Random walk on a graph is defined as:

**Example 1 - The triangle**:

Here are the matlab commands to create the transition matrix and it's powers:

K=(1/2)*(ones(3)-diag(ones(3,1))) K = 0 0.5000 0.5000 0.5000 0 0.5000 0.5000 0.5000 0 >> K2=K*K K2 = 0.5000 0.2500 0.2500 0.2500 0.5000 0.2500 0.2500 0.2500 0.5000 >> K3=K2*K K3 = 0.2500 0.3750 0.3750 0.3750 0.2500 0.3750 0.3750 0.3750 0.2500 K7 = 0.3281 0.3359 0.3359 0.3359 0.3281 0.3359 0.3359 0.3359 0.3281 >> sum(abs(K7(1,:)-(1/3)*ones(1,3))) ans = 0.0104

**Example 2 - The Bucky Ball**:

http://21net.com/difference/bucky100.htm

**With Matlab**:

Try the slide show `buckydem` and look at the incidence/adjacency matrix
created by:

[B V]= bucky; Bu=(1/3)*B;will be the transition matrix for the random walk on the graph.

We want to run experiments to find out how many steps an ant walking along the edges of the graph at random has to take in order to `get lost'.

Then we want to consider high powers of the transition matrix.

We write a little matlab function that does that:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function P=powermat(K,n) %Computes the 2^n power of K %powermat(K,n) outputs matrix or sparse %according to K's form prod=K; for i=2:n prod=prod*prod; end P=prod; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dist(Bu,(1/60)*ones(1,60),60) ans = 0.9500 >> B4=powermat(Bu,4); >> B4(1,:) ans = (1,1) 0.0925 (1,2) 0.0183 (1,3) 0.0544 (1,4) 0.0544 (1,5) 0.0183 (1,6) 0.0101 (1,7) 0.0663 (1,8) 0.0183 (1,9) 0.0183 ................. (1,53) 0.0034 (1,54) 0.0012 (1,55) 0.0003 (1,56) 0.0003 (1,57) 0.0009 (1,58) 0.0009 (1,59) 0.0003 dist(B4,(1/60)*ones(1,60),60) ans = 0.4435 >> B7=powermat(Bu,7); >> B7(1,:) ans = (1,1) 0.0169 (1,2) 0.0169 (1,3) 0.0168 (1,4) 0.0168 (1,5) 0.0169 (1,6) 0.0169 (1,7) 0.0168 (1,8) 0.0168 (1,9) 0.0168 (1,10) 0.0168 (1,11) 0.0168 (1,12) 0.0168 (1,13) 0.0167 (1,14) 0.0167 (1,15) 0.0168 ...... (1,54) 0.0165 (1,55) 0.0165 (1,56) 0.0165 (1,57) 0.0165 (1,58) 0.0165 (1,59) 0.0165 (1,60) 0.0164 >> dist(B7,(1/60)*ones(1,60),60) ans = 0.0034