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lecture 7 scribe_notes_feb02

lecture 7 scribe_notes_feb02 - CS 4850 Lecture Feb 2 2009...

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CS 4850 Lecture - Feb 2, 2009 Justin Prieto, jp393; Sujith Vidanapathirana, srv3 Increasing property: Q is an increasing property of a random graph G if when p 1 < p 2 , G ( n, p 2 ) almost surely has Q if G ( n, p 1 ) almost surely has Q . i.e., if we increase the probability of an edge, the property Q becomes more likely. Theorem: Every increasing property of N ( n, p ) has a threshold. (The combinatorial structure does not actually matter.) Proof: To show that p ( n ) is a threshold for property Q , we need to show that the probability of property Q goes from 0 to 1 within a range that is bounded by a multiplicative constant. We have to show that functions p ( ) and p (1 - ) are asymptotically equivalent. That is, we need to show that there exists as constant m such that p (1 - ) mp ( ). Notational note: Let p ( n, ) be the function p ( n ) such that the probability of Q is . We will also sometimes write N p for N ( n, p ). Start of Proof Let 0 < < 1 2 and let m be an integer such that (1 - ) m . We now show that p (1 - ) mp ( ).
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