A city's transit network has local stations of degree making up half of all stations…
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 city's transit network has local stations of degree 2 making up half of all stations and hub stations of degree 6 making up the other half. It is examined through two idealised models, one for local geometry and one for how much it can withstand loss of stations.
((a)) Model the local ring geometry of the low-degree stations as a Watts-Strogatz ring with k=4 neighbours before rewiring (two on each side); the network's measured clustering coefficient is C=0.35. This treats local clustering through an idealised uniform ring, a separate picture from the bimodal degree model used below. Derive C(0) for this ring and use C(p)≈C(0)(1−p)3 to estimate the fraction p of connections rewired away from strict ring neighbours.
((b)) Model the full network as a configuration model with degree distribution p2=p6=1/2 (no other degrees present). Derive the degree generating function G0(x), its excess-degree generating function G1(x)=G0′(x)/G0′(1), and hence the site-percolation threshold ϕc=1/G1′(1). State whether ϕ=0.5 (half the stations still operating after a service cut) leaves the network above or below threshold.
((c)) At ϕ=0.5, solve u=1−ϕ+ϕG1(u) and S=ϕ[1−G0(u)] for S, iterating from u=1−ϕ. Independent simulations of configuration-model networks with this degree distribution at ϕ=0.5 (20 seeded realisations per size, seed 55) give S=0.36±0.03 at N=103, S=0.388±0.009 at N=104 and S=0.394±0.003 at N=105. Compare these with the prediction and comment on the trend.