Foundations of Data Science

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Figure 8.8: insert caption
Exercise 8.18 What happens if we relax this restriction, for example, if we allow for S,
the entire set?
Exercise 8.19 Given the graph G = (V, E) of a social network where vertices represent
individuals and edges represent relationships of some kind, one would like to define the
concept of a community. A number of deferent definitions are possible.
1. A subgraph S = (V S , E S ) whose density E S
V 2 S
is greater than that of the graph E V 2 .
2. A subgraph S with a low conductance like property such as the number of graph edges
leaving the subgraph normalized by the minimum size of S or V − S where size is
measured by the sum of degrees of vertices in S or in V − S.
3. A subgraph that has more internal edges than in a random graph with the same
degree distribution.
Which would you use and why?
Exercise 8.20 A stochastic matrix is a matrix with non negative entries in which each
row sums to one. Show that for a stochastic matrix, the largest eigenvalue is one. Show
that the eigenvalue has multiplicity one if and only if the corresponding Markov Chain is
connected.
Exercise 8.21 Show that if P is a stochastic matrix and π satisfies π i p ij = π j p ji , then
for any left eigenvector v of P , the vector u with components u i = v i
π i
is a right eigenvector
with the same eigenvalue.
Exercise 8.22 In Theorem (??), how can one clustering C (0)
clustering? What if there are several proper clusterings?
be close to any proper
Exercise 8.23 Give an example of a clustering problem where the clusters are not linearly
separable in the original space, but are separable in a higher dimensional space.
Hint: Look at the example for Gaussian kernels in the chapter on learning.
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