# Affinity-dependent bound on the spectrum of stochastic matrices

## TL;DR

This paper proposes a new bound on the spectrum of stochastic matrices based on affinity, extending the Perron-Frobenius theorem, with proofs for unicyclic networks and numerical evidence for complex networks.

## Contribution

It introduces a conjectured affinity-dependent spectral bound for stochastic matrices, generalizing known results and providing proofs and numerical validation.

## Key findings

- Bound constrains all eigenvalues of stochastic matrices.
- Asymmetric random walk saturates the bound.
- Numerical evidence supports the bound in complex networks.

## Abstract

Affinity has proven to be a useful tool for quantifying the non-equilibrium character of time continuous Markov processes since it serves as a measure for the breaking of time reversal symmetry. It has recently been conjectured that the number of coherent oscillations, which is given by the ratio of imaginary and real part of the first non-trivial eigenvalue of the corresponding master matrix, is constrained by the maximum cycle affinity present in the network. In this paper, we conjecture a bound on the whole spectrum of these master matrices that constrains all eigenvalues in a fashion similar to the well known Perron-Frobenius theorem that is valid for any stochastic matrix. As in other studies that are based on affinity-dependent bounds, the limiting process that saturates the bound is given by the asymmetric random walk. For unicyclic networks, we prove that it is not possible to violate the bound by small perturbation of the asymmetric random walk and provide numerical evidence for its validity in randomly generated networks. The results are extended to multicyclic networks, backed up by numerical evidence provided by networks with randomly constructed topology and transition rates.