[SOLVED] Complex Networks MTH6142 Formative Assignment 5

Complex Networks (MTH6142)

Formative Assignment 5

• 1. Random networks in the G(N, p) ensemble

Assume that p = a/Nz
, where a > 0 and z ≥ 0, and a, z independent on N.

(a) Determine the average degree 〈k〉 in the limit N → ∞ for the following values of the parameters

(i) a = 0.5, z = 1;

(ii) a = 2, z = 1;

(iii) a > 0, z = 2;

(iv) a > 0, z = 0.5.

(b) In which of the above cases does the random network contain a giant component in the limit N → ∞?.

(c) Given p = a/Nz with generic values of a > 0, z ≥ 0 determine the average degree 〈k〉 in the large network limit N → ∞.

(d) Determine the conditions on a and z for these random networks to be subcritical, i.e. with a fraction S of nodes in the giant component given by S = 0 in the N → ∞ limit.

(e) Determine the conditions on a and z for these random networks to be supercritical, i.e. with a non vanishing fraction S of nodes in the giant component (S > 0) in the N → ∞ limit.

(f) Determine the conditions on a and z for which these random networks are critical, in the large network limit, i.e. in the limit N → ∞.2

• 2. Random networks in the G(N, p) ensemble with p = c/(N − 1) where c > 0.

(a) Calculate the average number of triangles in the network, by evaluating first the number of ways to select 3 nodes out of N nodes, and secondly the probability that the selected nodes are all connected to each other.

(b) Show that in the limit N → ∞ the average number of triangles in the network is

This means that the number of triangles is constant, neither growing or vanishing, in the limit of large N.