Foundations of Data Science

4.9 Growth Models
4.9.1 Growth Model Without Preferential Attachment
Many graphs that arise in the outside world started as small graphs that grew over
time. In a model for such graphs, vertices and edges are added to the graph over time.
In such a model there are many ways in which to select the vertices for attaching a
new edge. One is to select two vertices uniformly at random from the set of existing
vertices. Another is to select two vertices with probability proportional to their degree.
This latter method is referred to as preferential attachment. A variant of this method
would be to add a new vertex at each unit of time and with probability δ add an edge
where one end of the edge is the new vertex and the other end is a vertex selected with
probability proportional to its degree. The graph generated by this latter method is a tree.
Consider a growth model for a random graph without preferential attachment. Start
with zero vertices at time zero. At each unit of time a new vertex is created and with
probability δ, two vertices chosen at random are joined by an edge. The two vertices
may already have an edge between them. In this case, we will add another edge. So, the
resulting structure is a multi-graph, rather then a graph. But since at time t, there are t
vertices and in expectation only O(δt) edges where there are t 2 pairs of vertices, it is very
unlikely that there will be many multiple edges.
The degree distribution for this growth model is calculated as follows. The number of
vertices of degree k at time t is a random variable. Let d k (t) be the expectation of the
number of vertices of degree k at time t. The number of isolated vertices increases by one
at each unit of time and decreases by the number of isolated vertices, b(t), that are picked
to be end points of the new edge. b(t) can take on values 0,1, or 2. Taking expectations,
d 0 (t + 1) = d 0 (t) + 1 − E(b(t)).
Now b(t) is the sum of two 0-1 valued random variables whose values are the number
of degree zero vertices picked for each end point of the new edge. Even though the
two random variables are not independent, the expectation of b(t) is the sum of the
expectations of the two variables and is 2δ d 0(t)
. Thus,
t
d 0 (t + 1) = d 0 (t) + 1 − 2δ d 0(t)
.
t
The number of degree k vertices increases whenever a new edge is added to a degree k − 1
vertex and decreases when a new edge is added to a degree k vertex. Reasoning as above,
d k (t + 1) = d k (t) + 2δ d k−1(t)
t
− 2δ d k(t)
. (4.4)
t
Note that this formula, as others in this section, is not quite precise. For example, the
same vertex may be picked twice, so that the new edge is a self-loop. For k