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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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