Watts–Strogatz Small World Network

Imagine every person in a village knows only their 4 nearest neighbours — the people right next to them in a ring. To send a message across the village takes many hops. Now randomly swap a few of those local connections for long-distance friendships. Suddenly the whole village is reachable in just a few hops — yet most people still mostly know their neighbours.

That's the small world effect. It explains why social networks, power grids, and even brain connections all have this same property: highly clustered locally, but surprisingly short paths globally. Move the Rewiring (p) slider from 0 → 1 to watch the transformation happen live.

• Graph canvas: Nodes sit in a ring. Grey lines are original local connections. Glowing blue lines are rewired (long-distance) shortcuts. Hover any node to highlight its neighbours.
• L — Average Path Length: Average number of hops to get from any node to any other node. Watch this drop sharply as soon as even a few edges are rewired.
• C — Clustering Coefficient: How many of your neighbours also know each other (0 = none, 1 = all). This drops much more slowly than L — that's the key asymmetry.
• L/L₀ and C/C₀: Both metrics normalised against the pure lattice (p=0). The "sweet spot" is where L/L₀ is near 0.2 while C/C₀ is still near 0.8 — that's a small world.
• Phase transition chart: Precomputed L (blue) and C (green) across all rewiring values on a log scale. The wide gap between the two curves is exactly the small world regime.
• Find Shortest Path: Click the button, then click two nodes to highlight the shortest route in yellow. Compare path length at p=0 vs p=0.1.