whatTwo copies of the same ring of two hundred people, each knowing their six nearest neighbours. On the right-hand copy, every friendship is rewired with probability p to somebody random across the ring. The same rumour starts at the same person on both, spreading one handshake per beat: ticks lean outward once a person has heard it, the bold flame is this beat's frontier.

whyThe claim, from Watts and Strogatz: a handful of shortcuts collapses the path length while clustering barely moves. At p of one percent, six rewired friendships among six hundred, the rumour reaches the average person in half the hops, yet ninety-six percent of the cliquishness survives. The chart collects both against p: vertical ticks are path length, horizontal ticks clustering, each as a fraction of the ordered ring. The verticals dive a full decade before the horizontals sag. That gap is the small world: cliquish, yet six degrees wide.

howA ring lattice of two hundred nodes, degree six, rewired Watts-Strogatz style: one end of each edge moves with probability p. Every race resamples the rewiring, so the chart averages over graphs, not one lucky wiring. Path length is measured by the race itself, the mean hop at which each person hears; clustering is counted exactly, triangles over possible triangles. Slide p along three decades, or click either ring to start the rumour where you point. One integer seeds every rewiring and every race.