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This implies that z with probability 1 and

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Unformatted text preview: tical (µ > 1) Subcritical: µ < 1 Since E[Zn ] = µn , we have �∞ � �� ∞ 1 E[Z ] = E ∑ Zn = ∑ E Zn = < ∞, 1−µ n =1 n =1 (some care is needed in the second equality). This implies that Z < ∞ with probability 1 and P(extinction ) = 1. 7 Networks: Lecture 3 Introduction Branching Processes (Continued) Supercritical: µ > 1 Recall p0 = P(ξ = 0). If p0 = 0, then P(extinction ) = 0. Assume p0 > 0. We have ρ = P(extinction ) ≥ P(Z1 = 0) = p0 > 0. We can write the following fixed-point equation for ρ: ρ= ∞ ∑ pk ρ k = E [ ρ ξ ] ≡ Φ ( ρ ) . k =0 We have Φ(0) = p0 (using convention 00 = 1) and Φ(1) = 1 Φ is a convex function (Φ (ρ) ≥ 0 for all ρ ∈ [0, 1]), and Φ (1) = µ > 1. 1 Φ ( ρ* )= ρ* 0.8 µ 0.6 0.4 0.2 p0 0 0 ρ* 0.2 0.4 ρ 0.6 0.8 1 Figure: The generating function Φ has a unique fixed point ρ∗ ∈ [0, 1). 8 Networks: Lecture 3 Introduction Phase Transitions for Erd¨s-Renyi Model o Erd¨s-Renyi model is completely specified by the link formation probability o p (n ). For a given property A (e.g. connectivity), we define a threshold function t (n ) as a function that satisfies: P(property A) → 0 P(property A) → 1 p (n ) → 0, and t (n ) if if p (n ) → ∞. t (n ) This definition makes sense for “monotone or increasing properties,” i.e., properties such that if a given network satisfies it, any supernetwork (in the sense of set inclusion) satisfies it. When such a threshold function exists, we say that a phase transition occurs at that threshold. Exhibiting such phase transitions was one of the main contributions of the seminal work of Erd¨s and Renyi 1959. o 9 Networks: Lecture 3 Introduction Phase Transition Example Define property A as A = {number of edges > 0}. We are looking for a threshold for the emergence of the first edge. Recall E[number of edges] = p (n ) n (n −1) p (n ) 2 ≈ n2 p (n ). 2 Assume 2/n2 → 0 as n → ∞. Then, E[number of edges]→ 0, which implies that P(number of...
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