Foundations of Data Science

Figure 4.14: In selecting a component at random, each of the two components is equally
likely to be selected. In selecting the component containing a random vertex, the larger
component is twice as likely to be selected.
Consider n 1 (t), the expected number of isolated vertices at time t. At each unit of
time, an isolated vertex is added to the graph and an expected 2δn 1(t)
many isolated
t
vertices are chosen for attachment and thereby leave the set of isolated vertices. Thus,
n 1 (t + 1) = n 1 (t) + 1 − 2δ n 1(t)
.
t
For k >1, n k (t) increases when two smaller components whose sizes sum to k are joined
by an edge and decreases when a vertex in a component of size k is chosen for attachment.
The probability that a vertex selected at random will be in a size k component is kn k(t)
.
t
Thus,
∑k−1
n k (t + 1) = n k (t) + δ
j=1
jn j (t) (k − j)n k−j (t)
t t
− 2δ kn k(t)
.
t
To be precise, one needs to consider the actual number of components of various sizes,
rather than the expected numbers. Also, if both vertices at the end of the edge are in the
same k-vertex component, then n k (t) does not go down as claimed. These small inaccuracies
can be ignored.
Consider solutions of the form n k (t) = a k t. Note that n k (t) = a k t implies the number
of vertices in a connected component of size k is ka k t. Since the total number of
vertices at time t is t, ka k is the probability that a random vertex is in a connected
component
∑
of size k. The recurrences here are valid only for k fixed as t → ∞. So
∞
k=0 ka k may be less than 1, in which case, there are nonfinite size components whose
sizes are growing with t. Solving for a k yields a 1 = 1 and a 1+2δ k = δ
k−1 ∑
j(k − j)a
1+2kδ
j a k−j .
Consider the generating function g(x) for the distribution of component sizes where
the coefficient of x k is the probability that a vertex chosen at random is in a component
of size k.
∞∑
g(x) = ka k x k .
k=1
Now, g(1) = ∑ ∞
k=0 ka k is the probability that a randomly chosen vertex is in a finite sized
component. For δ = 0, this is clearly one, since all vertices are in components of size
118
j=1