Foundations of Data Science

for all δ greater than δ critical . If δ is less than δ critical , then all vertices are in finite components.
In this case g(1) = 1 and both the numerator and the denominator approach zero.
Appling L’Hopital’s rule
lim
x→1 g′ (x) = 1 2δ
xg ′ (x)−g(x)
x 2
g ′ (x)
or
(
(g ′ (1)) 2 = 1 2δ g ′ (1) − g(1) ) .
The quadratic (g ′ (1)) 2 − 1 2δ g′ (1) + 1 g(1) = 0 has solutions
2δ
√
1
± 1
− 4
g ′ 2δ 4δ
(1) =
2 2δ
= 1 ± √ 1 − 8δ
. (4.7)
2
4δ
The two solutions given by (4.7) become complex for δ > 1/8 and thus can be valid only
for 0 ≤ δ ≤ 1/8. For δ > 1/8, the only solution is g ′ (1) = 1 and a nonfinite component
2δ
exists. As δ is decreased, at δ = 1/8 there is a singular point where for δ < 1/8 there are
three possible solutions, one from (4.6) which implies a giant component and two from
(4.7) which imply no giant component. To determine which one of the three solutions is
valid, consider the limit as δ → 0. In the limit all components are of size one since there
are no edges. Only (4.7) with the minus sign gives the correct solution
g ′ (1) = 1 − √ 1 − 8δ
= 1 − ( 1 − 18δ − 1 2 4 64δ2 + · · ·)
= 1 + 4δ + · · · = 1.
4δ
4δ
In the absence of any nonanalytic behavior in the equation for g ′ (x) in the region
0 ≤ δ < 1/8, we conclude that (4.7) with the minus sign is the correct solution for
0 ≤ δ < 1/8 and hence the critical value of δ for the phase transition is 1/8. As we shall
see, this is different from the static case.
As the value of δ is increased, the average size of the finite components increase from
one to
1 − √ 1 − 8δ
4δ ∣ = 2
δ=1/8
when δ reaches the critical value of 1/8. At δ = 1/8, the average size of the finite components
jumps to 1 ∣
2δ δ=1/8
= 4 and then decreases as 1 as the giant component swallows
2δ
up the finite components starting with the larger components.
Comparison to static random graph
Consider a static random graph with the same degree distribution as the graph in the
growth model. Again let p k be the probability of a vertex being of degree k. From (4.5)
p k =
(2δ) k
(1 + 2δ) k+1 .
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∣
∣
x=1