MTHM750-MTH750U/P
Graphs and Networks
Solutions to Assignment 7 (Year 2016/2017)
Prof. Vito Latora
1 (Characteristic path length L for circle graphs)*
(a) Calculate the value of the characteristic path length L for a (N, m) circle graph
with N = 9 nodes and m = 2.
(b) Are you able to generalise your result, i.e. can you find an expression for L as a
function of N and m valid for any (N, m) circle graph? For simplicity assume N is
an odd number
(a) The value of the characteristic path length for such a graph is L = 3/2. This
result can be obtained in the following way. All the nodes of the graph are equivalent,
then we just need to calculate the distances from a node i to the remaining 8 nodes.
Since m = 2 there will be two nodes to the left of i and two nodes to the right of i with
distance 1. Also, there will be two nodes to the left of i and two nodes to the right of i
with distance 2. Therefore the characteristic path length L is equal to:
3
1
L = (4 · 1 + 4 · 2) =
8
2
(b) Since all nodes are equivalent, we can evaluate as before L by considering the
distances from one node to all the others. For simplicity, we can assume N is an odd
d=1
d=1
d=2
d=2
d=3
d=3
number. For symmetry, we only consider the (N − 1)/2 nodes to the left of node i,
and we sort them into groups of nodes having the same distance from i as shown in
1
Figure. The first group has m nodes at distance d = 1, the second group has m nodes
at distance d = 2 from i, and so on. Let us define the quantity:
Ng =
N −1
2m
If Ng is an integer number, then each of the Ng groups contains m nodes, with distances
going from 1 to Ng . Otherwise, there will be bNg c groups with m nodes, and the
remaining N − 1 − 2bNg c ∗ m nodes at distance bNg c + 1. In the simplest case in which
Ng is integer, the sum of the distances from node i to the other (N − 1)/2 nodes is
equal to:
Ng
Ng (Ng + 1) N − 1
=
(N − 1 + 2m)
m∑g=m
2
8m
g=1
dividing this number by the number of nodes we are considering, namely (N − 1)/2,
we get the following expression for the characteristic path length:
Lcircle =
N + 2m − 1
4m
Notice that, for m = 1, we get a formula for Lcircle , that is different from that of a linear
chain without the periodic boundary conditions found in Assignment 6. If Ng is not an
integer, one has to take into account the nodes that are not in the groups (nodes in the
red dashed set in figure). An exact expression for Lcircle is given by:
m
circle
bNg c (bNg c + 1) .
L
= 1−
N −1
For large N and for 1 m N, both expressions reduce to Equation:
Lcircle ≈
N
4m
that we have used in the lecture when we studied WS small-world model.
2 (Power-law degree distributions)*
(a) Consider the degree distribution p(k) = ck−γ , where k ∈ ℜ, k ≥ kmin , γ > 1, and
γ−1
the normalisation constant is c = (γ − 1)kmin . Prove that the moments of order m,
for γ > m + 1, are finite quantities equal to:
hkm i =
γ −1 m
k
γ − 1 − m min
while moments of order m are infinite in an infinitely large network when γ ≤ m + 1.
2
In order to get the various moments of a given degree distribution p(k), with k ∈ ℜ
and k ≥ kmin , we need to solve the integrals:
m
hk i =
Z ∞
km p(k)dk
kmin
for m = 1, 2, . . .. By substituting the power-law in the expression above, and solving
the integrals, we get:
h
i∞

c
Z ∞
 m+1−γ
for γ 6= m + 1
km+1−γ
h i∞ kmin
hkm i = c
k−γ+m dk =

kmin
c ln k
for γ = m + 1
kmin
where m is a positive integer. The terms in the brackets depend in general on the values
of the moment order, m, and of the degree exponent, γ. If γ ≤ m + 1 they diverge at
infinity. Thus, moments of order m are infinite in an infinitely large network when
γ ≤ m + 1, while, for γ > m + 1, they are finite and equal to:
hkm i =
γ −1 m
k
γ − 1 − m min
This means that the average of a power-law degree distribution is finite and well defined only for γ > 2, and is given by the expression:
hki =
γ −1
kmin .
γ −2
The second moment is finite and well-defined only for γ > 3, and is given by the
expression:
γ −1 2
hk2 i =
k .
γ − 3 min
In many real-world networks, as we have seen in the lecture, we extract exponents
2 < γ < 3. This means that we are in a situation such that the first moment hki is finite,
while the second moment hk2 i diverges with the number of nodes N in the system.
3