  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.
