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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
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
0.05 p (n ) = 0.04
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
We are interested in conditions and with what probabilities these
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.
- Fall '09