Foundations of Data Science

1 − o(1)
Probability
of a giant
component
o(1)
1 − ε 1 + ε
Expected number of friends per person
Figure 4.1: Probability of a giant component as a function of the expected number of
people each person knows directly.
chain of connections, such as A knows B, B knows C, C knows D, ..., and Y knows Z, we
say that A indirectly knows Z. Thus, all people belonging to a connected component of
the graph indirectly know each other. Suppose each pair of people, independent of other
pairs, tosses a coin that comes up heads with probability p = d/n. If it is heads, they
know each other; if it comes up tails, they don’t. The value of d can be interpreted as the
expected number of people a single person directly knows. The question arises as to how
large are sets of people who indirectly know each other ?
If the expected number of people each person knows is more than one, then a giant
component of people, all of whom indirectly know each other, will be present consisting
of a constant fraction of all the people. On the other hand, if in expectation, each person
knows less than one person, the largest set of people who know each other indirectly is a
vanishingly small fraction of the whole. Furthermore, the transition from the vanishing
fraction to a constant fraction of the whole, happens abruptly between d slightly less than
one to d slightly more than one. See Figure 4.1. Note that there is no global coordination
of who knows whom. Each pair of individuals decides independently. Indeed, many large
real-world graphs, with constant average degree, have a giant component. This is perhaps
the most important global property of the G(n, p) model.
4.1.1 Degree Distribution
One of the simplest quantities to observe in a real graph is the number of vertices of
given degree, called the vertex degree distribution. It is also very simple to study these
distributions in G (n, p) since the degree of each vertex is the sum of n − 1 independent
random variables, which results in a binomial distribution. Since we will be dealing with
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