ECE 158A - Data Networks

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Colleen Watts
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1 ECE 158A - Data Networks Homework 2 - due Tuesday Nov 5 in class Problem 1 - Clustering coefficient and diameter In this problem, we will compute the diameter and the clustering coefficient of a set of simple networks. Given a network N, recall that the diameter d(n) is defined as follows. For a pair of nodes u, v in N, we define their minimum distance d min (u, v) as the length of the shortest path connecting them. The diameter is then defined as d(n) = max d min(u, v), u,v that is, the maximum of the minimum distance over all pairs of nodes. Given a network N, recall that the clustering coefficient is defined as C(N) = # of closed triangles # of connected triples, where a triangle is a triple of nodes u, v, w such that there is an edge between each pair of them (edge (u, v), edge (u, w), edge (v, w)), and a triple of nodes u, v, w is connected if there exist at least two edges among (u, v), (u, w), (v, w). We argued in class that two desirable properties of networks are small diameter large clustering. For this reason, it is a good idea to make sure you understand how to evaluate these two properties. This first problem checks your ability to perform such a computation. 1) Compute the diameter and the clustering coefficient of the two networks in Fig.1. 2) Compute the diameter and the clustering coefficient of the two networks in Fig.2. 3) For each n 4, define the network N n as follows. There are n nodes positioned on a circle. Each node is connected to its 4 nearest neighbors (2 on its 1

2 Figure 1: Two networks. Figure 2: Other two networks. left, 2 on its right). Fig.3 depicts N 30. Compute and plot the diameter and the clustering coefficient of the network N n as a function of n. (Hint: start from small values of n, and then generalize). 2

3 Figure 3: The network N 30. Problem 2 - Clustering coefficient in small-world networks We now consider the model of small-world random network proposed by Watts and Strogatz, and we will observe how the clustering coefficient changes when the randomness increases. This model is only slightly different than the one examined in class as links are rewired rather than added at random. The parameters of interest are the number of nodes n, the number of neighbors d and the rewiring probability p. A network is constructed according to the following process. n nodes, indexed by 1, 2,..., n, are positioned on a ring. Each node is connected with an edge to its d-nearest neighbors on each of its sides, for a total of 2d edges for each node. The 2d nearest neighbors of node i {1,..., n} are {(i d)modn, (i d + 1)MODn,..., (i 1)MODn, (i+1)modn,..., (i+d)modn}, where amodb is the remainder of the integer division of a by b. The resulting network is denoted by N 0 (see Fig.4a, for n = 16 and d = 4), whose structure is regular. Each edge is rewired with probability p, independently of the others. That is, for each edge e = (i, j) in the network N 0, with probability p one of the extremes of e (say i) is rewired to a node k i, j that is not currently connected to i. This originates the network N p, two instances of which are represented in Fig.4b and Fig.4c (in the latter the rewiring parameter p is larger). Tuning the rewiring parameter p allows to gradually transform the regular network N 0 (for p = 0, see the leftmost network in Fig.5), into a random network N 1 (for p = 1, see the rightmost network in Fig.5). For small values of p the network is highly clustered but also has a large diameter, while for values of p 3

4 Figure 4: Small-world network. close to one the network has a small diameter but low clustering. The networks N p for intermediate values 0 < p < 1 are the ones of interest (see the central network in Fig.5) that have both high clustering and low diameter. Figure 5: Increasing the value of p increases randomness. 1) Using a programming language of your choice, write a computer program that takes n, d, p as inputs, and generates a random small-world network parameters n, d and p. Notice that a network with n nodes can be represented by a n n matrix A (called the adjacency matrix), whose element a i,j (i-th row, j-th column) is 1 is nodes i and j are connected and 0 otherwise. Proceed as follows: The adjacency matrix A 0 of the network N 0 can be easily generated by starting from a matrix with all zero entries, and setting to 1 the elements corresponding to the d-nearest neighbors (observe that A 0 is symmetric, and that each row and each column has exactly 2d nonzero entries). The adjacency matrix A p of the network N p can be generated as follows. Scan all entries a i,j = 1 of A 0 (corresponding to edges e = (i, j)) for all i < j, one by one (we need to scan only half of its elements because A 0 must be symmetric). For each considered a i,j with probability 1 p do nothing. With probability p select one among i and j (each with probability 1/2) 4

