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Unformatted text preview: h that the probability that an
infected node will infect a susceptible neighbor before the infected node is
removed can be described by the probability of transmission t .
Assume that the infection process is independent across links between
susceptible and infected nodes.
The independence assumption is clearly violated in many cases.
12 Networks: Lecture 8 SIR Model–2 To analyze the reach of infection, we can remove links (in an independent
and identical manner) with probability 1 − t , and compute the resulting
component size.
The analysis then is analogous to the analysis of diﬀusion with immune nodes (with t in place of 1 − π ). How do we determine the transmission probability t ?
Model 1: An infected node is removed within 1 time step (deterministic). A node infects each of its susceptible neighbor i independently within time Ti that is exponentially distributed with parameter β. We have t = P(Ti ≤ 1) = 1 − e − β . Model 2: An infected node is removed within time T ∼ exp(γ). A node infects each of its susceptible neighbor i independently within time Ti ∼ exp( β). β
We have t = P(Ti ≤ T ) = β+γ .
13 Networks: Lecture 8 SIS Model–1 In the SIS model, susceptible nodes can become infected, and then recover in
such a way that they become susceptible again (rather than being removed).
Models diseases such as certain variations of the common cold.
Consider a degreebased random meeting model: nodes interact randomly
according to their degree di .
Let P (d ) be the degree distribution in the society.
The probability that a meeting of node i is with a degree d node is P (d )d .
�d �
It is essential to keep track of nodes degrees since nodes with diﬀerent degrees tend to have diﬀerent infection rates. Let ρd (t ) denote the fraction of nodes of degree d infected at time t .
Let θ (t ) denote the probability that a given meeting is with an infected
individual. Then:
∑ P (d ) ρd...
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This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.
 Fall '09
 Acemoglu

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