Breakdown of random matrix universality in Markov models

Abstract

Biological systems need to react to stimuli over a broad spectrum of timescales. If and how this ability can emerge without external fine-tuning is a puzzle. We consider here this problem in discrete Markovian systems, where we can leverage results from random matrix theory. Indeed, generic large transition matrices are governed by universal results, which predict the absence of long timescales unless fine-tuned. We consider an ensemble of transition matrices and motivate a temperature-like variable that controls the dynamic range of matrix elements, which we show plays a crucial role in the applicability of the large matrix limit: as the dynamic range increases, a phase transition occurs whereby the random matrix theory result is avoided, and long relaxation times ensue, in the entire 'ordered' phase. We furthermore show that this phase transition is accompanied by a drop in the entropy rate and a peak in complexity, as measured by predictive information (Bialek, Nemenman, Tishby Neural Computation 13(21) 2001). Extending the Markov model to a Hidden Markov model (HMM), we show that observable sequences inherit properties of the hidden sequences, allowing HMMs to be understood in terms of more accessible Markov models. We then apply our findings to fMRI data from 820 human subjects scanned at wakeful rest. We show that the data can be quantitatively understood in terms of the random model, and that brain activity lies close to the phase transition when engaged in unconstrained, task-free cognition -supporting the brain criticality hypothesis in this context.

"Spectrally gapped" random walks on networks: a Mean First Passage Time formula

“…Choosing "too big" a variance σ 2 N has therefore two harmful effects: (i) positivity of the matrix entries of A is no longer guaranteed 3 , and (ii) all the eigenvalues of A become of the same order, with the circular bulk swallowing up the outlier and annihilating the spectral gap. A similar clash between the positivity constraint (leading to a Perron-Frobenius outlier) and the standard circular law for Gaussian matrices -leading to a phase transitionwas recently noted in [75].…”

“…Alternatively, one could consider a "soft" version of the positivity constraint for a deformed Ginibre ensemble, where one bounds the probability of having negative entries and studies in more details what constraints this poses on the spectrum in the complex plane. The investigation of a "phase transition" whereby the Perron-Frobenius outlier is swallowed by the spectral bulk as the variance of δA ij increases is particularly interesting and timely (see [75]).…”