The file “friends.mat” in the Assignments folder contains the node adjacency matrix of the friendship graph of 100 Dartmouth students. That means that matrix element friends(i; j) = 1 if student i and student j are friends and friends(i; j) = 0 if student i and student j are not friends. Note that friends(i; i) = 0 and friends(i; j) = friends(j; i): What is the probability that two randomly selected students, i and j, are separated by no more that 5 students through the friends relationship? That is, if i and j are not friends but have a mutual friend, k, they are separated by one student, etc. If i and j are friends they are separated by 0 people, etc.
using the same friends.mat file as above, people start passing around a single token (like a chain letter of which there can only be one copy at a time). You are one of the people in the friends graph. You pass the token by uniformly randomly selecting one of your friends and passing it to that friend. So if you have 8 friends, you will pass it to any one of the 8 friends with probability 1/8. That can include passing the token back to the person who gave it to you! Everyone uses this same rule for passing the token as you do. You are person 27 in the friends graph.
(a) As the token is passed around more and more (approaching infinity), what fraction of the time do you have the token? (Assume one token pass per time unit.)
(b) The token starts with person 93. What is the expected number of token passes (that is, time) required for you to get it for the first time?
(c) The token starts with a random person among the 100, including possibly you. What is the expected number of token passes (that is, time) required for you to get it for the first time?