Beyond Zipf's Law: Equifinality and Mechanistic Inference from Scaling Laws

2026-08-10

AI Summary

This study addresses the mechanistic ambiguity inherent in Zipf’s power-law distributions, as multiple generative processes can yield identical scaling exponents, rendering mechanism identification from the exponent alone unreliable. Framing this ambiguity as an equifinality problem, the work systematically compares four distinct models—finite Zipf processes, persistent Markov chains, sample space reducing (SSR) processes, and latent-scale mixture models—using a multidimensional set of discriminative metrics, including lagged mutual information, transition directionality, and latent-variable conditioning. The analysis incorporates effective sample size correction and boundary sensitivity assessments, revealing that model misfit primarily stems from mismatches in sequential dependency structures. The findings demonstrate that merely reproducing a power-law distribution is insufficient for mechanistic inference; instead, model identifiability necessitates purposefully differentiated observational designs.

Abstract

Zipf-like rank--frequency scaling occurs in language, city sizes, biological data and animal communication. Because several generative processes can produce the same marginal pattern, the exponent alone has limited mechanistic content. We formulate this ambiguity as an equifinality problem and compare four constructions under a common observation design. An i.i.d. finite-Zipf process, a persistent Markov chain and canonical sample-space reduction (SSR) have the same stationary marginal, $p_j=(jH_V)^{-1}$, whereas a latent-scale mixture produces a similar marginal through aggregation. The first three constructions therefore isolate differences in sequence structure without changing the population rank distribution. Markov dependence changes the finite-sample distribution of fitted exponents; at moderate persistence, the shift is closely reproduced by a block-adjusted effective sample size. Excess lag-1 mutual information separates exchangeable from sequential processes, and transition direction separates reversible persistence from the directional contraction built into SSR. Conditioning on latent scale reveals the aggregation route, while fit-window, alphabet and sequence-boundary analyses show which conclusions depend on the observation design. Recent work on learned animal communication is used to formulate prospective tests rather than to validate the models empirically. Matching a scaling law is thus a compatibility condition; discriminating among mechanisms requires observations on which the candidate models differ.

Problem

• equifinality
• scaling laws
• Zipf's law
• mechanistic inference
• generative processes

Innovation

• equifinality
• scaling laws
• mechanistic inference
• sample-space reduction
• Markov dependence