A fibre-optic junction network is built as a configuration model in which half the…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Random graphs, the giant component and percolation on networks

Problem

A fibre-optic junction network is built as a configuration model in which half the junctions have degree

2

and half have degree

4

(every junction has at least two fibres, so

p_2 = 0.5

p_4 = 0.5

, and all other

p_k = 0

). ((a)) Write the degree generating function

G_0(x) = p_2 x^2 + p_4 x^4

and find the excess degree generating function

G_1(x) = G_0'(x)/G_0'(1)

. Read off the excess degree probabilities

q_1

and

q_3

. ((b)) Solve

u = G_1(u)

for the probability

u

that a junction reached by following a fibre does not lead into the giant component, and hence find the fraction

S = 1 - G_0(u)

of junctions in the giant component.

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