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Duncan J. Watts studies graphs and collective behavior using mathematical models, experiments, and large-scale data. His 1998 work with Steven H. Strogatz introduced the Watts-Strogatz model, showing how a few rewired connections can greatly shorten paths in an otherwise locally clustered graph.^[ref]

This gave a concrete explanation for how high clustering and short path lengths can coexist. His institutional biography describes research on social and organizational graphs, collective dynamics, and web-based experiments, alongside training in physics and theoretical and applied mechanics.^[ref]

For the Watts-Strogatz model, choose $n$ vertices on a ring, an even initial vertex degree $K<n$, and a rewiring probability $p$. Connect each vertex to its $K/2$ nearest neighbours on each side. Consider each original clockwise edge once: with probability $p$, retain its starting endpoint and replace its other endpoint uniformly, excluding self-loops and duplicate edges. The edge count remains $nK/2$, although individual vertex degrees change.

Increasing $n$ at fixed $K$ lengthens unrewired-ring graph distances. Increasing $K$ adds edges and shortens distances. Increasing $p$ introduces shortcuts, sharply reducing average shortest path length at first, then progressively reducing the clustering coefficient by destroying local triangles.

The small-world network regime occurs at intermediate $p$ with sufficient shortcuts, substantial clustering, and $K\ll n$. At $p=0$ distances are long; near $p=1$ the clustering advantage disappears. The degree distribution remains concentrated around $K$, so this is not a scale-free network.^[ref]

The Watts-Strogatz model starts with $n$ vertices arranged on a ring, each joined to a fixed even number $k$ of nearest neighbors. It then rewires one orientation of each original ring edge independently with probability $\beta$, choosing a new endpoint while avoiding loops and duplicate edges. The number of edges is preserved; individual degrees need not be.^[ref]

At $\beta =0$ the graph is a regular ring lattice. A small positive rewiring probability introduces shortcuts that can sharply reduce typical path lengths while retaining substantial local clustering. This offers a mechanism for small-world networks with both properties. It is a model of a structural effect, not a claim that every observed network was generated by literal rewiring.