We have seen two ways of solving the end-stage distribution of cities in the example of Amy’s random walk to her friends. Method #1 is to multiply the transition matrix with itself n times and finally multiply with the initial-state distribution. Method #2 is to solve the equation, P X* = X*.
In the matrix form
0.0 0.33 0.0 0.25 0.0
0.5 0.00 0.5 0.25 0.0
0.0 0.33 0.0 0.25 0.0
0.5 0.33 0.5 0.00 1.0
0.0 0.0 0.0 0.25 0.0There is a third method: find the first eigenvector of the transition matrix and normalise it (i.e., divide it with the sum of elements to make the total distribution 1). We will manage it using an R program that has an inbuilt function to calculate the higher vector.
pM <- matrix(c(0.0, 0.5, 0.0, 0.50, 0.0,
0.333, 0.0, 0.333, 0.333, 0.0,
0.0, 0.5, 0.0, 0.5, 0.0,
0.25, 0.25, 0.25, 0.0, 0.25,
0.0, 0.0, 0.0, 1.0, 0.0), nrow = 5)
eigenvec <- eigen(pM)$vectors
first_eigenvec <- eigenvectors[,1]
first_eigenvec/sum(first_eigenvec)0.16663714 0.25003386 0.16663714 0.33333681 0.08335504