5 and set a i,j = a j,i = 0 (edge deleted); if i was chosen select a random k i, j such that (i, k) is not currently an edge ad set a i,k = a k,i = 1 (edge added); if j was chosen select a random k i, j such that (j, k) is not currently an edge ad set a j,k = a k,j = 1 (edge added). Your program needs to output the matrix A p. You need to turn in your code, and to plot instances of A p in the case of n = 16, d = 2 and p {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}. 2) We now want to check that increasing the randomness of a small-world network (that is, increasing p) reduces its clustering coefficient. Remember that the clustering coefficient of a network N p (with adjacency matrix A p )is defined as # of closed triangles C(N p ) = # of connected triples, where a triangle is a triple of nodes i, j, k such that a i,j = a i,k = a j,k = 1, and a triple i, j, k is connected is at least two among a i,j, a i,k, a j,k are equal to 1. Write a computer program that, given an adjacency matrix as the input, computes the clustering coefficient of the corresponding network. An easy way to do this is to consider all unordered triples of nodes i, j, k {1,..., n} (that is, if the triple i, j, k is considered then the triple i, k, j must not be considered), and count how many of them form a triangle and how many of them are connected triples. 3) With n = 16, d = 2 and p {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}, plot C(N p ) as a function of p. To obtain a smooth plot, you may want to run the steps 1) and 2) multiple times. 5

6 Problem 3 - geographic routing in small world networks We now consider another generative model of small world networks discussed in class. Start with a grid of n nodes and add to it random connections in the following way. For each pair of nodes (x, y) in the grid, add an edge between them with probability P (x, y) = 1 e 1/ x y α (1) where x y is the grid distance between x and y and α > 0 is the parameter of the model. Consider two nodes a and b at the opposite corners of the grid. 1. As n (our graph gets larger and larger) show mathematically that the random connections are added according to the power law P (a, b) 1 x y α. (2) 2. Using a programming language of your choice, implement the following local greedy algorithm to route a packet from a to b. Send the packet on the outgoing link connecting to the node closest to b as possible. Using an initial grid of size 100x100 and adding random connections as described above, count the number of hops required to reach b and average over multiple realizations of the random graph. Perform the experiment for different values of α and determine the value of α for which the average number of hops is minimum. Repeat the experiment for larger grid sizes. 3. (Extra points) Repeat the experiment at point 2, using Dijkstra s algorithm. Is the optimal value of α different in this case? Explain your findings. 4. (Extra points). What is the degree distribution of the model considered in this Problem? How does it differ from Kleinberg s model discussed in class? 6

7 Problem 4 - Heavy tail versus exponential tail In class we encountered the Pareto distribution, and the exponential distribution. In this problem we will visually understand why the former is said to be a heavy-tail distribution. A continuous random variable has the Pareto distribution with parameters α > 0 and x m > 0 if its probability density function is given by { x α α m f P (x; α, x m ) = x for x x α+1 m, 0 for x < x m. The mean is given by m P (α, x m ) = { xmα α 1 for α > 1, for α 1. A continuous random variable has the Exponential distribution with parameter λ > 0 if its probability density function is given by { λe λx for x 0, f E (x; λ) = 0 for x < 0. The mean is given by m E (λ) = 1/λ. 1) Fix a value of λ of your choice, and (using the programming language of your choice) plot f E (x; λ) in log-log scale. 2) For α {1, 2, 3, 4, 5, 6, 7, 8}, consider the distribution f P (x; α, x m ) such that m P (α, x m ) = m E (λ). That is, for each α you have to set xmα α 1 = 1/λ, and compute the corresponding value of x m (call it x m (α)). 3) For α {1, 2, 3, 4, 5, 6, 7, 8}, plot f P (x; α, x m (α)), for the value of x m (α) computed in part 2), in the same figure were you plotted f E (x; λ). You should now have an understating why the Pareto distribution is said to have an heavy tail. Explain your understanding. 7

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