Mathematics
A few shortcuts make a huge network small
Every social network is mostly local friendships, yet a few random long-distance links are enough to shrink it to just a handful of hops between strangers. Watts and Strogatz proved this in 1998: rewiring only 1 percent of a network's links this way collapses its average path length while barely touching local clustering.
Sources & further reading
Nature (Springer Nature)Collective dynamics of 'small-world' networksThe 1998 paper that discovered small-world networks and tested the idea against a real power grid, a film-actor collaboration network, and a worm's brain.Stanford University (CS224W course readings)Collective dynamics of 'small-world' networks (preprint PDF)A university-hosted copy confirming the original paper's findings.Brain Connectivity (Mary Ann Liebert) via PMC/NIHThe Ubiquity of Small-World NetworksA follow-up study confirming small-world structure shows up across brains, social networks, and the internet.
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