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We have seen that internet has a power law

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Unformatted text preview: sappears. Under such degree distributions, there are enough very high degree nodes that many nodes are connected to and the network has a giant component even when many nodes are eliminated uniformly at random. Immunized nodes can be viewed as nodes that are removed from the system. We have seen that Internet has a power-law distribution with exponent ∼ 2.1 − 2.7. The preceding shows that Internet is robust: remove 98% of the nodes, you still have connectivity. However, a targeted removal of highest-degree nodes implies a much lower threshold: If γ = 2.5, then π = 0.056!! (removing 5% of the nodes disconnects the network). Leads to the catchy phrase “Internet is robust, yet fragile.” 10 Networks: Lecture 8 Size of the Infected Population Compute size of giant component, gives the size of infected population. Consider a node and the event that this node is in the giant component, or equivalently the event that the branching process does not die out. Let q denote the probability that the branching process does not die out ˜ starting from a neighboring node: ∞ 1 − q = π + (1 − π ) ˜ ˜ ˜ ∑ P (d )(1 − q )d −1 . d =1 Let q denote the probability that the branching process does not die out: ∞ 1−q = ˜ ∑ P (d )(1 − q )d . d =0 The size of the giant component is given by qn. 11 Networks: Lecture 8 SIR Model–1 In the SIR model, a node can be in one of 3 states: Susceptible: Before the node has caught the disease, it is susceptible to infection from the neighbors. Infected: Once the node has caught the disease, it is infectious and has some probability of infecting each of its susceptible neighbors. Removed: After the disease has run its course, the node either dies or becomes completely immune (no longer susceptible). A good model for diseases such as chickenpox. Assume individuals are connected through a network generated under the configuration model with degree distribution P (d ). Suppose that the infection process is suc...
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