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6.262.PS11

# 6.262.PS11 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262 – Discrete Stochastic Processes Problem Set # 11 Issued: May 3, 2010 Due: May 7, 2010 Extensions until May 12 readily granted. 1) The Voter Problem (Part II). Consider once again the voter model from Problem 2 in Problem Set #10, defined on a connected graph G with n vertices. Given an initial condition in which each vertex has a specified opinion (0 or 1), find the probability that the Markov process eventually settles into a state in which every vertex’s opinion is 1. Though this Markov process has 2 n states, give an expression for this probability that can be evaluated for any such graph in at most O(n 2 ) steps. This continues the line of thought we had begun to develop in Problem 2 of Pset #9 and Problem 2 of Pset #10. However, it seems essentially impossible to solve this without using the following insight, which we introduce via an example of a 4-vertex graph with initial opinions at the 4 vertices of a, b, c, and d.

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