Heterogeneity dominates irreversibility in random Markov models

TL;DR

This paper introduces a two-parameter ensemble of random Markov models that captures critical slowing down and broken detailed balance, revealing how heterogeneity and asymmetry influence relaxation and entropy production, with applications to human brain data.

Contribution

It extends heterogeneous Markov ensembles by incorporating correlations via an asymmetry parameter and analyzes critical behavior using random matrix theory, linking criticality to biological data.

Findings

Relaxation times diverge at a critical heterogeneity level.

Empirical brain data operate near the predicted critical locus.

Heterogeneity strongly influences observables, while asymmetry has a weaker effect.

Abstract

We introduce a two-parameter ensemble of random discrete-time Markov models that simultaneously captures critical slowing down and broken detailed balance. Extending a previously studied heterogeneous Markov ensemble, we incorporate correlations between forward and backward transition rates through a single asymmetry parameter $\gamma$, while heterogeneity is controlled by $\epsilon$. Using results from random matrix theory, we identify a critical locus $\epsilon_c(\gamma,N)$ at which relaxation times diverge and spectral universality breaks down. We characterize the behavior of entropy production, predictive information, and relaxation dynamics across the ensemble, showing that many observables depend strongly on heterogeneity but only weakly on asymmetry, except near the symmetric limit. Applying maximum-likelihood inference to human fMRI and EEG data, we find that both modalities operate near the predicted critical locus and occupy a similar region of the $\epsilon-\gamma$ plane, supporting a super-universality of human brain dynamics. While ensemble averages are well captured by the null model, empirical data exhibit substantially enhanced variability, indicating subject-specific structure beyond random expectations. Our results unify criticality and nonequilibrium measures within a single framework and clarify their intertwined role in the analysis of complex biological dynamics.