Minimal and maximal characteristic path lengths in connected sociomatrices

Abstract The characteristic path length is one of the most important and frequently-invoked characteristics of a social network. Given a specific network, it can be of considerable interest to know how the path length compares to the “best” or “worst” possible configuration for networks with the same number of agents and lines. Heretofore, the literature has relied heavily on asymptotic results for random networks to suggest a lower path length benchmark, and no upper benchmark is commonly used. We provide easily computed bounds for these best and worst cases. The bounds are constructive, so they can be attained, and we identify the network structures that attain them. Probabilistic analyses confirm short path lengths in connected random networks, provide additional insights for small networks not revealed by asymptotic limits, and suggest when maximal path lengths are attained with non-negligible probabilities.