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Unformatted text preview: individual lies in a “giant
component” of the network with the immune nodes removed.
We have studied this problem when the underlying network is an Erd¨sRenyi network. o
We will now generalize to arbitrary degree distributions. 7 Networks: Lecture 8 Diﬀusion with General Degree Distributions Recall that the degree distribution of a neighboring node is given by
dP (d )
˜
P (d ) =
.
�d �
From this, we showed that the expected number of children is given by �d 2 � − �d �
˜
E[number of children] =
.
�d �
Hence the expected number of infected children (basic reproductive number
of the disease) is
�d 2 � − �d �
λ ≡ (1 − π )
.
�d �
8 Networks: Lecture 8 Diﬀusion with General Degree Distributions
Recall the branching process analysis: If λ < 1, then with probability one, the disease dies out after a ﬁnite number of stages.
If λ > 1, then with positive probability, the disease persists by infecting a
large portion of the population.
This yields the following threshold for the probability of immune π : a giant
�d 2 � − 2�d �
component will emerge if
π<
,
�d 2 � − �d �
i.e., if the fraction of immune nodes is below this threshold.
¯
For example, for a regular network (each node with degree d ), this leads to
¯
d −2
π= ¯
.
d −1
¯
If d = 2, giant component never emerges.
¯
If d = 3, giant component emerges if < half the population is immune.
For the Erd¨sRenyi graph, we have �d 2 � = �d �2 + �d � and �d � = (n − 1)p . o
1
This yields the threshold π = 1 − (n−1)p or p (n − 1)(1 − π ) = 1, as before. 9 Networks: Lecture 8 Diﬀusion with General Degree Distributions
For a powerlaw degree distribution (or a scalefree network) with
P (d ) ∼ d −γ , γ < 3, we have that the �d 2 � is diverging in n.
Therefore, the contagion threshold for this case is π = 1, i.e., all nodes have
to be immune before the giant component of susceptible nodes di...
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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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