Foundations of Data Science

Exercise 5.40 Suppose that the clique in Exercise 5.39 was replaced by an arbitrary graph
with m − 1 edges. What would be the return time to A in terms of m, the total number of
edges.
Exercise 5.41 Suppose that the clique in Exercise 5.39 was replaed by an arbitrary graph
with m − d edges and there were d edges from A to the graph. What would be the expected
length of a random path starting at A and ending at A after returning to A exactly d
times.
Exercise 5.42 Given an undirected graph with a component consisting of a single edge
find two eigenvalues of the Laplacian L = D −A where D is a diagonal matrix with vertex
degrees on the diagonal and A is the adjacency matrix of the graph.
Exercise 5.43 A researcher was interested in determining the importance of various
edges in an undirected graph. He computed the stationary probability for a random walk
on the graph and let p i be the probability of being at vertex i. If vertex i was of degree
d i , the frequency that edge (i, j) was traversed from i to j would be 1 d i
p i and the frequency
that the edge was traversed in the opposite direction would be 1 d j
p j . Thus, he assigned an
∣ importance of ∣ 1 d i
p i − 1 ∣∣
d j
p j to the edge. What is wrong with his idea?
Exercise 5.44 Prove that two independent random walks starting at the origin on a two
dimensional lattice will eventually meet with probability one.
Exercise 5.45 Suppose two individuals are flipping balanced coins and each is keeping
tract of the number of heads minus the number of tails. Will both individual’s counts ever
return to zero at the same time?
Exercise 5.46 Consider the lattice in 2-dimensions. In each square add the two diagonal
edges. What is the escape probability for the resulting graph?
Exercise 5.47 Determine by simulation the escape probability for the 3-dimensional lattice.
Exercise 5.48 What is the escape probability for a random walk starting at the root of
an infinite binary tree?
Exercise 5.49 Consider a random walk on the positive half line, that is the integers
0, 1, 2, . . .. At the origin, always move right one step. At all other integers move right
with probability 2/3 and left with probability 1/3. What is the escape probability?
Exercise 5.50 Consider the graphs in Figure 5.18. Calculate the stationary distribution
for a random walk on each graph and the flow through each edge. What condition holds
on the flow through edges in the undirected graph? In the directed graph?
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