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Graphs: Expected Number of Triangles and Variance

  1. Nov 7, 2011 #1

    jgens

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    Gold Member

    1. The problem statement, all variables and given/known data

    Let [itex]G[/itex] be a random graph on [itex]n[/itex] vertices:
    1) What is the expected number of triangles in [itex]G[/itex]?
    2) What is the variance in the number of triangles?

    2. Relevant equations

    N/A

    3. The attempt at a solution

    I think I can do (1) by using indicator variables. In particular, let [itex]X[/itex] denote the number of triangles in [itex]G[/itex], suppose [itex]\theta_{ijk}[/itex] is the indicator variable that the triple of vertices [itex]i,j,k[/itex] make a triangle, and let [itex]A_{ijk}[/itex] denote the event that the triple of vertices make a triangle. Then, we know
    [tex]E(X) = \sum_{i,j,k}P(A_{ijk})[/tex]
    where the sum extends over all triples [itex]i,j,k[/itex]. Since there are [itex]\binom{n}{3}[/itex] such triples and since [itex]P(A_{ijk}) = \frac{1}{8}[/itex], this means
    [tex]E(X) = \binom{n}{3}\frac{1}{8}[/tex]
    which gives us the expected number of triangles.

    Now, I just need help getting started with the variance problem. I think I'm just supposed to utilize the fact that [itex]Var(X) = E(X^2) - E(X)^2[/itex], but I'm not sure. Any help is appreciated.
     
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