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lecture3 notes - 6.207/14.15 Networks Lecture 3 Erds-Renyi...

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6.207/14.15: Networks Lecture 3: Erd¨os-Renyi graphs and Branching processes Daron Acemoglu and Asu Ozdaglar MIT September 16, 2009 1
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Networks: Lecture 3 Introduction Outline Erd¨os-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
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�� Networks: Lecture 3 Introduction Erd¨ os-Renyi Random Graph Model We use G ( n , p ) to denote the undirected Erd¨os-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 Erd¨os-Renyi model, random variables I ij are independent and 1 with probability p , = I ij 0 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 > 0 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
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Networks: Lecture 3 Introduction Properties of Erd¨ os-Renyi model Recall statistical properties of networks: Degree distributions Clustering Average path length and diameter For Erd¨os-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 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 power-law distribution . Individual clustering coefficient Cl i ( p ) = p . Interest in p ( n ) 0 as n , implying Cl i ( p ) 0. Diameter:? 4
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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 0 in the limit. Consider an Erd¨os-Renyi model with link formation probability p ( n ) (again interest in p ( n ) 0 as n ).
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