PFACornell University ECE6970/SYSEN5420: Network Systems and Games Prof. F. Parise Homework 2, due...

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PFACornell University ECE6970/SYSEN5420: Network Systems and Games Prof. F. Parise Homework 2, due 6pm Oct 1, 2021 Rules: - Students taking ECE4960 or SYSEN 5420: to get full grades you need 12 points in total. - Students taking ECE5960: to get full grades you need 15 points in total. For all students, to achieve the points you need you can solve whichever problem or part of a problem you like. The wording advanced denotes problems that are more challenging and are suggested for ECE5960 students, but all students are welcome to try them. You can even attempt to solve more problems than you need. Note however that you will never get more than the points you need (e.g., if you are in ECE4960 and solve all problems in this homework at best you will still only get 12 points, yet by attempting more problems you will have some bonus points in case you made mistakes.) ALGEBRAIC GRAPH THEORY Problem 1 (Leslie Model) [3 points] Consider the Leslie population model described in class where a population is divided in n age- classes (ordered so that i = 1 are newborns and i = n are elderly) and xi(k) denotes the number of individuals in age-class i at time k. At every time period k each of the xi(k) individuals • produces αi offsprings (αi ≥ 0 fertility rate) • progress to the next age class with survival rate βi ∈ [0, 1] leading to the dynamics x(k + 1) =  x1(k + 1) x2(k + 1) x3(k + 1) ... xn(k + 1)  =  α1 α2 . . . αn−1 αn β1 0 0 0 0 0 β2 . . . . . . 0 ... . . . . . . . . . ... 0 0 . . . βn−1 0  ︸ ︷︷ ︸ Leslie matrix  x1(k) x2(k) x3(k) ... xn(k)  = Ax(k) Assume that αi ≥ 0 and 1 ≥ βi ≥ 0 for all i. 1. Find necessary and sufficient conditions on αi, βi so that the Leslie matrix is irreducible. (2p) 1 2. For an irreducible matrix as in the previous point find a sufficient condition on the parameters αi, βi that ensures that the population will not get extinct. (1p) DISCRETE TIME AVERAGING Problem 2 (Friedkin and Johnsen opinion dynamics model) [4 points] Let A be a row-stochastic matrix whose associated graph describes an interpersonal influence network (as in the DeGroot model). Let each individual possess an openness level λi ∈ [0, 1] for i ∈ {1, . . . , n}, describing how open the individual is to changing her initial opinion about a subject; set Λ = diag(λ1, ..., λn). Consider the Friedkin-Johnsen model of opinion dynamics x(k + 1) = ΛAx(k) + (In − Λ)x(0). In this model, xi(k) represents the current opinion of individual i and each individual i exhibits an attachment (1− λi) to its initial opinion xi(0). Consider the following two assumptions: (A1) at least one individual has a strictly positive attachment to its initial opinion, that is, λi A and AA> are symmetric and have the same eigenvalues. (1p) 2. Compute the hub and authority weights for the following graph: when α = β = 1√ λ and λ = ρ(AA>) = ρ(A>A). Normalize the vectors a and b such that a>1 = 1 and h>1 = 1. (1p) 3. What can you say about the weights of node 2? Is it more of a hub or an authority? Does this correspond to your intuition? (1p) COMPUTATIONAL Problem 4 (US airplane) [5 points] The following figure illustrates mint flights offered by jet blue. (Source: JetBlue website) 3 Consider both the current and future mint service to answer the following questions (i.e. consider both blue and green links, here a link between hub i and j should be intended as two arrows: one pointing from i to j and one pointing from j to i respectively, as each route can be taken in both directions). 1. Can a passenger go from any hub to any other? What is the name for this property? (1p) Next write the unweighted adjacency matrix A of this transportation network (note that hubs have already been assigned a node number) and relying upon A and its powers answer the following: 2. Is the graph acyclic? Is it periodic? If yes, what is the period? (1p) 3. What is the number of links on the shortest path connecting Aruba and Fort Lauderdale? . (1p) 4. Is it possible to go from Boston to St.Maarten in 4 links? and in 5? (1p) 5. How many different routes with strictly less than 9 links and possibly visiting the same hub more than once start from Boston and end in NYC? (1p) Problem 5 (Centralities - Computational) [3 points] Download the adjacency matrix for the KarateClub network from canvas. 1. Compute the following centrality measures (as defined in class) and report a plot where on the x-axis you have the node labels and on the y-axis the centrality measure. Put all of the centralities in the same plot normalized such that each centrality vector sums up to 1. (2p) • Degree, Closeness, Betweenness, Eigenvector Hint: You can either implement your own code or use the function centrality from Matlab. 2. Which agent would you remove if your objective was to create two almost disconnected groups? Which agent would you pick if you wanted to diffuse some information to the rest of the agents with as few hops as possible? (1p) 4

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