Foundations of Data Science

than log n vertices. The number of components containing a cycle is a constant independent
of n. Thus, the graph consists of a forest of trees plus a few components that have
a single cycle with no Ω(log n) size components.
At p equal to 1/n, a phase transition occurs in which a giant component emerges. The
transition consists of a double jump. At p = 1/n, components of n 2/3 vertices emerge,
which are almost surely trees. Then at p = d/n, d > 1, a true giant component emerges
that has a number of vertices proportional to n. This is a seminal result in random graph
theory and the main subject of this section. Giant components also arise in many real
world graphs; the reader may want to look at large real-world graphs, like portions of the
web and find the size of the largest connected component.
When one looks at the connected components of large graphs that appear in various
contexts, one observes that often there is one very large component. One example is
a graph formed from a data base of protein interactions 13 where vertices correspond to
proteins and edges correspond to pairs of proteins that interact. By an interaction, one
means two amino acid chains that bind to each other for a function. The graph has 2735
vertices and 3602 edges. At the time we looked at the data base, the associated graph
had the number of components of various sizes shown in Table ??. There are a number
of small components, but only one component of size greater than 16, and that is a giant
component of size 1851. As more proteins are added to the data base the giant component
will grow even larger and eventually swallow up all the smaller components.
Size of
component
1 2 3 4 5 6 7 8 9 10 11 12 · · · 15 16 · · · 1851
Number of
components
48 179 50 25 14 6 4 6 1 1 1 0 0 0 1 0 1
Table 2: Size of components in the graph implicit in the database of interacting proteins.
The existence of a giant component is not unique to the graph produced from the
protein data set. Take any data set that one can convert to a graph and it is likely that
the graph will have a giant component, provided that the ratio of edges to vertices is a
small number greater than one. Table 12.13 gives two other examples. This phenomenon,
of the existence of a giant component in many real world graphs, deserves study.
Returning to G(n, p), as p increases beyond d/n, all non isolated vertices are absorbed
into the giant component, and at p = 1 ln n, the graph consists only of isolated vertices
2 n
plus a giant component. At p = ln n , the graph becomes completely connected. By
n
p = 1/2, the graph is not only connected, but is sufficiently dense that it has a clique of
13 Science 1999 July 30 Vol. 285 No. 5428 pp751-753.
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