AM 5080: Complex Networks
Assignment - 4
Due date: Oct 13, 2023
Answer all questions
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1. Consider a growing network.
(a) If the network grows from an initial number of nodes m by the preferki
i
ential attachment rule dk
dt = t , show that this leads to a stationary
degree distribution P (k) = k .
(b) If the preferential attachment rule is modified to
↵ > 0, find the stationary distribution.
dki
dt
=
ki↵
t ,
where
(c) Sketch the expected degree distribution if the probability of attachment of a new node to any given node in the network is a constant.
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2. To generate a scale free undirected network using the preferential attachment algorithm, one starts with an m-clique, and one attaches a new node
to m preexisting nodes. The probability of attachment is proportional to
the current degree ki of a pre-existing node. It has been shown that the
exact degree distribution of the produced network is
P [ki = k] = pk =
2m(m + 1)
.
k(k + 1)(k + 2)
In other words, if a node i is selected randomly from the network, the
probability that this node is connected to exactly k nodes is equal to pk .
If the graph has N nodes, k ranges in the set {m, hdots, N }.
(a) Show that
PN
k=m
pk = 1.
(b) Suppose that N ! 1. Then k 2 {m, . . . , 1}. What is the average
degree < k >, given by
N
lim X
< k >=
kpk
N !1
k=m
AM 5080: Complex Networks
Assignment - 4
Due date: Oct 13, 2023
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3. Write a computer code to generate networks of size N with power law
degree distribution with degree exponent . Generate three netwroks with
= 2.2 and with N = 103 m N = 104 and N = 105 nodes respectively.
What is the percentage of multi-link and self loops in each network? Do
the same for networks with = 3.
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4. Using a software such as MATLAB, Mathematica or Numpy in python,
generate three synthetic datasets, each containing 10, 000 integers that
follow a power-law distribution with
= 2.2, = 2.5 and
3. Use
kmin = 1. Use statistical tests to fit the three distributions and estimate
the degree exponent statistically. Please read Chapter 4 from Barabasi’s
book.
5. The degree distribution pk expresses the probability that a randomly selected node has k neighbours. However, if we randomly select a link, the
probability that a node at one of its ends has degree k is qk = Akpk , where
A is a normalization factor.
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(a) Find the normalization factor A, assuming that the network has a
power law degree distribution with 2 < < 3, with minimum degree
kmin and maximum degree krmmax .
(b) In the configuration model qk is also the probability that a randomly
chosen node has a neighbour with degree k. What is the average
degree of the neighbours of a randomly chosen node?
(c) Calculate the average degree of the neighbours of a randomly chosen
node in a network with N = 104 , = 2.3, krmmin = 1 and kmax =
103 . Compare the results with the average degree of the network
< k >.
(d) How do you explain the paradox of (c), that is, a node’s friends have
more friends than the node itself?
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6. Please visit http://networkrepository.com which is a repository of scientific network data. Please download the data set in moodle that has been
downloaded from this repository. This represents data from a network of
artists in facebook. Nodes are labelled from 0 continuously. Each line contains two node labels A and B, representing either a directed link A > B,
or an undirected link A B. Undirected links appear once in the file.
(a) Using networks of Gephi, draw the topology of the network. Make
appropriate use of colors, size and layout to create a clear and informative visualization.
(b) Estimate the important characterizers associated with this network.
This includes degree distributions, centrality measures etc.