Changes of graph structure of transition probability matrices indicate the slowest kinetic relaxations

TL;DR

This paper investigates how the structure of graphs derived from transition probability matrices changes over time, revealing a kinetic threshold that indicates the slowest relaxation modes in complex systems.

Contribution

It introduces a graph-based approach to identify slowest kinetic relaxations and provides a method to accurately estimate eigenvalues from transition graph merging patterns.

Findings

Existence of a kinetic threshold τg for graph structure changes

Recombination patterns reveal slow relaxation eigenmodes

Evaluation formula improves eigenvalue accuracy from merging data

Abstract

Graphs of the most probable transitions for a transition probability matrix, $e^{\tau K}$, i.e., the time evolution matrix of the transition rate matrix $K$ over a finite time interval $\tau$, are considered. We study how the graph structures of the most probable transitions change as functions of $\tau$, thereby elucidating that a kinetic threshold $\tau_g$ for the graph structures exists. Namely, for $\tau<\tau_g$, the number of connected graph components are constant. In contrast, for $\tau\geq \tau_g$, recombinations of most probable transitions over the connected graph components occur multiple times, which introduce drastic changes into the graph structures. Using an illustrative multi-funnel model, we show that the recombination patterns indicate the existence of the eigenvalues and eigenvectors of slowest relaxation modes quite precisely. We also devise an evaluation formula that enables us to correct the values of eigenvalues with high accuracy from the data of merging processes. We show that the graph-based method is valid for a wide range of kinetic systems with degenerate, as well as non-degenerate, relaxation rates.