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This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Small-world_network


00:03:04 1 Properties of small-world networks
00:09:22 2 Examples of small-world networks
00:10:29 3 Examples of non-small-world networks
00:11:32 4 Network robustness
00:13:16 5 Construction of small-world networks
00:16:11 6 Applications
00:18:24 6.1 Applications to sociology
00:18:34 6.2 Applications to earth sciences
00:20:17 6.3 Applications to computing
00:21:09 6.4 Small-world neural networks in the brain
00:22:05 7 Small world with a distribution of link length
00:23:19 8 See also
00:23:49 9 References



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SUMMARY
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A small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but the neighbors of any given 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 required) grows proportionally to the logarithm of the number of nodes N in the network, that is:




L
∝
log
⁡
N


{\displaystyle L\propto \log N}
while the clustering coefficient is not small. In the context of a social network, this results in the small world phenomenon of strangers being linked by a short chain of acquaintances. Many empirical graphs show the small-world effect, including social networks, wikis such as Wikipedia, gene networks, and even the underlying architecture of the Internet. It is the inspiration for many network-on-chip architectures in contemporary computer hardware.A certain category of small-world networks were identified as a class of random graphs by 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 (also known 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 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 crossover in the Watts–Strogatz model between a "large world" (such as a lattice) and a small world was first 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 Mendes; Barmpoutis and Murray, 2010). Braunstein found that for weighted ER networks where the weights have a very broad distribution, the optimal path scales becomes significantly longer and scales as N1/3.