A research-collaboration network of physicists is modelled
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Complex and collective systems
Many Parts, One Behaviour, Ch Random graphs, the giant component and percolation on networks
Problem
A research-collaboration network of physicists is modelled, for the purpose of estimating how much it can withstand a wave of author departures, 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=0.45. Model its local structure as a Watts-Strogatz ring with k=6 nearest neighbours (three on each side) before rewiring. Derive C(0) for this ring, then use C(p)≈C(0)(1−p)3 (valid for small rewiring probability p, neglecting the small chance that a shortcut reconnects two of its own former neighbours) to estimate the rewiring probability p that would give the measured clustering.
((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=6. Derive the site-percolation threshold ϕc=1/G1′(1) and state whether the fraction ϕ=0.35 of authors retained after a wave of departures leaves the network above or below threshold.
((c)) At ϕ=0.35, solve the self-consistency equations u=1−ϕ+ϕec(u−1) and S=ϕ(1−ec(u−1)) for the fraction S of authors in the giant component, iterating from u=1−ϕ. Then simulate: with a seeded random-number generator, build Erdos-Renyi graphs with c=6 at N=103, 104 and 105, keep each node independently with probability ϕ=0.35, and measure the fraction of all N nodes in the largest surviving component, over 20 realisations per size. Report the mean and its standard error at each size, compare them with the prediction and comment on the trend.