Why does my random walk simulation come out so weird?

Why does my graph come out so weird? I've provided the code:

clearvars
N = 1000;
M = 500;
P = cumsum(full(sparse(1:N, randi(M,1,N), [0 2*randi([0 1],1,N-1)-1], N, M))) ; 
figure (2);
hold on ;
for S=1:size(P,2)
    plot(1:size(P,1),P(:,S),'.-');
end
title('Random walk 2')
xlabel('#Step') ;
ylabel('Distance') ;
hold off;

I was told I'm plotting along the wrong dimension of S, but what does that mean? When I alter it, the graph either crashes or gives ,e a straight line.

Answer 1

[William Rose]

@Kevin Nelson,

Your formula introduces an increaing probability of "stay the same on each step" as M, the number of walks, grows. Is that your intent? I doubt it. Here are plots of what happ[ens with your code when N=100, and M=1, M=2, and M=5. Notice the increase in the probability of making a "step" of 0, and M grows.

You can fix it by modifying (simplifying) your cumsum() statement.

[William Rose]

@Kevin Nelson,

[edit: add explanation of code with and without 0 as a step option]

The line

P = cumsum(full(sparse(1:N, randi(M,1,N), [0 2*randi([0 1],1,N-1)-1], N, M))) ;

causes (M-1)/M of the random steps (which would otherwise be +1 or -1) to become zero.

Therefore, when M=1, none of the steps get reset to zero, and all steps are therefore +1 or -1. When M=5, you get 4/5=80% zeros. When M=500, as in your original example, you get 499/500=99.8% zeros. This is why your random walks look weird.

If the above behavior is not desired, and if you want random steps of -1, 0, or +1 at every time point of every walk, then do:

P=cumsum(zeros(1,M);randi([-1 1],N-1,M));

Try it. It yields a nice looking set of random walks. See below for N=300,M=20 and N=1000,M=500.

If you want only -1 or +1 steps, then do

P=cumsum([zeros(1,M);2*randi([-1 1],N-1,M)-1]);  % steps: -1, +1

This set of walks spreads out a bit faster than the earlier set, since 0 is not an option.