How few random shortcuts does it take to halve the time a message needs to reach everyone? On the pure ring the message moves as two orange fronts, one each way round. A violet shortcut lets it jump across and start new fronts far from person 0.
Try this Set rewire-p to 0 and press Reset (it is read only at setup). The message crawls both ways round the ring and the run stops at steps-elapsed 16. Back at 10, most runs stop between 7 and 12.
About this model Where it ends up
Watts and Strogatz (1998) showed that rewiring a small fraction of a ring lattice's links at random gives it the short paths of a random graph while it keeps most of its clustering. This version shows the path-length half of that result as a message spreading across the ring.
Setup places nodes people on a circle. Each person makes k/2 links to its nearest clockwise neighbors, or, with probability rewire-p, makes all k/2 of them to random other people instead (violet shortcuts). Person 0 starts with the message. Each tick, every uninformed person with an informed link-neighbor becomes informed, so the message moves exactly one hop per tick and steps-elapsed at the stop is one more than the largest distance from person 0. All three params are read only at setup.
Two differences from Watts and Strogatz: they rewire each link independently, while here a person's k/2 links are rewired together, so shortcuts come in groups from the same person. And nothing here measures clustering or average path length, which NetLogo's Small Worlds model reports.
