Environmental sensors are placed at equal spacing around a circular boundary and
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
N=500 environmental sensors are placed at equal spacing around a circular boundary and, before any rewiring, each is wired to its k=6 nearest neighbours around the ring (three on each side), giving the Watts-Strogatz model at p=0.
((a)) By counting, for a node on the ring, how many of the (2k) pairs of its k neighbours are themselves joined by a ring edge, derive the general formula C(0)=4(k−1)3(k−2) for the clustering coefficient of the unrewired ring, and evaluate it for k=6.
((b)) Each edge of the ring is now independently rewired to a uniformly random other node with probability p=0.05. Using the small-p approximation C(p)≈C(0)(1−p)3, which holds because a rewired edge destroys only the triangles at its own two ends, estimate C(0.05).
((c)) The unrewired ring has mean shortest path length ℓ(0)≈N/(2k). Give this value for the sensor ring, and explain, without further calculation, why even p=0.05 collapses ℓ towards lnN/lnk while C has fallen by less than 15%.