Foundations of Data Science

Size of
nonfinite
component
grown
static
1/8 1/4 δ
Figure 4.15: Comparison of the static random graph model and the growth model. The
curve for the growth model is obtained by integrating g ′ .
Recall the Molloy Reed analysis of random graphs with given degree distributions which
∑
asserts that there is a phase transition at ∞ i(i − 2)p i = 0. Using this, it is easy to see
that a phase transition occurs for δ = 1/4. For δ = 1/4,
i=0
and
p k =
(2δ)k =
(1+2δ) k+1
( 1
2
) k
(
1+ 1 ) k+1 =
2
( 1
2
) k
(
3 3 k
=
2 2) 2 3
( 1
3) k
∞∑
i(i − 2) 2 3
i=0
( 1
) i
3 =
2
3
∞∑
i=0
i ( ) 2 1 i
3 −
4
3
∞∑
i=0
Recall that 1 + a + a 2 + · · · = 1
1−a , a + 2a2 + 3a 3 · · · =
i ( )
1 i
3 =
2
× 3 − 4 × 3 = 0.
3 2 3 4
a , and a + 4a 2 + 9a 3 · · · = a(1+a) .
(1−a) 2 (1−a) 3
See references at end of the chapter for calculating the fractional size S static of the
giant component in the static graph. The result is
{ 0 δ ≤
1
4
S static =
1
1 −
δ+ √ δ > 1 δ 2 +2δ
4
4.9.2 Growth Model With Preferential Attachment
Consider a growth model with preferential attachment. At each time unit, a vertex is
added to the graph. Then with probability δ, an edge is attached to the new vertex and
to a vertex selected at random with probability proportional to its degree. This model
generates a tree with a power law distribution.
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