Please try to tackle the following problems. You need not be able to answer them correctly, but rather feel comfortable approaching such a problem and expect to complete tasks like these for the class. If you are interested in taking this class, but are concerned that you may not have a sufficient technical background, please contact the professor, Lada Adamic (ladamic@umich.edu).
1. The inhabitants of circle-world are evenly spaced on a ring. Most are poor and do not own a telephone. They are pretty lazy, so instead of walking over to deliver a message, they shout the message to their neighbor, who shouts it to their neighbor, until the message reaches the intended recipient. There are
*n = k*i*
inhabitants in all. Every
*k*^{th} inhabitant is well off and has a phone and can call any of the *i* other telephone owners. The inhabitants are called ‘one’, ‘two’, ‘three’, .., according to where they are on the ring. Each is aware where the closest telephone is, so given an intended recipient, say “ninety-six”, they know which way to route a message so that it reaches the recipient in the minimum number of hops (a shout or a phone call count as one hop). The network looks like this:

Questions:

1 a) What is the maximum number of hops a message has to make to go between any two individuals (assume *k* is even).

1 b) The inhabitants are sometimes forgetful, and if they are not directly the sender or intended recipient of a message, there is a 10% chance that they’ll forget to pass the message on. If two inhabitants are 7 hops removed, what is the probability that a message from one reaches the other?

2. You just got your hands on a very juicy and very large network data set. It is given to you in the form of an adjacency list. Person 1 connects to person 2 and person 104. Person 2 connects to 1, 33 and 48, etc.

1: 2 104

2: 1 33 48

3: 5 1027 2573

….

However, AwesomeGraph™, the software you would like to use to do the analysis, requires that the input data be of the form:

1,2

1,104

2,1

2,33

2,48

..

Can transform the data from one format to the other? How?

Answers are given here.