Foundations of Data Science

4.8 Nonuniform and Growth Models of Random Graphs
4.8.1 Nonuniform Models
So far we have considered the random graph G(n, p) in which all vertices have the
same expected degree and showed that the degree is concentrated close to its expectation.
However, large graphs occurring in the real world tend to have power law degree
distributions. For a power law degree distribution, the number f(d) of vertices of degree
d plotted as a function of d satisfies f(d) ≤ c/d α , where α and c are constants.
To generate such graphs, we stipulate that there are f(d) vertices of degree d and
choose uniformly at random from the set of graphs with this degree distribution. Clearly,
in this model the graph edges are not independent and this makes these random graphs
harder to analyze. But the question of when phase transitions occur in random graphs
with arbitrary degree distributions is still of interest. In this section, we consider when
a random graph with a nonuniform degree distribution has a giant component. Our
treatment in this section, and subsequent ones, will be more intuitive without providing
rigorous proofs.
4.8.2 Giant Component in Random Graphs with Given Degree Distribution
Molloy and Reed address the issue of when a random graph with a nonuniform degree
distribution has a giant component. Let λ i be the fraction of vertices of degree i. There
∑
will be a giant component if and only if ∞ i(i − 2)λ i > 0.
i=0
To see intuitively that this is the correct formula, consider exploring a component
of a graph starting from a given seed vertex. Degree zero vertices do not occur except
in the case where the vertex is the seed. If a degree one vertex is encountered, then
that terminates the expansion along the edge into the vertex. Thus, we do not want to
encounter too many degree one vertices. A degree two vertex is neutral in that the vertex
is entered by one edge and left by the other. There is no net increase in the size of the
frontier. Vertices of degree i greater than two increase the frontier by i − 2 vertices. The
vertex is entered by one of its edges and thus there are i − 1 edges to new vertices in the
frontier for a net gain of i − 2. The iλ i in i (i − 2) λ i is proportional to the probability of
reaching a degree i vertex and the i − 2 accounts for the increase or decrease in size of
the frontier when a degree i vertex is reached.
Example: Consider applying the Molloy Reed conditions to the G(n, p) model, and use
p i to denote the probability that a vertex has degree i, i.e., to analog to λ i . It turns out
that the summation ∑ n
i=0 i(i − 2)p i gives value zero precisely when p = 1/n, the point
at which the phase transition occurs. At p = 1/n, the average degree of each vertex is
one and there are n/2 edges. However, the actual degree distribution of the vertices is
binomial, where the probability that a vertex is of degree i is given by p i = ( n
i)
p i (1−p) n−i .
n∑
We now show that lim i(i − 2)p i = 0 for p i = ( n
i)
p i (1 − p) n−i when p = 1/n.
n→∞ i=0
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