Question 2 20 points Consider a social network that allows accounts to be

# Question 2 20 points consider a social network that

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Question 2: (20 points) Consider a social network that allows accounts to be secured with a 6-digit passcode (any sequence of exactly six digits between 0-9 is valid). Assume the network has m users includ- ing you, and that all users choose one of the valid 6-digit passcodes uniformly at random. A user’s passcode is considered safe if no other user has the same passcode. a) As a function of m , what is the probability that your own passcode is safe? b) How many users must there be for there to be a 50% or greater chance that your own passcode is not safe? c) As a function of m , what is the probability that all users have a safe passcode? d) How many users must there be for there to be a 50% or greater chance that at least one user’s passcode is not safe? 1

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Question 3: (20 points) Consider a set of n people who are members of an online social network. Suppose that each pair of people are linked as “friends” independently with probability 1/2. We can think of their relationships as a graph with n nodes (one for each person), and an undirected edge between each pair that are friends. A clique is a fully connected subset of the graph, or equivalently a subset of people for which all pairs are friends. a) A clique of size 2 is simply a pair of nodes that are linked by an edge. Find the expected number of edges as a function of the number of nodes, n . What is the expected number of friend relationships among n = 10 people? b) A clique of size 3 is a triplet of nodes within which all three pairs are linked by an edge. Find the expected number of 3-cliques as a function of the number of nodes, n . What is the expected number of 3-cliques among n = 10 people?
• Fall '08
• Smyth,P
• Variance, Probability theory, probability density function

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