Multifractal nonlinearity as a robust estimator of multiplicative cascade dynamics

Natural and behavioral sciences increasingly model measurement time series as random multiplicative cascades-nonlinear processes that divide, branch, or aggregate structures across generations, creating multiplicative interactions that break ergodicity. Multifractal formalisms provide a framework for studying cascades, where multifractal spectrum width, Δα, fluctuates with the number of estimable power-law relationships. However, multifractality without surrogate comparison can be ambiguous, as the original measurement series' multifractal spectrum width, Δα_{original}, is sensitive to series length, ergodicity-breaking linear correlations (e.g., fractional Gaussian noise, fGn), or additive dynamics. To address these challenges, we constructed random cascades varying in length, noise type (additive white Gaussian noise, awGn, or fGn), mixtures of awGn and fGn across generations (progressively more awGn, progressively more fGn, or random mixing of awGn and fGn across generations), and noise operations (addition versus multiplication). We found that "multifractal nonlinearity," t_{MF} (a t-statistic comparing Δα_{original} to surrogate spectra width, Δα_{surrogates}), reliably identifies random multiplicative cascades regardless of series length or noise type-t_{MF} is more sensitive to interactivity and number of generations than series length. Unlike random additive cascades, random multiplicative cascades exhibit stronger ergodicity breaking due to their heavy histogram tails and show progressively greater ergodicity breaking with more correlated noise (e.g., fGn as opposed to awGn) and additional generations-independent of series length. Consequently, t_{MF} robustly identifies multiplicative cascade processes, quantifies the depth of cross-scale interactivity, and pinpoints sources of ergodicity breaking while remaining independent of ergodicity-breaking and series length. These properties highlight the potential of t_{MF} as a powerful metric for advancing our understanding of the mechanisms and origins of cascading processes in natural and behavioral sciences. The simulations further suggest its utility in extending findings from group-level analyses to the level of individuals.