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Unformatted text preview: 6.207/14.15: Networks Lecture 3: Erd¨osRenyi graphs and Branching processes Daron Acemoglu and Asu Ozdaglar MIT September 16, 2009 1 Networks: Lecture 3 Introduction Outline Erd¨osRenyi random graph model Branching processes Phase transitions and threshold function Connectivity threshold Reading: Jackson, Sections 4.1.1 and 4.2.14.2.3. 2 Networks: Lecture 3 Introduction Erd¨ osRenyi Random Graph Model We use G ( n , p ) to denote the undirected Erd¨osRenyi graph. Every edge is formed with probability p ∈ ( 0, 1 ) independently of every other edge. Let I ij ∈ { 0, 1 } be a Bernoulli random variable indicating the presence of edge { i , j } . For the Erd¨osRenyi model, random variables I ij are independent and 1 with probability p , = I ij with probability 1 − p . E [ number of edges ] = E [ ∑ I ij ] = n ( n 2 − 1 ) p Moreover, using weak law of large numbers, we have for all α > n ( n − 1 ) 2 n ( n − 1 ) 2 ∑ I ij − P ≥ α 0, p → as n ∞ . Hence, with this random graph model, the number of edges is a → random variable, but it is tightly concentrated around its mean for large n . 3 Networks: Lecture 3 Introduction Properties of Erd¨ osRenyi model Recall statistical properties of networks: Degree distributions Clustering Average path length and diameter For Erd¨osRenyi model: Let D be a random variable that represents the degree of a node. D is a binomial random variable with E [ D ] = ( n − 1 ) p , i.e., P ( D = d ) = ( n − 1 ) p d ( 1 − p ) n − 1 − d . d Keeping the expected degree constant as n ∞ , D can be → approximated with a Poisson random variable with λ = ( n − 1 ) p , e − λ λ d P ( D = d ) = , d ! hence the name Poisson random graph model . This degree distribution falls off faster than an exponential in d , hence it is not a powerlaw distribution . Individual clustering coeﬃcient ≡ Cl i ( p ) = p . Interest in p ( n ) as n ∞ , implying Cl i ( p ) 0. → → → Diameter:? 4 Networks: Lecture 3 Introduction Other Properties of Random Graph Models Other questions of interest: Does the graph have isolated nodes? cycles? Is it connected? For random graph models, we are interested in computing the probabilities of these events , which may be intractable for a fixed n . Therefore, most of the time, we resort to an asymptotic analysis, where we compute (or bound) these probabilities as n ∞ . → Interestingly, often properties hold with either a probability approaching 1 or a probability approaching in the limit. Consider an Erd¨osRenyi model with link formation probability p ( n ) (again interest in p ( n ) → as n → ∞ )....
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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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