Constrained interactions and sublinear multifractality in multiscale systems.

Abstract

Multifractal formalisms characterize complex interactions across scales in natural and behavioral systems. They capture an excess form of power-law scaling beyond the single power-law relationship between fluctuation and scale in monofractality. Because monofractality fits neatly within the general linear model via fractional integration, multifractality can provide evidence of nonlinear correlations across scales. That is, multifractality might reflect variability exceeding the correlations articulable by the best linear model. The classic expectation under nonlinear correlations across scales is that multifractal spectra for the original series, Δα_{orig}, will show statistically significantly greater width than multifractal spectrum widths for linear surrogates, Δα_{surr}, yielding large positive t statistics, t_{MF}. Thus, Δα_{orig}≫Δα_{surr} supports the conclusion of nonlinear correlations across scales consistent with multiplicative cascade dynamics. By contrast, the interpretation of sublinear multifractality-Δα_{orig}≪Δα_{surr}, and hence negative t_{MF}-has been less clear, including whether it reflects or rules out nonlinear correlations across scales consistent with multiplicative cascades. We identify five distinct mechanisms by which random multiplicative cascades and related linear processes can in principle produce nonlinear correlations with significantly negative t_{MF}, that is, Δα_{orig}≪Δα_{surr}, and we characterize which mechanisms do so robustly under stringent surrogate-testing protocols. We show how various nonmultiplicative constraints on the nonlinear correlations across scales of a multiplicative cascade might produce sublinear multifractality. These constraints may serve as analogues to the physical limits characteristic of homeostatic regulation and feedback control in complex systems. Altogether, this framework builds expectations for how cascade dynamics collaborate with regulatory processes in physics, biology, and engineered systems.