A scale free network is heterogeneous as the degrees of the nodes have a

# A scale free network is heterogeneous as the degrees

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A scale-free network is heterogeneous, as the degrees of the nodes, have a negative expo- nential distribution: p ( k ) = 1 ζ ( γ, k min ) k - γ . (21) In this expression k min is the smallest degree of any vertex and for the applications we discuss in this paper k min = 1; ζ is the Hurvitz zeta function. p ( k ) is the probability that a vertex has degree k , in other words it is connected to other k vertices. Thus, the number of nodes with a high degree of connectivity is relatively small as we can see from Table 6. Table 6: A power-law distribution with degree γ = 2 . 5; the probability, p ( k ), and N k , the number of vertices with degree k , for a network with a the total number of vertices N = 10 8 . k p ( k ) N k k p ( k ) N k 1 0.745 74 . 5 × 10 6 6 0.009 0 . 9 × 10 6 2 0.131 13 . 1 × 10 6 7 0.006 0 . 6 × 10 6 3 0.049 4 . 9 × 10 6 8 0.004 0 . 4 × 10 6 4 0.023 2 . 3 × 10 6 9 0.003 0 . 3 × 10 6 5 0.013 1 . 3 × 10 6 10 0.002 0 . 2 × 10 6 An organization where these nodes are selected as controllers has several advantages: There is a natural way to choose controllers, c i is a controller if deg( c i ) > κ , with κ a relatively large number, e.g. κ = 10. There is a natural way to cluster the nodes around the controllers; a node q joins the cluster built around the controller c i at the minimum distance d ( q, c i ). Thus, the average distance between the controller and the servers in the cluster is minimal. A number of studies have shown that scale-free networks have remarkable properties such as: robustness against random failures [12], favorable scaling [2, 3], resilience to congestion [32], tolerance to attacks [80], small diameter [18], and small average path length [11]. Problem 7. Use the start-time fair queuing (SFQ) scheduling algorithm to compute the virtual startup and the virtual finish time for two threads a and b with weights w a = 1 and w b = 5 when the time quantum is q = 15 and thread b blocks at time t = 24 and wakes up at time t = 60 . Plot the virtual time of the scheduler function of the real time. 51