Foundations of Data Science

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2 3
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Figure 4.8: A graph (left) and the breadth first search of the graph (right). At vertex 1
the algorithm queried all edges. The solid edges are real edges, the dashed edges are edges
that were queried but do not exist. At vertex 2 the algorithm queried all possible edges to
vertices not yet discovered. The algorithm does not query whether the edge (2,3) exists
since vertex 3 has already been discovered when the algorithm is at vertex 2. Potential
edges not queried are illustrated with dotted edges.
Thus the probability that a vertex is discovered after i steps is approximately id. The n
expected number of discovered vertices grows as id and the expected size of the frontier
grows as (d − 1) i. As the fraction of discovered vertices increases, the expected rate of
growth of newly discovered vertices decreases since many of the vertices adjacent to the
vertex currently being searched have already been discovered. For d > 1, once d−1n
vertices
have been discovered, the growth of newly discovered vertices slows to one at each
d
step. Eventually for d >1, the growth of discovering new vertices drops below one per
step and the frontier starts to shrink. For d 1, the expected size of the frontier grows as (d−1)i for small i. The actual size
of the frontier is a random variable. What is the probability that the actual size of the
frontier will differ from the expected size of the frontier by a sufficient amount so that the
actual size of the frontier is zero? To answer this, we need to understand the distribution
of the number of discovered vertices after i steps. For small i, the probability that a vertex
has been discovered is 1 − (1 − d/n) i ≈ id/n and the binomial distribution for the number
of discovered vertices, binomial(n, id ), is well approximated by the Poisson distribution
n
with the same mean id. The probability that a total of k vertices have been discovered in
i steps is approximately e −di (di) k
. For a connected component to have exactly i vertices,
k!
the frontier must drop to zero for the first time at step i. A necessary condition is that
exactly i vertices must have been discovered in the first i steps. The probability of this is
approximately
−di (di)i
e = e −di di i i
e i = e −(d−1)i d i = e −(d−1−ln d)i .
i! i i
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