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Unformatted text preview: Stat206b Random Graphs Spring 2003 Lecture 3: January 28 Lecturer: David Aldous Scribe: Samantha Riesenfeld Random graphs with a prescribed degree distribution A degree distribution ( d ,d 1 ,d 2 ,... ) of a graph specifies for each possible degree the fraction of vertices of that degree, that is d i = # vertices of degree i # vertices . Let the random variable D be the degree of a vertex chosen uniformly at random. Then d i = Pr [ D = i ]. For a connected graph on n vertices, clearly D ≥ 1. The sum of the degrees over all the vertices is at least 2( n- 1), and therefore the expected value of D is E [ D ] ≥ 2( n- 1) n , i.e. in a connected graph, the mean degree is at least 2, asymptotically. Theme (imprecise): Specify a degree distribution ( d 1 ,d 2 ,... ) with mean degree at least 2. (Note that we assume d = 0.) There is a model for a random graph G n such that the graph • has n vertices • is connected • the degree D n of a vertex chosen uniformly at random from the vertex set V n satisfies Pr [ D n = i ] → d i as n → ∞ • G n is “completely random” subject to these constraints, and all such models are essentially equivalent. The model also has the following properties: • It is “locally tree-like”; in particular, the cluster coefficient C n → 0 as n → ∞ . • If d i = i- (3+ + ω (1)) ) as i → ∞ , then for the random variable L that is the distance between two vertices chosen uniformly at random, E [ L ] ∼ c log n for a constant c depending on ( d i )....
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This note was uploaded on 04/20/2008 for the course STAT 260 taught by Professor Davidaldous during the Spring '07 term at University of California, Berkeley.
- Spring '07