Emergence of generic first-passage time distributions for large Markovian networks

TL;DR

This paper demonstrates that the limiting distributions of first-passage times in large Markovian networks are universally determined by the eigenvalue spectrum of the generator matrix, revealing fundamental asymmetries and robustness conditions.

Contribution

It establishes a general eigenvalue-based framework explaining the emergence of delta and exponential first-passage time distributions in large Markovian networks.

Findings

Deterministic peaks arise from many contributing eigenvalues.

Exponential distributions result from a dominant eigenvalue.

Exponential limit is robust in reversible networks with backward bias.

Abstract

First-passage times are often the most relevant aspect of a complex Markovian network, because they signify when information processing has resulted in a definite decision. Previous studies have shown that for kinetic proofreading networks in the limit of large network size the first-passage time distribution converges either to a delta or to an exponential distribution. Remarkably, these two forms correspond to the two extreme distributions of minimal and maximal entropy for a fixed mean, respectively. Here we build on the connection between first-passage times and graph theory to show that these two limits are not model-specific, but arise generically in Markovian networks from the distribution of the eigenvalues of the generator matrix. A deterministic peak emerges when infinitely many eigenvalues contribute, while the exponential limit arises from a single dominant eigenvalue. We also show that the exponential limit emerges robustly for reversible networks when a backward bias exists. In contrast, the deterministic limit is not obtained from a simple reversal of this condition, but under structurally tighter conditions, revealing a fundamental asymmetry between both regimes. Our theoretical analysis is illustrated and validated by computer simulations of one-step master equations and random networks.