Small-world Network - Examples of Non-small-world Networks

Examples of Non-small-world Networks

Networks are less likely to have the small-world properties if links between nodes arise mainly from spatial or temporal proximity, because there may be no short path between two "distant" nodes.

For example, the famous theory of "six degrees of separation" between people tacitly presumes that the domain of discourse is the set of people alive at any one time. The number of degrees of separation between Albert Einstein and Alexander the Great is almost certainly greater than 30 and this network does not have small-world properties. A similarly constrained network would be the "went to school with" network: if two people went to the same college ten years apart from one another, it is unlikely that they have acquaintances in common amongst the student body.

Similarly, the number of relay stations through which a message must pass was not always small. In the days when the post was carried by hand or on horseback, the number of times a letter changed hands between its source and destination would have been much greater than it is today. The number of times a message changed hands in the days of the visual telegraph (circa 1800–1850) was determined by the requirement that two stations be connected by line-of-sight.

Tacit assumptions, if not examined, can cause a bias in the literature on graphs in favor of finding small-world networks.