The small-world phenomenon refers to the fact that even in a very large population, it takes relatively few connections to go from any element 1 to another random element. Amongst people, we know this concept as the “six degrees of separation” game. Any population of objects with connections can be conceptualized this way. Examples include crickets communicating by audible chirps, websites with links, electrical elements with wiring, board members with common members, or authors on mutual scientific papers. All of the examples I list have been examined in various scientific studies.
In a small-world network, elements are first connected in a regular lattice; for example, each element is connected to one or two nearby neighbors on each side. The leftmost picture below shows a regular lattice of elements. A connection between element i and element j is then removed. Then we add a connection between element i and any other element x, like the middle picture below. If x is across the network from i, then the number of steps between i and x has been reduced from some large number to 1. All of the elements connected to i are now 2 steps from element x. This reduces the diameter of the network, which is the maximum number of steps between any two elements, although the number of connections remains constant. In the six-degrees of separation game, the diameter would be 6. As we replace more of the lattice connections with random ones, the network becomes more and more random. We quantify a small-world network by its randomness, as in the picture below.
The small-world network has been explored as a means of sending information efficiently through a population. As the diameter reduces, the time it takes information to spread through the entire network reduces. Neurons in the brain have been explored as small-world networks; certain regions of the brain are highly interconnected with a few long distance connections to other regions of the brain. Protein networks and gene transcription networks have also been described with the small-world model. Further information with scholarly references is available on the scholarpedia page (which is generally a great resource for complex systems problems).
Here you can read a good scientific paper by Steven Strogatz, one of the premier scientists in the area. This is a paper published in Nature, one of the highest scientific publications. There are some equations, but the figures are also excellent if you are uncomfortable with the math. The paper models the power grid, boards of directors, and coauthorship using network ideas. I mention this paper specifically because I find Strogatz a very relatable and clear writer. Also consider reading his recent nontechnical book about math, The Joy of X, for more math fun.