Small-world network
A small-world network is a type of mathematical graph in which almost nodes are not neighbors of one another, but the neighbors of any condition node are likely to be neighbors of each other and most nodes can be reached from every other node 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 requested grows proportionally to the logarithm of the number of nodes N in the network, that is:
while the global clustering coefficient is non small.
In the context of a social network, this results in the small world phenomenon of strangers being linked by a short institution of acquaintances. many empirical graphs show the small-world effect, including social networks, wikis such(a) as Wikipedia, gene networks, as living as even the underlying architecture of the Internet. it is for the inspiration for numerous network-on-chip architectures in innovative computer hardware.
A certain variety of small-world networks were subject as a a collection of matters sharing a common attribute of random graphs by Duncan Watts & Steven Strogatz in 1998. They covered that graphs could be classified according to two self-employed person structural features, namely the clustering coefficient, & average node-to-node distance also requested as average shortest path length. Purely random graphs, built according to the 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 score a small average shortest path length, but also a clustering coefficient significantly higher than expected by random chance. Watts and Strogatz then delivered 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 crossover in the Watts–Strogatz service example between a "large world" such(a) as a lattice and a small world was number one described by Barthelemy and Amaral in 1999. This relieve oneself was followed by many studies, including exact results Barrat and Weigt, 1999; Dorogovtsev and Mendes; Barmpoutis and Murray, 2010.
Properties of small-world networks
Small-world networks tend to contain cliques, and near-cliques, meaning sub-networks which gain connections between almost any two nodes within them. This follows from the build property of a high clustering coefficient. Secondly, most pairs of nodes will be connected by at least one short path. This follows from the determining property that the mean-shortest path length be small. Several other properties are often associated with small-world networks. Typically there is an over-abundance of hubs – nodes in the network with a high number of connections known as high degree nodes. These hubs serve as the common connections mediating the short path lengths between other edges. By analogy, the small-world network of airline flights has a small mean-path length i.e. between all two cities you are likely to have to take three or fewer flights because many flights are routed through hub cities. This property is often analyzed by considering the fraction of nodes in the network that have a specific number of connections going into them the degree distribution of the network. Networks with a greater than expected number of hubs will have a greater fraction of nodes with high degree, and consequently the degree distribution will be enriched at high degree values. This is known colloquially as a fat-tailed distribution. Graphs of very different topology qualify as small-world networks as long as they satisfy the two definitional indications above.
Network small-worldness has been quantified by a small-coefficient, 
, calculated by comparing clustering and path length of a assumption network to an equivalent random network with same degree on average.
Another method for quantifying network small-worldness utilizes the original definition of the small-world network comparing the clustering of a given network to an equivalent lattice network and its path length to an equivalent random network. The small-world measure 
is defined as
Where the characteristic path length L and clustering coefficient C are calculated from the network you are testing, Cℓ is the clustering coefficient for an equivalent lattice network and Lr is the characteristic path length for an equivalent random network.
Still another method for quantifying small-worldness normalizes both the network's clustering and path length relative to these characteristics in equivalent lattice and random networks. The Small World Index SWI is defined as
Both ω′ and SWI range between 0 and 1, and have been present to capture aspects of small-worldness. However, they follow slightly different conceptions of ideal small-worldness. For a given kind of constraints e.g. size, density, degree distribution, there exists a network for which ω′ = 1, and thus ω aims to capture the extent to which a network with given constraints as small worldly as possible. In contrast, there may not constitute a network for which SWI = 1, the thus SWI aims to capture the extent to which a network with given constraints approaches the theoretical small world ideal of a network where C ≈ Cℓ and L ≈ Lr.