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Unformatted text preview: e Transitions (Continued) Figure: Emergence of cycles: a random network on 50 nodes with p = 0.03. 14 Networks: Lecture 3 Introduction Phase Transitions (Continued) Figure: Emergence of a giant component: a random network on 50 nodes with
p = 0.05. 15 Networks: Lecture 3 Introduction Phase Transitions (Continued) Figure: Emergence of connectedness: a random network on 50 nodes with
p = 0.10. 16 Networks: Lecture 3 Introduction Threshold Function for Connectivity
(Erd¨s and Renyi 1961) A threshold function for the connectivity of the Erd¨s
and Renyi model is t (n ) = n .
log(n) To prove this, it is suﬃcient to show that when p (n ) = λ(n ) n
λ(n ) → 0, we have P(connectivity) → 0 (and the converse). with log(n) However, we will show a stronger result: Let p (n ) = λ n .
If λ < 1, P(connectivity) → 0, (1) If λ > 1, P(connectivity) → 1. (2) Proof:
We ﬁrst prove claim (1). To show disconnectedness, it is suﬃcient to show
that the probability that there exists at least one isolated node goes to 1.
17 Networks: Lecture 3 Introduction Proof (Continued)
Let Ii be a Bernoulli random variable deﬁned as
1 if node i is isolated,
We can write the probability that an individual node is isolated as
q = P(Ii = 1) = (1 − p )n−1 ≈ e −pn = e −λ log(n) = n−λ ,
where we use limn→∞ 1 − n
= e −a to get the approximation. (3) Let X = ∑n=1 Ii denote the total number of isolated nodes. Then, we have
E[X ] = n · n − λ . (4) For λ < 1, we have E[X ] → ∞. We want to show that this implies
P(X = 0) → 0.
In general, this is not true.
Can we use a Poisson approximation (as in the previous example)? No,
since the random variables Ii here are dependent.
We show that the variance of X is of the same order as its mean.
18 Networks: Lecture 3 Introduction Proof (Continued)
We compute the variance of X , var(X ):
= ∑ var(Ii ) + ∑
cov(Ii , Ij ) = var(X )...
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- Fall '09