MIT14_15JF09_lec03

MIT14_15JF09_lec03 - 6.207/14.15: Networks Lecture 3:...

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Unformatted text preview: 6.207/14.15: Networks Lecture 3: Erdos-Renyi graphs and Branching processes Daron Acemoglu and Asu Ozdaglar MIT September 16, 2009 1 Networks: Lecture 3 Introduction Outline Erdos-Renyi random graph model Branching processes Phase transitions and threshold function Connectivity threshold Reading: Jackson, Sections 4.1.1 and 4.2.1-4.2.3. 2 Networks: Lecture 3 Introduction Erd os-Renyi Random Graph Model We use G ( n , p ) to denote the undirected Erdos-Renyi 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 Erdos-Renyi 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 > n ( n 1 ) 2 n ( n 1 ) 2 I ij P 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 os-Renyi model Recall statistical properties of networks: Degree distributions Clustering Average path length and diameter For Erdos-Renyi 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 = ( n 1 ) p , e 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 power-law distribution . Individual clustering coecient 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 Erdos-Renyi model with link formation probability p ( n ) (again interest in p ( n ) as n )....
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MIT14_15JF09_lec03 - 6.207/14.15: Networks Lecture 3:...

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