Graph Transformation Method for Calculating Waiting Times in Markov Chains

TL;DR

This paper introduces a precise, non-iterative method for computing transition probabilities and waiting times in finite-state Markov chains, enabling efficient analysis of complex systems.

Contribution

The authors develop a graph transformation technique that allows exact calculation of waiting times and transition probabilities without stochastic simulation.

Findings

Efficient computation of mean first passage times.

Robustness demonstrated across multiple examples.

Applicable to discrete path sampling databases.

Abstract

We describe an exact approach for calculating transition probabilities and waiting times in finite-state discrete-time Markov processes. All the states and the rules for transitions between them must be known in advance. We can then calculate averages over a given ensemble of paths for both additive and multiplicative properties in a non-stochastic and non-iterative fashion. In particular, we can calculate the mean first passage time between arbitrary groups of stationary points for discrete path sampling databases, and hence extract phenomenological rate constants. We present a number of examples to demonstrate the efficiency and robustness of this approach.