A small-world network is a type of Graph (mathematics)|mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps. Specifically, a small-world network is defined to be a network where the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes N in the network, that is:
:L \propto \log N
In the context of a social network, this results in the small world phenomenon of strangers being linked by a mutual acquaintance. Many empirical graphs are well-modeled by small-world networks. Social networks, the connectivity of the Internet, wikis such as Wikipedia, and gene regulatory network|gene networks all exhibit small-world network characteristics.
A certain category of small-world networks were identified as a class of random graphs by Duncan J. Watts|Duncan Watts and Steven Strogatz in 1998. They noted that graphs could be classified according to two independent structural features, namely the clustering coefficient, and average node-to-node distance (graph theory)|distance (also known as average shortest path length). Purely random graphs, built according to the Erdős–Rényi model|Erdős–Rényi (ER) model, exhibit a small average shortest path length (varying typically as the logarithm of the number of nodes) along with a small clustering coefficient. Watts and Strogatz measured that in fact many real-world networks have a small average shortest path length, but also a clustering coefficient significantly higher than expected by random chance. Watts and Strogatz then proposed a novel graph model, currently named the Watts and Strogatz model, with (i) a small average shortest path length, and (ii) a large clustering coefficient. The first description of the crossover in the Watts-Strogatz model between a large world (such as a lattice) and a small-world was described by Barthelemy and Amaral in 1999. This work was followed by a large number of studies, including exact results (Barrat and Weigt, 1999; Dorogovtsev and José Fernando Ferreira Mendes|Mendes; Barmpoutis and Murray, 2010).

A small-world network is a type of Graph (mathematics)|mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps. Specifically, a small-world network is defined to be a network where the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes N in the network, that is:
:L \propto \log N
In the context of a social network, this results in the small world phenomenon of strangers being linked by a mutual acquaintance. Many empirical graphs are well-modeled by small-world networks. Social networks, the connectivity of the Internet, wikis such as Wikipedia, and gene regulatory network|gene networks all exhibit small-world network characteristics.
A certain category of small-world networks were identified as a class of random graphs by Duncan J. Watts|Duncan Watts and Steven Strogatz in 1998. They noted that...