Editing Small-world experiment (section)

===Network models===
[[File:Watts Strogatz graph.svg|alt=There are three graphs side by side. The titles on top from left to right are: "Regular Ring Graph (p = 0)", "Small-World Graph (p = 0.2), and "Random Graph (p = 1)".|thumb|282x282px|Comparison of [[Watts–Strogatz model|Watts-Strogatz graphs]] with different randomization probability. A regular ring graph (left), a small-world graph with some edges randomly rewired (center), and a random graph with all edges randomly rewired (right).]]
In 1998, [[Duncan J. Watts]] and [[Steven Strogatz]] from [[Cornell University]] published the first network model on the small-world phenomenon. They showed that networks from both the natural and man-made world, such as [[power grid]]s and the neural network of ''[[Caenorhabditis elegans|C. elegans]]'', exhibit the small-world phenomenon. Watts and Strogatz showed that, beginning with a regular lattice, the addition of a small number of random links reduces the diameter&mdash;the longest direct path between any two vertices in the network&mdash;from being very long to being very short.<ref>{{Cite journal |last1=Watts |first1=Duncan J. |last2=Strogatz |first2=Steven H. |date=June 1998 |title=Collective dynamics of 'small-world' networks |url=https://www.nature.com/articles/30918 |journal=Nature |language=en |volume=393 |issue=6684 |pages=440–442 |doi=10.1038/30918 |pmid=9623998 |bibcode=1998Natur.393..440W |issn=1476-4687|url-access=subscription }}</ref> The research was originally inspired by Watts' efforts to understand the synchronization of [[Cricket (insect)|cricket]] [[stridulation|chirps]], which show a high degree of coordination over long ranges as though the insects are being guided by an invisible conductor. The mathematical model which Watts and Strogatz developed to explain this phenomenon has since been applied in a wide range of different areas. In Watts' words:<ref>{{cite web| url=http://discovermagazine.com/1998/dec/frommuhammadalit1553 | title=From Muhammad Ali to Grandma Rose | date =1 December 1998 | first=Polly | last=Shulman | publisher=DISCOVER magazine | access-date=13 August 2010}}</ref>

{{quote|I think I've been contacted by someone from just about every field outside of English literature. I've had letters from mathematicians, physicists, biochemists, neurophysiologists, epidemiologists, economists, sociologists; from people in marketing, information systems, civil engineering, and from a business enterprise that uses the concept of the small world for networking purposes on the Internet.}}

Generally, their model demonstrated the truth in [[Mark Granovetter]]'s observation that it is "the strength of weak ties"<ref>{{cite journal | url=https://www.jstor.org/stable/2776392 | jstor=2776392 | last1=Granovetter | first1=Mark S. | title=The Strength of Weak Ties | journal=American Journal of Sociology | date=1973 | volume=78 | issue=6 | pages=1360–1380 | doi=10.1086/225469 | url-access=subscription }}</ref> that holds together a social network. Although the specific model has since been generalized by [[Jon Kleinberg]]{{citation needed|date=June 2022}}, it remains a canonical case study in the field of [[complex network]]s. In [[network theory]], the idea presented in the [[small-world network]] model has been explored quite extensively. Indeed, several classic results in [[random graph]] theory show that even networks with no real topological structure exhibit the small-world phenomenon, which mathematically is expressed as the diameter of the network growing with the logarithm of the number of nodes (rather than proportional to the number of nodes, as in the case for a lattice). This result similarly maps onto networks with a power-law degree distribution, such as [[scale-free networks]].

In [[computer science]], the small-world phenomenon (although it is not typically called that) is used in the development of secure peer-to-peer protocols, novel routing algorithms for the Internet and [[ad hoc]] wireless networks, and search algorithms for communication networks of all kinds.