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Watts and Strogatz showed that a regular ring lattice only needs a tiny fraction of random rewiring to acquire the short path lengths of a random graph while keeping most of its local clustering. That regime - highly clustered yet only a few hops across - is the small world of the six-degrees experiments.
Here people sit on a ring, each wired to their k nearest neighbors, and every edge is rewired to a random partner with probability rewire-p; rewired edges live in a violet shortcuts breed so they stand out. One person starts with a message and each tick informed people pass it to their link neighbors. Compare rewire-p at 0 and at 10 percent: a handful of shortcuts roughly halves the steps-elapsed monitor when the run stops.