What is the expected rerouting path in Crowds, given that each node in the middle chooses the next hop to be the final destination with probability p?
Added by Shawn B.
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In the Crowds system, users send messages through a network of nodes (or peers) to maintain anonymity. Each node can either forward the message to another node or send it directly to the final destination. Show more…
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1.23. Consider a particle that moves along the set of integers in the following manner If it is presently at i then it next moves to i + 1 with probability p and to i - 1 with probability 1 - p Starting at 0, let ̑ denote the probability that it ever reaches 1. (a) Argue that ̑ = p + (1 - p)̑". (b) Show that ̑ = { 1 if p ≥ 1/2 { p/(1 - p) if p < 1/2 (c) Find the probability that the particle ever reaches n, n > 0 (d) Suppose that p < 1/2 and also that the particle eventually reaches n, n > 0 If the particle is presently at i, i < n, and n has not yet been reached, show that the particle will next move to i + 1 with probability 1 - p and to i - 1 with probability p That is, show that P{next at i + 1 | at i and will reach n} = 1 - p (Note that the roles of p and 1 - p are interchanged when it is given that n is eventually reached )
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