Foundations of Data Science

Exercise 4.61 Consider an n × n grid in the plane.
1. Prove that for any vertex u, there are at least k vertices at distance k for 1 ≤ k ≤
n/2.
2. Prove that for any vertex u, there are at most 4k vertices at distance k.
3. Prove that for one half of the pairs of points, the distance between them is at least
n/4.
Exercise 4.62 Recall the definition of a small-world graph in Section 4.10. Show that
in a small-world graph with r ≤ 2, that there exist short paths with high probability. The
proof for r = 0 is in the text.
Exercise 4.63 Change the small worlds graph as follows. Start with a n × n grid where
each vertex has one long-distance edge to a vertex chosen uniformly at random. These are
exactly like the long-distance edges for r = 0. But the grid edges are not present. Instead,
we have some other graph with the property that for each vertex, there are Θ(t 2 ) vertices
at distance t from the vertex for t ≤ n. Show that, almost surely, the diameter is O(ln n).
Exercise 4.64 Consider an n-node directed graph with two random out-edges from each
node. For two vertices s and t chosen at random, prove that with high probability there
exists a path of length at most O(ln n) from s to t.
Exercise 4.65 Explore the concept of small world by experimentally determining the answers
to the following questions:
1. How many edges are needed to disconnect a small world graph? By disconnect we
mean at least two pieces each of reasonable size. Is this connected to the emergence
of a giant component?
2. How does the diameter of a graph consisting of a cycle change as one adds a few
random long distance edges?
Exercise 4.66 In the small world model, would it help if the algorithm could look at edges
at any node at a cost of one for each node looked at?
Exercise 4.67 Consider the n × n grid in the section on small world graphs. If the
probability of an edge from vertex u to vertex v is proportional to d −r (u, v), show that the
constant of proportionality c r (u) is
θ(n 2−r ) for r > 2
θ(ln n) for r = 2
θ(1) for r < 2
Exercise 4.68 Show that for r < 2 in the small world graph model that short paths exist
but a polylog length path is unlikely to encounter a long distance edge whose end point is
close to the destination.
Exercise 4.69 Make a list of the ten most interesting things you learned about random
graphs.
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