Foundations of Data Science

4.12 Exercises
Exercise 4.1 Search the World Wide Web to find some real world graphs in machine
readable form or data bases that could automatically be converted to graphs.
1. Plot the degree distribution of each graph.
2. Compute the average degree of each graph.
3. Count the number of connected components of each size in each graph.
4. Describe what you find.
5. What is the average vertex degree in each graph? If the graph were a G(n, p) graph,
what would the value of p be?
6. Spot differences between your graphs and G(n, p) for p from the last part. [Look at
sizes of connected components, cycles, size of giant component.]
Exercise 4.2 In G(n, p) the probability of a vertex having degree k is ( n
k)
p k (1 − p) n−k .
1. Show by direct calculation that the expected degree is np.
2. Compute directly the variance of the distribution.
3. Where is the mode of the binomial distribution for a given value of p? The mode is
the point at which the probability is maximum.
Exercise 4.3
1. Plot the degree distribution for G(1000, 0.003).
2. Plot the degree distribution for G(1000, 0.030).
Exercise 4.4 To better understand the binomial distribution plot ( n
k)
p k (1 − p) n−k as a
function of k for n = 50 and k = 0.05, 0.5, 0.95. For each value of p check the sum over
all k to ensure that the sum is one.
Exercise 4.5 In G ( n, n) 1 , argue that with high probability there is no vertex of degree
greater than
6 log n (i.e. ,the probability that such a vertex exists goes to zero as n goes
log log n
to infinity). You may use the Poisson approximation and may wish to use the fact that
k! ≥ ( k e )k .
Exercise 4.6 The example of Section 4.1.1 showed that if the degrees in G(n, 1 n ) were
independent there would almost surely be a vertex of degree Ω(log n/ log log n). However,
the degrees are not independent. Show how to overcome this difficulty.
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