A regional water-distribution network is modelled

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 regional water-distribution network is modelled, for the purpose of estimating how much it can withstand pump failures, first as a Watts-Strogatz ring and then as an Erdos-Renyi graph with the same mean degree. ((a)) The network's measured clustering coefficient is

C=0.47

. Model its local structure as a Watts-Strogatz ring with

k=8

pipes per pumping station (four on each side) before some pipes are rerouted at random. Derive

C(0)

for this ring and use

C(p)\approx C(0)(1-p)^3

to estimate the fraction

p

of pipes that have been rerouted. ((b)) Ignoring this clustering (an idealisation used only to isolate the effect of the degree distribution), model the same network as an Erdos-Renyi random graph with mean degree

c=8

. Derive the site-percolation threshold

\phi_{\rm c}=1/G_1'(1)

and state whether the fraction

\phi=0.5

of stations left operating after a fault leaves the network above or below threshold. ((c)) At

\phi=0.5

, solve

u=1-\phi+\phi\,e^{c(u-1)}

and

S=\phi\big(1-e^{c(u-1)}\big)

for

S

, iterating from

u=1-\phi

. Independent simulations of Erdos-Renyi graphs with

c=8

\phi=0.5

(

20

seeded realisations per size, seed

771

) give

S=0.46\pm0.03

N=10^3

S=0.485\pm0.008

N=10^4

and

S=0.489\pm0.003

N=10^5

. Compare these with the prediction and comment on the trend.

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