small-world: one construct, many systems

In social psychology, the small-world problem is the observation that any two people, however distant, tend to be connected by a surprisingly short chain of personal acquaintances. Travers and Milgram's 1969 An experimental study of the small world problem had people forward a packet toward a target through people they knew on a first-name basis, and the completed chains averaged only a handful of steps, the origin of the 'six degrees' idea. The referent is the short social distance between strangers in a human acquaintance network, the same structural fact later formalised for graphs and brains.[1]

Unit

The number of acquaintance steps between two people

Recovery means

Not applicable

Measured by

Estimated from the length of completed referral chains between random source and target individuals

In network science a small-world network is a graph that combines the dense local clustering of a regular lattice with the short average path length of a random graph, a balance produced when a few long-range shortcuts are added to an otherwise locally connected structure. Watts and Strogatz's 1998 Collective dynamics of 'small-world' networks gave the model and the two measures (clustering coefficient and characteristic path length) that define the regime. It is the same construct as Milgram's acquaintance chains, now a precise and general property of any network.[2]

Unit

Clustering coefficient together with characteristic path length

Recovery means

Not applicable

Measured by

Diagnosed by high clustering alongside a short mean shortest-path length relative to a random graph of the same size

In network neuroscience the brain's anatomical and functional connectivity shows small-world organisation: regions form tight local clusters for specialised processing while a sparse set of long-range connections keeps the whole network only a few steps across, giving efficient integration at low wiring cost. Bassett and Bullmore's 2006 Small-world brain networks reviewed the evidence for this architecture across imaging modalities. It is the identical clustering-plus-shortcuts construct as the Watts–Strogatz network and Milgram's social chains, realised in neural wiring.[3]

Unit

The small-world index of a brain graph (clustering versus path length)

Recovery means

Not applicable

Measured by

Assessed from connectivity graphs by comparing clustering and path length to matched random networks