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Unformatted text preview: 6.207/14.15: Networks Lecture 4: Erd¨osRenyi Graphs and Phase Transitions Daron Acemoglu and Asu Ozdaglar MIT September 21, 2009 1 Networks: Lecture 4 Outline Phase transitions Connectivity threshold Emergence and size of a giant component An application: contagion and diffusion Reading: Jackson, Sections 4.2.24.2.5, and 4.3. 2 Networks: Lecture 4 Phase Transitions for Erd¨osRenyi Model Erd¨osRenyi model is completely specified by the link formation probability 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 ) → if p ( n ) t ( n ) → 0, and P ( property A ) → 1 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¨os and Renyi 1959. 3 Networks: Lecture 4 Threshold Function for Connectivity Theorem (Erd¨os and Renyi 1961) A threshold function for the connectivity of the Erd¨os and Renyi model is t ( n ) = log n ( n ) . To prove this, it is suﬃcient to show that when p ( n ) = λ ( n ) log n ( n ) with λ ( n ) 0, we have P ( connectivity ) (and the converse). → → However, we will show a stronger result: Let p ( n ) = λ log n ( 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. 4 Networks: Lecture 4 Proof (Continued) Let I i be a Bernoulli random variable defined as 1 if node i is isolated, I i = otherwise. We can write the probability that an individual node is isolated as q = P ( I i = 1 ) = ( 1 − p ) n − 1 ≈ e − pn = e − λ log ( n ) = n − λ , (3) n where we use lim n ∞ 1 − n a = e − a to get the approximation. → Let X = ∑ i n = 1 I i 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....
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This note was uploaded on 06/12/2010 for the course EECS 6.207J taught by Professor Acemoglu during the Fall '09 term at MIT.
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