Computer Science Department
Networks PhD Depth Exam
1. Check that there are 5 questions over 6 pages including this page. Answer all questions. Turn in your
answers by Friday, November 19, 4:00pm, at MCS-140G.
2. You are free to consult any notes, books and papers during the examination—make sure to include
your references. Copying existing solutions or parts of them will be considered plagiarism, and in this
case you will fail the exam.
You must develop your own solutions, which might use existing ideas and techniques, as long as you
cite them and clearly explain how these existing ideas/techniques ﬁt within your own solution. Failure
to do so and a feel of cut-and-paste in your answers will receive zero credit and you may fail the exam.
You are NOT allowed to consult with any person during the examination.
3. You may typeset your answers (in which you should feel free to include hand-drawn ﬁgures), or you
may neatly write your answers. Please answer each question in a separate writeup.
4. Although questions are of diﬀerent length, they are of equal weight.
5. Please indicate the question number and your code number on each answer sheet. Do not write your
name on the answer sheet.
6. If you have any doubt as to the interpretation of a question, make a reasonable assumption and explain
your interpretation in your answer. No explanations will be given during the exam.
7. Your answers should show research maturity and depth, so you may support your answers not only by
analysis, but possibly by measurements or simple simulations. Be creative!
8. The copyright to any new research question is owned by its author :)
Networks PhD Depth Exam
You are given a set of paths traversing a network. The paths are organized into a binary matrix Ain
which Aij = 1 if link iis traversed by path j. The number of paths is (as usual) considerably larger than
the number of links.
1. What information about the network can you obtain from the rank of A?
2. What information about the network can you obtain from the eigenspectrum of A(ie., the eigenvalues
3. What information about the network can you obtain from the eigenvectors of ATA?
4. What information about the network can you obtain from the eigenvectors of AAT?
5. Assume you would like to extract the most commonly-used path segments in the set of traceroutes.
What linear-algebraic operations could you perform on Ato assist you?
6. Assume you are given some measurements on the set of links ~y and are told that the corresponding
measurements on the set of paths follow ~y =A~x. Give an example of a commonly used metric for
which this is true.
7. Is estimating ~x hard in this situation? Explain why or why not. Describe a general strategy for
8. Now consider the matrix G=AT. Assume you are given some measurements on the set of paths ~y
and are told that the corresponding measurements on the set of links follow ~y =G~x. Give an example
of a commonly used metric for which this is true.
9. Is estimating ~x hard in this situation? Explain why or why not. Describe a general strategy for
10. Now assume that the equation ~y =G~x holds, but that matrix multiplication should be interpreted in
terms of (min, ×) algebra. Give an example metric for which this is true.
11. Now consider the problem of estimating or discovering A, given ~x and ~y. Which of the three metrics
(from answer 6, 8, or 10) would you use to attack this problem? Describe how you would attack this
problem using that metric.
12. Now consider this problem: you can only measure some of the paths in ~y and you want to estimate the
values for the unmeasured paths. Assume we are back to standard linear algebra (ie., (+,×) algebra);
how would you use the fact that ~y =G~x to help you here?
13. Now consider this problem: Go back to the case of ~y =A~x. Assume you can only measure some of the
links in ~y and you want to estimate the values for the for the unmeasured links. How would you use
the fact that ~y =A~x to help you? Consider also using the structure of ATA.