Scaling Homogeneity Approach

• Scaling homogeneity is a framework that defines how systems, functions, and algorithms transform under varying scales, distinguishing invariant behavior from scale-dependent inhomogeneity.
• It underpins multifractal analysis in cosmic structure, elucidating transitions to homogeneity and validating principles like ΛCDM through rigorous scaling diagnostics.
• The approach drives algorithmic calibration in ADCs and robust optimization in control and reinforcement learning, ensuring consistent performance across diverse scaling regimes.

The scaling homogeneity approach encompasses a diverse set of methodologies and theoretical frameworks that formalize, quantify, and apply the concepts of scaling and homogeneity in mathematical, physical, and applied domains. This approach centers on the rigorous characterization of how systems, functions, distributions, or algorithms behave under transformations of scale—spatial, temporal, or parametric—and provides refined diagnostic and analytic tools that distinguish true homogeneity from scale-dependent inhomogeneity, as well as mechanisms for exploiting scaling invariance in practical settings.

1. Foundational Definitions and Structural Principles

Scaling homogeneity is rooted in precise transformations that define how an object responds to changes in scale. Four principal notions of functional homogeneity have been formalized:

• Additive Homogeneity: $f(x+t) = a_f(x, t) + f(x)$
• Multiplicative Homogeneity: $f(tx) = m_f(x, t) \cdot f(x)$
• Exponential Homogeneity: $f(x+t) = e_f(x, t) \cdot f(x)$
• Logarithmic Homogeneity: $f(tx) = \ell_f(x, t) + f(x)$

These generalize the classical Eulerian homogeneity $f(tx) = t^p f(x)$, admitting scale-dependent exponents and functional dependencies in the homogeneity function itself. Such broad formulations enable analysis of functional symmetries, invariants under algebraic operations, and the algebraic closure properties of sums, products, and quotients within each homogeneity class. For instance, multiplicative homogeneity is preserved under multiplication and quotients, while additive homogeneity is preserved under summation but generally not under multiplication (Himmel, 24 Sep 2025).

2. Scaling Homogeneity in Cosmic Structure and Fractals

In cosmic large-scale structure, the scaling homogeneity approach operationalizes statistical homogeneity via fractal and multifractal analyses:

• Counts-in-spheres and Generalized Dimensions: For point distributions $x_i$ in 3D, one evaluates local neighbor counts $n_i(r)$ and generalized correlation integrals $C_q(r) = \frac{1}{MN}\sum_i n_i(r)^{q-1}$. The scaling of $C_q(r)$ versus $r$ yields the spectrum of generalized (Minkowski–Bouligand) dimensions,

$D_q(r) = \frac{1}{q-1}\frac{d\ln C_q(r)}{d\ln r}.$

Statistical homogeneity is diagnosed when $D_q(r) \to 3$ for all $q$ at large $r$, matching the ambient dimension (Goyal et al., 14 Apr 2024, Sarkar et al., 2016).

• Transition to Homogeneity: The scale of homogeneity $r_h$ is robustly identified as the point where measured $D_q(r)$ becomes consistent with the random (homogeneous) distribution within statistical error ($1\sigma$ of mocks or random catalogues). For the SDSS-IV eBOSS DR16 quasar catalog, transition is observed at $r_h \approx 110~h^{-1}$ Mpc, with strong clustering only on smaller scales (Goyal et al., 14 Apr 2024). Multifractal spectra provide a more nuanced probe, distinguishing overdense and underdense regions through varying $q$.
• Angular Homogeneity Indices: For photometric catalogues, analogs such as the angular homogeneity index $H_2(\theta) = d\ln G_2(\theta)/d\ln V(\theta)$ are used, with statistical homogeneity signaled as $H_2(\theta) \to 1$ (Alonso et al., 2014, Gonçalves et al., 2017).
• Cosmic Implications: These analyses provide stringent tests of the Cosmological Principle and yield a quantitative homogeneity scale, which correlates with but typically exceeds the baryon acoustic oscillation scale. They underpin the empirical validation of ΛCDM on the largest observable scales.

3. Scaling Homogeneity in Mathematical and Statistical Frameworks

Scaling homogeneity has been generalized beyond physical space to effect measures and statistical models:

• Homogeneity in Effect Measures: For binary outcome tables, homogeneity (no-interaction) constraints define three-dimensional submanifolds within the four-dimensional outcome probability simplex, corresponding to the risk-difference, risk-ratio, and odds-ratio scales. The differential geometry of these manifolds allows direct volume computation:

$$V[\text{risk difference}] < V[\text{risk ratio}] < V[\text{odds ratio}]$$

Quantitatively, the odds-ratio constraint allows the greatest homogeneity across subpopulations, providing a principled metric for effect scale transportability and the comparative restrictiveness of interaction-null constraints (Ding et al., 2015).

