τ = 3/2 for any B) and (ii) a hierarchically constrained class (Routes 2–5, τ = 1 + d_c/D_f, B-dependent). The classical τ = 3/2 appears as the special case B* = e where both classes agree. The physical condition determining class membership is not derived in this paper and remains an open problem — identified as the primary direction for future work. Five complementary analytical perspectives demonstrate structural over-determination: Route 1 (classical Galton-Watson, rigorous), Routes 2 and 4 (coordination dimension geometry, conditional), Routes 3 and 5 (RG fixed point and entropy balance, heuristic). Novel falsifiable prediction: Binary-branching coordination trees (B = 2) should exhibit τ ≈ 1.72, quantitatively distinguishable from τ = 3/2 — a direct empirical test of whether cascade propagation is geometrically constrained by hierarchical structure. This paper is Series Paper II of the Deficit-Fractal Governance (DFG) Framework, connecting the self-tuned criticality of Paper I to the intervention dynamics of Paper III through the cascade universality structure of near-critical hierarchical systems.
Author: Seol, Bin
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7135378472
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