Foundations of Data Science

or
The two triangles of Part 1 are either
disjoint or share at most one vertex
The two triangles
of Part 2 share an
edge
The two triangles in
Part 3 are the same triangle
Figure 4.4: The triangles in Part 1, Part 2, and Part 3 of the second moment argument
for the existence of triangles in G(n, d n ).
Let’s calculate E(x 2 ) where x is the number of triangles. Write x as x = ∑ ijk ∆ ijk,
where ∆ ijk is the indicator variable of the triangle with vertices i, j, and k being present.
Expanding the squared term
( ∑ ) 2 ( ∑
)
E(x 2 ) = E ∆ ijk = E ∆ ijk ∆ i ′ j ′ k ′ .
i,j,k
i, j, k
i ′ ,j ′ ,k ′
Split the above sum into three parts. In Part 1, let S 1 be the set of i, j, k and i ′ , j ′ , k ′
which share at most one vertex and hence the two triangles share no edge. In this case,
∆ ijk and ∆ i ′ j ′ k ′ are independent and
( ∑ )
E ∆ ijk ∆ i ′ j ′ k ′ = ∑ (
E(∆ ijk )E(∆ i ′ j ′ k ′) ≤ ∑ ) ( ∑ )
E(∆ ijk ) E(∆ i ′ j ′ k ′) = E 2 (x).
S 1 S 1 all
all
ijk
i ′ j ′ k ′
In Part 2, i, j, k and i ′ , j ′ , k ′ share two vertices and hence one edge. See Figure 4.4.
Four vertices and five edges are involved overall. There are at most ( n
4)
∈ O(n 4 ), 4-vertex
subsets and ( 4
2)
ways to partition the four vertices into two triangles with a common edge.
The probability of all five edges in the two triangles being present is p 5 , so this part sums
to O(n 4 p 5 ) = O(d 5 /n) and is o(1). There are so few triangles in the graph, the probability
of two triangles sharing an edge is extremely unlikely.
In Part 3, i, j, k and i ′ , j ′ , k ′ are the same sets. The contribution of this part of the
summation to E(x 2 ) is ( )
n
3 p 3 = d3 . Thus, putting all three parts together, we have:
6
which implies
E(x 2 ) ≤ E 2 (x) + d3
6 + o(1),
Var(x) = E(x 2 ) − E 2 (x) ≤ d3
6 + o(1).
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