• Dimension Spectra in Fractal Geometry: The introduction of the Assouad dimension spectrum,

$$\dim_{\mathrm{A}}^\theta F = \limsup_{R \to 0} \sup_{x \in F} \frac{\log N(B(x,R) \cap F, R^{1/\theta})}{(1-1/\theta)\log R},$$

provides a continuum of scaling diagnostics, interpolating between classical (box-counting) and extreme-scaling (Assouad) dimensions. Dimension spectra reveal inhomogeneous scaling at various relative scales, are continuous in $\theta$, and are invariant under bi-Lipschitz maps, offering refined invariants in fractal and metric geometry (Fraser et al., 2016).

4. Algorithmic and Applied Perspectives

The scaling homogeneity paradigm has practical algorithmic implications and enables advanced techniques in applied fields:

• Strong Homogeneity in Global Optimization: For univariate Lipschitz algorithms (GS-scheme), strong homogeneity guarantees that the sequence of trial points is invariant under arbitrary positive scaling and additive shifts of the objective, including infinite or infinitesimal values (when using the Infinity Computing paradigm). This property ensures that the search trajectory is unaffected by the choice of units or numerical scale, even avoiding ill-conditioning in extreme regimes by employing specialized positional numeral systems (Sergeyev et al., 2018).
• Homogeneity-Enforced Self-Calibration: In pipelined ADCs, homogeneity-enforced calibration (HEC) exploits the expected functional relationship $f(\alpha x) = \alpha f(x)$ of an ideal ADC. By feeding both $x$ and $\alpha x$, and enforcing self-consistent homogeneity in the digital correction, the system identifies and cancels nonlinearity without requiring a known test signal. The bi-linear extension (BL-HEC) further compensates for scale mismatches, maintains spectral performance, and is implementable on-chip (Wagner et al., 2023).
• Dual-Scale Homogeneity for Control and RL: For parameter-dependent nonlinear dynamic systems, dual-scale homogeneity enables exact lifting of a fixed policy to parameter-varying regimes through explicit time and amplitude rescaling of state and control variables. Any certificate (such as closed-loop stability or integral cost) for a nominal system instantaneously extends to all admissible parameterizations after dual-scale transformation, yielding robust zero-shot policy generalization in reinforcement learning benchmarks (Haddad et al., 2023).

5. Scaling Homogeneity in Critical Phenomena and Field Theories

Homogeneity-based finite-size scaling remains foundational in statistical mechanics and critical phenomena:

• Homogeneous Scaling Ansatz: Singular parts of the free energy density admit scaling forms

$f_s(t, h, L) = L^{-d} F(t L^{y_t}, h L^{y_h}, 1),$

for reduced temperature $t$, field $h$, and system size $L$. Introduction of a size-dependent scaling perturbation $A L^{-p}$ modifies the scaling via a third RG variable $u = A L^{y_\Delta - p}$, with regimes of relevance (irrelevant/marginal/relevant) predicted via comparison of $p$ and $y_\Delta$ (Turban, 2023).

• Scaling Invariance of Density Functionals: In electronic-structure theory, functionals of electron density $F[n]$ may be homogeneous of arbitrary degree under uniform scaling $$n_{\lambda m}(\mathbf{r}) = \lambda^m n(\lambda \mathbf{r})$$, with integral representations and local PDE constraints dictated by homogeneity and invariance. These conditions guide the construction of physically consistent functionals in density functional theory (Calderín, 2014).

6. Scaling Diagnostics, Tomography, and Data-Driven Inference

Scaling homogeneity approaches are essential for diagnosing nontrivial scaling relations in empirical systems:

• Scaling Tomography via Local Exponents: In systems hypothesized to follow power-law scaling $Y \sim P^\beta$, local scaling exponents

$$\beta_{\mathrm{loc}}(P_1, P_2) = \frac{\ln(Y_2/Y_1)}{\ln(P_2/P_1)}$$

provide granular diagnostics of scaling across size ratios, revealing nonlinearity, crossover regimes, or threshold behaviors. Aggregating $\langle \beta_{\mathrm{loc}}(r) \rangle$ across all pairs in data yields a “tomography” of scaling, enabling anomaly detection and effective exponent estimation without requiring global parametric fits (Barthelemy, 2019).

7. Implications, Extensions, and Open Problems

The scaling homogeneity approach yields a robust analytic and computational toolkit for dissecting, quantifying, and leveraging the symmetries manifest in scale transformations. It underpins the modern understanding of cosmic structure, informs effect-measure transportability in epidemiology, optimizes algorithms for robustness, and guides the design of physical functionals and control policies. Open directions include the extension of these methodologies to non-Euclidean and dynamic contexts, the rigorous handling of non-uniform priors and empirical weighting in statistical spaces, the fusion with information-theoretic diagnostics, and the intersection with learning algorithms in high-dimensional parameter spaces.