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Unformatted text preview: edges > 0) → 0.
p (n ) Assume next that 2/n2 → ∞ as n → ∞. Then, E[number of edges]→ ∞.
This does not in general imply that P(number of edges > 0) → 1.
Here it follows because the number of edges can be approximated by a
Poisson distribution (just like the degree distribution), implying that
�
e − λ λk �
�
P(number of edges = 0) =
= e −λ .
�
k! �
k =0 Since the mean number of edges, given by λ, goes to inﬁnity as n → ∞, this
implies that P(number of edges > 0) → 1.
10 Networks: Lecture 3 Introduction Phase Transitions Hence, the function t (n ) = 1/n2 is a threshold function for the emergence
of the ﬁrst link, i.e.,
When p (n ) << 1/n2 , the network is likely to have no edges in the
limit, whereas when p (n ) >> 1/n2 , the network has at least one edge
with probability going to 1.
How large should p (n ) be to start observing triples in the network?
We have E[number of triples] = n3 p 2 , using a similar analysis we can
1
show t (n ) = n3/2 is a threshold function.
How large should p (n ) be to start observing a tree with k nodes (and k − 1
arcs)?
We have E[number of trees] = nk p k −1 , and the function t (n ) = nk /1 −1 is a threshold function. k
1
The threshold function for observing a cycle with k nodes is t (n ) = n
Big trees easier to get than a cycle with arbitrary size!
11 Networks: Lecture 3 Introduction Phase Transitions (Continued)
Below the threshold of 1/n, the largest component of the graph includes no
more than a factor times log(n ) of the nodes.
Above the threshold of 1/n, a giant component emerges, which is the
largest component that contains a nontrivial fraction of all nodes, i.e., at
least cn for some constant c .
The giant component grows in size until the threshold of log(n )/n, at which
point the network becomes connected. 12 Networks: Lecture 3 Introduction Phase Transitions (Continued) Figure: A ﬁrst component with more than two nodes: a random network on 50 nodes with p = 0.01. 13 Networks: Lecture 3 Introduction Phas...
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 Fall '09
 Acemoglu

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