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Cycles is it connected for random graph models we are

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Unformatted text preview: n 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¨s-Renyi model with link formation probability p (n ) (again o interest in p (n ) → 0 as n → ∞). 0.11 0.1 0.09 0.08 P ( con n ect ed ) ≈ 1 p(n) 0.07 0.06 0.05 p (n ) = 0.04 0.03 0.02 0.01 l og n n P ( con n ect ed ) ≈ 0 50 100 150 200 250 300 350 400 450 500 n The graph experiences a phase transition as a function of graph parameters (also true for many other properties). 5 Networks: Lecture 3 Introduction Branching Processes To analyze phase transitions, we will make use of branching processes. The Galton-Watson Branching process is defined as follows: Start with a single individual at generation 0, Z0 = 1. Let Zk denote the number of individuals in generation k . Let ξ be a nonnegative discrete random variable with distribution pk , i.e., P (ξ = k ) = pk , E[ ξ ] = µ, var (ξ ) �= 0. Each individual has a random number of children in the next generation, which are independent copies of the random variable ξ . This implies that Z1 = ξ , Z2 = Z1 ∑ ξ (i ) (sum of random number of rvs). i =1 and therefore, E[Z1 ] = µ, and E[Zn ] = µn . E[Z2 ] = E[E[Z2 | Z1 ]] = E[µZ1 ] = µ2 , 6 Networks: Lecture 3 Introduction Branching Processes (Continued) Let Z denote the total number of individuals in all generations, ∞ Z = ∑ n = 1 Zn . We consider the events Z < ∞ (extinction) and Z = ∞ (survive forever). We are interested in conditions and with what probabilities these events occur. Two cases: Subcritical (µ < 1) and supercri...
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This document was uploaded on 03/18/2014 for the course EECS 6.207J at MIT.

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