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10 16 networks lecture 3 introduction threshold

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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 Theorem (Erd¨s and Renyi 1961) A threshold function for the connectivity of the Erd¨s o o log(n) 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, Ii = 0 otherwise. 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 , � �n a 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 i 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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