Foundations of Data Science

Exercise 5.10 Let p be a probability vector (nonnegative components adding up to 1) on
the vertices of a connected graph which is sufficiently large that it cannot be stored in a
computer. Set p ij (the transition probability from i to j) to p j for all i ≠ j which are
adjacent in the graph. Show that the stationary probability vector is p. Is a random walk
an efficient way to sample according to a distribution close to p? Think, for example, of
the graph G being the n × n × n × · · · n grid.
Exercise 5.11 Construct the edge probability for a three state Markov chain where each
pair of states is connected by an edge so that the stationary probability is ( 1
2 , 1 3 , 1 6)
.
Exercise 5.12 Consider a three state Markov chain with stationary probability ( 1
2 , 1 3 , 1 6)
.
Consider the Metropolis-Hastings algorithm with G the complete graph on these three
vertices. What is the expected probability that we would actually make a move along a
selected edge?
Exercise 5.13 Try Gibbs sampling on p (x) =
Metropolis Hasting Algorithm do?
( 1
2
0
0
1
2
)
. What happens? How does the
Exercise 5.14 Consider p(x), where, x = (x 1 , . . . , x 100 ) and p (0) = 1 2 , p (x) = 1
(2 100 −1)
x ≠
0. How does Gibbs sampling behave?
Exercise 5.15 Construct, program, and execute an algorithm to compute the volume of
a unit radius sphere in 20 dimensions by carrying out a random walk on a 20 dimensional
grid with 0.1 spacing.
Exercise 5.16 Given a connected graph G and an integer k how would you generate
connected subgraphs of G with k vertices with probability proportional to the number of
edges in the subgraph induced on those vertices? The probabilities need not be exactly
proportional to the number of edges and you are not expected to prove your algorithm for
this problem.
Exercise 5.17 Suppose one wishes to generate uniformly at random regular, degree three
undirected, connected multi-graphs each with 1,000 vertices. A multi-graph may have
multiple edges between a pair of vertices and self loops. One decides to do this by a Markov
Chain Monte Carlo technique. They design a network where each vertex is a regular degree
three, 1,000 vertex multi-graph. For edges they say that the vertices corresponding to two
graphs are connected by an edge if one graph can be obtained from the other by a flip of a
pair of disjoint edges. In a flip, a pair of edges (a, b) and (c, d) are replaced by (a, c) and
(b, d).
1. Prove that a swap on a connected multi-graph results in a connected multi-graph.
2. Prove that the network whose vertices correspond to the desired graphs is connected.
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