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Complex Networks (MTH6142)

Formative Assignment 4

• 1. Degree distribution of random graphs

A random graph ensemble G(N, p) with  has a binomial degree distribution

that in the limit of N >> 1 can be approximated by a Poisson distribution PP (k) given by

(a) Calculate the generating function

for the binomial degree distribution PB(k) given by Eq. (1).

(b) Using the properties of the generating functions, evaluate the first moment h ki and the second moment of the degree distribution PB(k) given by Eq. (1).

(c) Calculate the generating function

for the Poisson degree distribution PP (k) given by Eq. (2).

(d) Using the properties of the generating functions, evaluate the first moment h ki and the second moment h k(k − 1)i of the degree distribution PP (k) given by Eq. (2).

(e) Show that the first h ki and second moment of the binomial distribution PB(k) obtained in (b) are the same as the first h ki and second moments of the Poisson distribution PP (k) obtained in (d), as long as  with c constant and N → ∞.

• 2. A given random network

Consider a random network in the ensemble G(N, p) with N = 4 × 106 nodes and a linking probability p = 10−4 .

(a) Calculate the average degree h ki of this network.

(b) Calculate the standard deviation σP using the approximated degree distribution given by Eq. (2).

(c) Assume that you observe a node with degree 2 × 103 . How many standard deviations is this observation from the mean? Is this an expected observation or is this an unexpected observation?