4
Random Graphs
Large graphs appear in many contexts such as the World Wide Web, the internet,
social networks, journal citations, and other places. What is different about the modern
study of large graphs from traditional graph theory and graph algorithms is that here
one seeks statistical properties of these very large graphs rather than an exact answer to
questions. This is akin to the switch physics made in the late 19
th
century in going from
mechanics to statistical mechanics.
Just as the physicists did, one formulates abstract
models of graphs that are not completely realistic in every situation, but admit a nice
mathematical development that can guide what happens in practical situations. Perhaps
the most basic such model is the
G
(
n, p
) model of a random graph. In this chapter, we
study properties of the
G
(
n, p
) model as well as other models.
4.1
The
G
(
n, p
)
Model
The
G
(
n, p
) model, due to Erd¨
os and R´
enyi, has two parameters,
n
and
p
. Here
n
is
the number of vertices of the graph and
p
is the edge probability. For each pair of distinct
vertices,
v
and
w
,
p
is the probability that the edge (
v
,
w
) is present. The presence of each
edge is statistically independent of all other edges.
The graphvalued random variable
with these parameters is denoted by
G
(
n, p
). When we refer to “the graph
G
(
n, p
)”, we
mean one realization of the random variable. In many cases,
p
will be a function of
n
such
as
p
=
d/n
for some constant
d
. In this case, the expected degree of a vertex of the graph is
d
n
(
n

1)
≈
d
. The interesting thing about the
G
(
n, p
) model is that even though edges are
chosen independently with no “collusion”, certain global properties of the graph emerge
from the independent choices. For small
p
, with
p
=
d/n
,
d <
1, each connected compo
nent in the graph is small. For
d >
1, there is a giant component consisting of a constant
fraction of the vertices. In addition, as
d
increases there is a rapid transition in proba
bility of a giant component at the threshold
d
= 1. Below the threshold, the probability
of a giant component is very small, and above the threshold, the probability is almost one.
The phase transition at the threshold
d
= 1 from very small
o
(
n
) size components to a
giant Ω(
n
) sized component is illustrated by the following example. Suppose the vertices
of the graph represents people and an edge means the two people it connects have met
and became friends. Assume that the probability two people meet and become friends is
p
=
d/n
and is statistically independent of all other friendships. The value of
d
can be
interpreted as the expected number of friends a person knows. The question arises as to
how large are the components in this friendship graph?
If the expected number of friends each person has is more than one, then a giant
component will be present consisting of a constant fraction of all the people.
On the
other hand, if in expectation, each person has less than one friend, the largest component
is a vanishingly small fraction of the whole. Furthermore, the transition from the vanishing