Foundations of Data Science

Exercise 4.53 Consider graph 3-colorability. Randomly generate the edges of a graph
and compute the number of solutions and the number of connected components of the
solution set as a function of the number of edges generated. What happens?
Exercise 4.54 Construct an example of a formula which is satisfiable, but the SC heuristic
fails to find a satisfying assignment.
Exercise 4.55 In G(n, p), let x k be the number of connected components of size k. Using
x k , write down the probability that a randomly chosen vertex is in a connected component
of size k. Also write down the expected size of the connected component containing a
randomly chosen vertex.
Exercise 4.56 In a G(n, p) graph with p asymptotically greater than 1 , show that
n
∞∑
i(i − 2)λ i > 0 where λ i is the fraction of vertices of degree i.
i=0
Exercise 4.57 Describe several methods of generating a random graph with a given degree
distribution. Describe differences in the graphs generated by the different methods.
Exercise 4.58 Consider generating a random graph adding one edge at a time. Let n(i,t)
be the number of components of size i at time t.
n(1, 1) = n
n(1, t) = 0 t > 1
n(i, t) = n(i, t − 1) + ∑ j(i − j)
n (j, t − 1) n (i − j, t − 1) − 2i
n 2
n n (i)
Compute n(i,t) for a number of values of i and t. What is the behavior? What is the
sum of n(i,t) for fixed t and all i? Can you write a generating function for n(i,t)?
Exercise 4.59 The global clustering coefficient of a graph is defined as follows. Let d v be
the degree of vertex v and let e v be the number of edges connecting pairs of vertices that
are adjacent to vertex v. The global clustering coefficient c is given by
c = ∑ v
2e v
. d v(d v−1)
In a social network, for example, it measures what fraction of pairs of friends of each
person are themselves friends. If many are, the clustering coefficient is high. What is c
for a random graph with p = d in the limit as n goes to infinity? For a denser graph?
n
Compare this value to that for some social network.
Exercise 4.60 Consider a structured graph, such as a grid or cycle, and gradually add
edges or reroute edges at random. Let L be the average distance between all pairs of
vertices in a graph and let C be the ratio of triangles to connected sets of three vertices.
Plot L and C as a function of the randomness introduced.
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