Spring 2016 Network Science. Solution of Quiz I

Transcription

1 Spring 2016 Network Science Department of Electrical and Computer Engineering National Chiao Tung University Solution of Quiz I Problem Points Your Score Total 100 Read all of the following information before starting the exam: This exam is closed book, but you are allowed to bring a calculator and an A-sized hand-written sheet which contains the information you need during the exam Show all your work clearly and in order if you want to get full credit I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct) This exam has problems ( pages) and is worth 100 points It is your responsibility to make sure that you have all of the pages! Good luck! 1

2 Problem 1 (5%) [Basic Short Questions] (a) Which of the following is NOT the necessary condition to identify whether a counting process N(t) is a Poisson process with rate λ or not? Ans: (iii) (i) N(0) = 0 (ii) It has stationary and independent increments (iii) The interarrival times are an exponential random variable (iv) P [N(h) 1] = λh + o(h) if h is small (b) Suppose there are two independent Poisson processes with the same rate λ, and we can merge them into a new Poisson process Then which of the following description about this new Poisson process is TRUE? Ans: (iv) (i) The new Poisson process has a rate λ/2 (ii) The expectation of the interarrival time for the new Poisson process is 1 λ (iii) The kth arrival time Y k of the new Poisson process has the Erlang pdf of order k as f Yk (y) = λk y k 1 e λy (k 1)! (iv) The variance of the kth arrival time Y k for the new Poisson process is (c) What is the average clustering coefficient for the following network g? k λ Ans: From the network, we can find Cl 1 (g) = Cl 2 (g) = 0, Cl (g) = 1, Cl (g) = 1 and Cl 5 (g) = 1 So the average clustering coefficient is Clavg = (1 + 1/ + 1/)/5 = 1 (d) For the following network g = which nodes between them have the walk of length 2? Ans: there are two walks of length: from nodes 2 to 2 and from nodes to 1 (e) As you may know that the Erdös-Renyi network model has short path and low clustering drawbacks wile using it for modeling a social network Clearly explain how to use a small world ring model to get rid of these two drawbacks Ans: See the lecture slides (f) What is the threshold function for starting to observe at least one connected triple in the Erdös-Renyi network model? Ans: t(n) = n /2, 2

3 (g) What is the threshold function for observing a cycle with edges in the Erdös-Renyi network model? Ans: t(n) = 1/n Problem 2 (20%) [SIS Model] Consider the SIS model of disease diffusion Let ρ d (t) denote the fraction of nodes of degree d infected at time t in a society and θ(t) denote the probability that a given meeting is with an infected individual at time t If P (d) is the degree distribution in the society, the probability that a meeting of node i is with a degree d node is P (d)/ d Let v denote the transmission rate of infection and δ denote the recovery rate of an infected individual (a) Assume the probability that a susceptible agent with degree d becomes infected in a period [t, t + ɛ] is ɛvθ(t)d α / d where α 1, and the fraction that recover to become susceptible again is ρ d (t)δ/d α Using the continuous-time mean field approximation, write down the differential equation of ρ d (t) with respect to t (ie, dρ d(t) =?) and solve this differential equation in the steady state (ie, find ρ d ( )) What is θ in the steady state? Sol:The differential equation of ρ d (t) wrt t is given by dρ d (t) = [1 ρ d (t)] vθdα d ρ d(t)δ d α In the steady state, we have dρ d(t) = 0 and thus ρ d ( ) = vθd 2α δ d + vθd 2α and θ = d P (d) d ρ d = d P (d)vθd 2α d (δ d + vθd 2α ) (b) Let λ v Find the range of λ if the degree distribution is regular δ Sol: If the degree distribution is regular, we have θ = 2α 1 λθ d ( d + λθ d 2α ) θ = 1 d 1 λ d > 0 λ > 1 2α 1 d 2α 2 (1 θ d) (c) What is the constrain on λ that results in a positive steady state infection rate θ? Sol: We know H (0) = λ d2α δ d > 1 2 is the condition to have a positive fixed point of H(θ) Hence, Problem (25%) [Game Theory I] λ > δ d 2 d 2α (a) Consider the following simultaneous move game:

4 (i) Suppose player 1 chooses her actions B and C with equal probability For what values of G, does this mixture strictly dominate her action A? Sol: If player 2 plays X, the mixture of (0, 05, 05) dominates A If player 2 plays Y, the mixture (0, 05, 05) yields an expected payoff of 1/2 So in order to dominate A, G < 1/2 Note this is a STRICT inequality (ii) Suppose G = 0 and find ALL Nash Equilibria Sol: If G is zero, it satisfies the inequality in (b) so we can eliminate A from consideration By the underlining method, we have 2 pure strategy Nash equilibria at (C, X) and (B, Y ) (iii) Are any of the pure strategy Nash Equilibria you found in part (c) strict Nash equilibria? If yes, which ones? Sol: (C, X) is a strict pure strategy Nash equilibrium (B, Y ) is not (b) Consider the following game of Cournot Competition, which models two firms producing the same homogeneous good and seeking to maximize their profits A utility function u i for each player i given by its total revenue minus its total cost, ie, u i (s 1, s 2 ) = s i p(s i, s i ) 2 s i, i = 1, 2, where s i is the amount of good that firm i produces, p(s i, s i ) = max{0, 2 (s i + s i )/2} is the price function for the total amount of good and 2s i / is the cost for firm i For firm i, the best-response correspondence B i (s i ) is given by B i (s i ) = arg max s i 0 { s i p(s i, s i ) 2 s i } Find B i (s i ) What is the Nash equilibrium for these two firms? Sol: B i (s i ) can be solved as follows { B i (s i ) = 1s 2 i, s i 8, for i = 1, 2 0, otherwise Thus we know the NE should be on the cross point of the two lines B 1 (s 2 ) and B 2 (s 1 ) and it is ( 8 9, 8 9 ) Problem (20%) [Game Theory II] Consider the following simultaneous move game:

5 (a) Is the following statement true or false? Explain your reasoning: Strategies that form a Nash Equilibrium have the property that they are optimal no matter the strategy of the other player Sol: False, strategies that are part of a NE are optimal given the strategy of the other player Not for all strategies of the other player A dominant strategy NE does have this property (b) In the game above, for what values of X is (T, R) a pure strategy Nash Equilibrium? Sol: X 1 Note the greater than or equal to relationship If player 1 plays T, player 2 must want to play R, ie X 2 and X 1 (c) For what values of Y does player 1 have a strictly dominated strategy where the strategy is dominated by a pure strategy? Sol: Y < Note again the strict inequality condition (d) Now assume X = 0 and Y = 5 Find all pure strategy Nash Equilibria Sol: (T, L) and (M, C) 5