The small-world phenomenon: an algorithmic perspective
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The "algorithmic small-world hypothesis" states that not only are pairs of individuals in a large social network connected by short paths, but that ordinary individuals can find these paths. Although theoretically plausible, empirical evidence for the hypothesis is limited, as most chains in "small-world" experiments fail to complete, thereby biasing estimates of "true" chain lengths. Using data from two recent small-world experiments, comprising a total of 162,328 message chains, and directed at one of 30 "targets" spread across 19 countries, we model heterogeneity in chain attrition rates as a function of individual attributes. We then introduce a rigorous way of estimating true chain lengths that is provably unbiased, and can account for empirically-observed variation in attrition rates. Our findings provide mixed support for the algorithmic hypothesis. On the one hand, it appears that roughly half of all chains can be completed in 6-7 steps--thus supporting the "six degrees of separation" assertion--but on the other hand, estimates of the mean are much longer, suggesting that for at least some of the population, the world is not "small" in the algorithmic sense. We conclude that search distances in social networks are fundamentally different from topological distances, for which the mean and median of the shortest path lengths between nodes tend to be similar.