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# PS1 - University of Illinois Spring 2011 ECE 567 Problem...

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University of Illinois Spring 2011 ECE 567: Problem Set 1 Countable State Markov Processes Due: Tuesday, February 1, beginning of class Reading: ECE 567 Course Notes, Chapter 1, including the problems and solutions. 1. [Random trees] Let G = ( V, E ) be an undirected, connected graph with n vertices and m edges (so | V | = n and | E | = m ). Suppose that m n, so the graph has at least one cycle. A spanning tree of G is a subset T of E with cardinality n - 1 and no cycles. Let S denote the set of all spanning trees of G. We shall consider a Markov process with state space S ; the one-step transition probabilities are described as follows. Given a state T , an edge e is selected at random from among the m - n + 1 edges in E - T, with all such edges having equal probability. The set T ∪ { e } then has a single cycle. One of the edges in the cycle (possibly edge e ) is selected at random, with all edges in the cycle having equal probability of being selected, and is removed from T ∪ { e } to produce the next state, T 0 . (a) Is the Markov process irreducible (for any choice of G satisfying the conditions given)? Justify your answer. (b) Is the Markov process aperiodic (for any choice of G satisfying the conditions given)? (c) Show that the one-step transition probability matrix P = ( p T,T 0 : T, T 0 ∈ S ) is symmetric.

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The text passage is a problem set from a university course, specifically dealing with Markov processes and spanning trees in graph theory. It contains several questions but no answers are provided within the text.
multiple_questions
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    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

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Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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# PS1 - University of Illinois Spring 2011 ECE 567 Problem...

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University of Illinois Spring 2011 ECE 567: Problem Set 1 Countable State Markov Processes Due: Tuesday, February 1, beginning of class Reading: ECE 567 Course Notes, Chapter 1, including the problems and solutions. 1. [Random trees] Let G = ( V, E ) be an undirected, connected graph with n vertices and m edges (so | V | = n and | E | = m ). Suppose that m n, so the graph has at least one cycle. A spanning tree of G is a subset T of E with cardinality n - 1 and no cycles. Let S denote the set of all spanning trees of G. We shall consider a Markov process with state space S ; the one-step transition probabilities are described as follows. Given a state T , an edge e is selected at random from among the m - n + 1 edges in E - T, with all such edges having equal probability. The set T ∪ { e } then has a single cycle. One of the edges in the cycle (possibly edge e ) is selected at random, with all edges in the cycle having equal probability of being selected, and is removed from T ∪ { e } to produce the next state, T 0 . (a) Is the Markov process irreducible (for any choice of G satisfying the conditions given)? Justify your answer. (b) Is the Markov process aperiodic (for any choice of G satisfying the conditions given)? (c) Show that the one-step transition probability matrix P = ( p T,T 0 : T, T 0 ∈ S ) is symmetric.

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### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern
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The text passage is a problem set from a university course, specifically dealing with Markov processes and spanning trees in graph theory. It contains several questions but no answers are provided within the text.

TYPE: MULTIPLE_QUESTIONS