Editing Small-world experiment (section)

==Current research on the small-world problem==
The small-world question is still a popular research topic today, with many experiments still being conducted. For instance, Peter Dodds, Roby Muhamad, and Duncan Watts conducted the first large-scale replication of Milgram's experiment, involving 24,163 e-mail chains and 18 targets around the world.<ref>{{cite journal | doi=10.1126/science.1081058 | title=An Experimental Study of Search in Global Social Networks | date=2003 | last1=Dodds | first1=Peter Sheridan | last2=Muhamad | first2=Roby | last3=Watts | first3=Duncan J. | journal=Science | volume=301 | issue=5634 | pages=827–829 | pmid=12907800 | bibcode=2003Sci...301..827D }}</ref>

Dodds ''et al''. also found that the mean chain length was roughly six, even after accounting for attrition. A similar experiment using popular social networking sites as a medium was carried out at [[Carnegie Mellon University]]. Results showed that very few messages actually reached their destination. However, the critiques that apply to Milgram's experiment largely apply also to this current research.{{citation needed|date=August 2012}}

Recent research suggests that the small-world effect is a phenomenon that appeared rather recently in human history, leading to a drastic reduction in the average chain distance in social and physical networks. This can be justified by studying evolution patterns of infectious diseases throughout history, notably the [[Black Plague]] in Medieval Europe. Past epidemics have been noticed to spread in waves from well-defined central points, which can be explained through the localized nature of interactions of medieval populations. More recent [[epidemics]] have exhibited qualitatively different properties, as diseases no longer spread from one location outward, but rather with many starting clusters, due to travel and long-range physical (and social) interactions. This means that new long-distance connections were made through the development of transportation and communication technologies and that the likelihood of two individuals knowing each other if they live far away from each other has increased enough to drastically change the pattern of disease spread. This serves as an indication that the graph of physical and social connections in the world’s population has structurally changed.<ref>{{cite arXiv |eprint=1310.2636 |title=The small-world effect is a modern phenomenon |date=9 October 2013 |first1=Seth |last1=Marvel |first2=Travis |last2=Martin |first3=Charles R. |last3=Doering |author-link3=Charles R. Doering |first4=David |last4=Lusseau |first5=M. E. J. |last5=Newman |class=physics.soc-ph |author-link5=Mark Newman}}</ref>

===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.

===Modern Studies and Digital Networks===
With the rise of [[digital communication]] and [[online social networks]], researchers have revisited the small-world phenomenon in large-scale, real-world contexts. Modern studies indicate that the degrees of separation have significantly decreased, particularly due to the widespread use of social media platforms.

One of the most extensive studies on digital networks was conducted by Facebook and the University of Milan. In 2011, researchers analyzed the connections between 721 million active Facebook users—over 10% of the global population at the time. They found that the average number of intermediaries between any two users was 4.74, suggesting a much smaller world than previously estimated.<ref>{{cite news|author=John D. Sutter|url=https://edition.cnn.com/2011/11/22/tech/social-media/facebook-six-degrees/index.html?|title=On Facebook, it's now 4.74 degrees of separation|work= |location= |publisher=CNN Business |access-date=22 November 2011}}</ref>
By 2016, an updated study by Facebook revealed that this number had further decreased to just 3.57 degrees of separation, highlighting the growing interconnectedness of individuals through digital platforms. <ref>{{cite news|author=Jonah Engel Bromwich|url=https://www.nytimes.com/2016/02/05/technology/six-degrees-of-separation-facebook-finds-a-smaller-number.html|title=Six Degrees of Separation? Facebook Finds a Smaller Number|work= The New York Times|date=4 February 2016 |location= |access-date=4 February 2016}}</ref>

The increasing reach of digital networks has profound implications across various domains:
*Networking and Employment: Online professional platforms enable job seekers and employers to connect across geographic boundaries, facilitating career opportunities beyond traditional networks.
*Marketing and Business: Social media allows businesses to reach global audiences, using targeted advertising and personalized content to engage consumers more effectively.
*Information Dissemination: News, trends, and social movements spread rapidly across digital networks, sometimes within minutes, reshaping the way societies consume and react to information.
While digital connectivity has brought people closer, it also presents challenges such as misinformation spread, privacy concerns, and the impact of online interactions on real-world relationships. Nonetheless, these studies demonstrate how technology continues to reshape social structures, reducing the degrees of separation and further validating the small-world phenomenon in the digital age.

===Linking Social Capital to the Small-World Phenomenon===
The small-world phenomenon, originally demonstrated by [[Stanley Milgram]]'s experiment, suggests that individuals in large social networks are connected through surprisingly short chains of acquaintances. This structural property has significant implications for [[social capital]], which refers to the resources and benefits that individuals or groups can access through their social connections. Research has shown that [[small-world networks]] optimize both local clustering and global reach, facilitating the efficient flow of information and trust. In such networks, social capital is enhanced as [[weak ties]]—bridges between otherwise distant clusters—enable access to diverse resources and opportunities. These weak ties, often described in [[Mark Granovetter]]'s strength of weak ties theory, act as conduits for novel information and [[social mobility]]. Moreover, small-world structures support both bonding social capital, by reinforcing strong community ties, and bridging social capital, by connecting disparate [[social groups]].<ref>{{cite journal | title=Linking Social Capital to Small-worlds: A look at local and network-level processes and structure | date=2009 | first1=Christina | last1=Prell | journal=Methodological Innovations Online | volume=4 | pages=8–22 | doi=10.1177/205979910900400102 | doi-access=free }}</ref>

Empirical studies have linked the small-world topology to [[innovation diffusion]], job-market efficiency, and collective action, demonstrating that network structure plays a crucial role in shaping social capital at both individual and societal levels.<ref>{{cite journal | url=https://www.sciencedirect.com/science/article/pii/S0378437114000946 | title=The impact of network characteristics on the diffusion of innovations | date=11 February 2014 | first1=Renana| last1=Peres| journal=Physica A | volume=402 | pages=330–343 | author-link1=Renana Peres | doi=10.1016/j.physa.2014.02.003 | bibcode=2014PhyA..402..330P | access-date=1 April 2025| url-access=subscription }}</ref>