Explicit formulae for stochastic equilibria

TL;DR

This paper derives explicit formulas for stochastic equilibria of finite-state matrices, providing analytical insights and practical formulas for matrices up to size 4, with a general method for larger matrices, and applies these results to graph random walks.

Contribution

It introduces a systematic approach to explicitly compute stochastic equilibria for any size matrices using adjugate matrices and minors, simplifying the eigenvector problem.

Findings

Explicit formulas for 2x2 and 3x3 matrices are derived.

A general method using adjugate matrices for n x n matrices is presented.

Application to random walks on graphs demonstrates practical utility.

Abstract

Finding the stochastic equilibria for finite-state stochastic matrices amounts to solving an eigen\-vector problem $\pi = \pi P$. Various techniques for doing so are known, some extremely computationally intensive. Herein we shall aim to extract a number of relatively simple analytic results that shed light on this problem. It is very easy to find an explicit general formula for the equilibrium vector (when it is unique) of a $2\times 2$ stochastic matrix. The corresponding explicit general formula for the equilibrium vector (when it is unique) of a $3\times 3$ stochastic matrix is a somewhat messier four-line result. (Though with a bit of work you can shoe-horn it into one line of text.) An explicit general formula for the equilibrium vector (when it is unique) of a $4\times 4$ stochastic matrix requires a paragraph of text. Ultimately, for $n\times n$ stochastic matrices a general and fully explicit construction of the equilibrium vector (when it is unique) can be developed in terms of a suitable adjugate (classical adjoint) matrix, and can subsequently be reduced to the computation of $n$ principal matrix minors. Finally, an application to random walks on graphs is presented.