Random walks on directed graph in R - r

Question

I am new in R programming. I have a directed graph which has 6 nodes and also provided a probability matrix of 6 rows and 6 columns. If a random walker walk 100,000 steps on the graph should end up the output vector like the following:
0.1854753, 0.1301621,0.0556688, 0.1134808, 0.15344649, 0.3617481
corresponding to the probabilities of 6 nodes being visited in this random walk experiment(counts divided by the total number of steps, in this case, 100,000).
I need to create a function for this task and to demonstrate how to use it. The function takes a graph and number of steps as input.
The provided matrix as follows:
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.0 0.5 0.3 0.0 0.0 0.2
[2,] 0.1 0.2 0.0 0.4 0.1 0.2
[3,] 0.5 0.0 0.0 0.0 0.0 0.5
[4,] 0.0 0.1 0.0 0.0 0.6 0.3
[5,] 0.0 0.0 0.0 0.4 0.0 0.6
[6,] 0.4 0.0 0.0 0.0 0.2 0.4
Can someone help me to solve the problem?

Answer 1

Here is a step-by-step implementation using a Markov chain (through R library markovchain).
We start by loading the library.
library(markovchain);
We define the transition matrix and states (here simply 1...6 for the graph nodes)
mat <- matrix(c(
0.0, 0.5, 0.3, 0.0, 0.0, 0.2,
0.1, 0.2, 0.0, 0.4, 0.1, 0.2,
0.5, 0.0, 0.0, 0.0, 0.0, 0.5,
0.0, 0.1, 0.0, 0.0, 0.6, 0.3,
0.0, 0.0, 0.0, 0.4, 0.0, 0.6,
0.4, 0.0, 0.0, 0.0, 0.2, 0.4), ncol = 6, byrow = T)
states <- as.character(1:6);
We define a Markov chain object.
mc <- new(
"markovchain",
states = states,
byrow = TRUE,
transitionMatrix = mat,
name = "random_walk");
We now simulate a random walk consisting of nSteps (here 1e6) and obtain asymptotic probabilities for every state (node) with prop.table(table(...))
nSteps <- 1e6;
random_walk <- markovchainSequence(nSteps, mc, t0 = "1");
prop.table(table(random_walk));
#random_walk
# 1 2 3 4 5 6
#0.185452 0.129310 0.055692 0.113410 0.153787 0.362349
Note that asymptotic probabilities might change slightly if you re-run the code.
Wrapping this in a single function is straight-forward and I'll leave that up to you.

Answer 2

Assuming you are giving probability matrix (prob_mat) for the directed graph and no of steps (no_of_steps) as input. This should do:
set.seed(150)
find_pos_prob <- function(prob_mat, no_of_steps){
x <- c(1:nrow(prob_mat)) # index for nodes
position <- 1 # initiating from 1st Node
occured <- rep(0,nrow(prob_mat)) # initiating occured count
for (i in 1:no_of_steps) {
# update position at each step and increment occurence
position <- sample(x, 1, prob = prob_mat[position,])
occured[position] <- occured[position] + 1
}
return (occured/no_of_steps)
}
find_pos_prob(prob_mat, 100000)
#[1] 0.18506 0.13034 0.05570 0.11488 0.15510 0.35892
Data:
prob_mat <- matrix( c(0.0, 0.5, 0.3, 0.0, 0.0, 0.2,
0.1, 0.2, 0.0, 0.4, 0.1, 0.2,
0.5, 0.0, 0.0, 0.0, 0.0, 0.5,
0.0, 0.1, 0.0, 0.0, 0.6, 0.3,
0.0, 0.0, 0.0, 0.4, 0.0, 0.6,
0.4, 0.0, 0.0, 0.0, 0.2, 0.4), byrow = TRUE, ncol = 6)
Note: Simulation results will differ from analytical solutions. Ideally you should remove the seed, run the function 15-20 times and take the average of probabilities over the runs