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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 powerlaw 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 highestdegree 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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 Fall '09
 Acemoglu

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