Correlation Functions, Mean First Passage Times and the Kemeny Constant

TL;DR

This paper unifies and clarifies the relationships between key quantities in kinetic network models, such as MFPTs, correlation functions, and the Kemeny constant, providing new insights and practical methods for analysis and reduction.

Contribution

It offers a unified framework linking important kinetic network quantities, introduces a new interpretation of the Kemeny constant, and proposes methods for inferring parameters and reducing network complexity.

Findings

Provides a physical interpretation of the Kemeny constant.

Introduces a method to infer equilibrium distributions from MFPTs.

Proposes a protocol for dimensionality reduction consistent with existing approaches.

Abstract

Markov processes are widely used models for investigating kinetic networks. Here we collate and present a variety of results pertaining to kinetic network models, in a unified framework. The aim is to lay out explicit links between several important quantities commonly studied in the field, including mean first passage times (MFPTs), correlation functions and the Kemeny constant, and highlight some of the subtleties which are often overlooked in the literature, while providing new insights. Results include (i) a simple physical interpretation of the Kemeny constant, (ii) a recipe to infer equilibrium distributions and rate matrices from measurements of MFPTs, potentially useful in applications, including milestoning in molecular dynamics, and (iii) a protocol to reduce the dimensionality of kinetic networks, based on specific requirements that the MFPTs in the coarse-grained system should satisfy. It is proven that this protocol coincides with the one proposed by Hummer and Szabo in [1] and it leads to a variational principle for the Kemeny constant. We hope that this study will serve as a useful reference for readers interested in theoretical aspects of kinetic networks, some of which underpin useful applications, including milestoning and coarse-graining.