Foundations of Data Science

Prob(vertex has degree k) = ( )
n−1
k p k (1 − p) n−k−1 ≈ ( n
k)
p k (1 − p) n−k .
The quantity ( )
n−1
k is the number of ways of choosing k edges, out of the possible n − 1
edges, and p k (1 − p) n−k−1 is the probability that the k selected edges are present and the
remaining n−k−1 are not. Since n is large, replacing n−1 by n does not cause much error.
The binomial distribution falls off exponentially fast as one moves away from the mean.
However, the degree distributions of graphs that appear in many applications do not exhibit
such sharp drops. Rather, the degree distributions are much broader. This is often
referred to as having a “heavy tail”. The term tail refers to values of a random variable
far away from its mean, usually measured in number of standard deviations. Thus, although
the G (n, p) model is important mathematically, more complex models are needed
to represent real world graphs.
Consider an airline route graph. The graph has a wide range of degrees, from degree
one or two for a small city, to degree 100, or more, for a major hub. The degree distribution
is not binomial. Many large graphs that arise in various applications appear to have power
law degree distributions. A power law degree distribution is one in which the number of
vertices having a given degree decreases as a power of the degree, as in
Number(degree k vertices) = c n k r ,
for some small positive real r, often just slightly less than three. Later, we will consider
a random graph model giving rise to such degree distributions.
The following theorem claims that the degree distribution of the random graph G (n, p)
is tightly concentrated about its expected value. That is, the probability that the degree of
a vertex differs from its expected degree, np, by more than λ √ np, drops off exponentially
fast with λ.
Theorem 4.1 Let v be a vertex of the random graph G(n, p). Let α be a real number in
(0, √ np).
Prob(|np − deg(v)| ≥ α √ np) ≤ 3e −α2 /8 .
Proof: The degree deg(v) is the sum of n − 1 independent Bernoulli random variables,
y 1 , y 2 , . . . , y n−1 , where, y i is the indicator variable that the i th edge from v is present. So
the theorem follows from Theorem 12.6.
Although the probability that the degree of a single vertex differs significantly from
its expected value drops exponentially, the statement that the degree of every vertex is
close to its expected value requires that p is Ω( ln n ). That is, the expected degree grows
n
as ln n.
Corollary 4.2 Suppose ε is a positive constant. If p is Ω( ln n
nε 2 ), then, almost surely, every
vertex has degree in the range (1 − ε)np to (1 + ε)np.
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