Small-world networks revisited

Small-world networks have received much attention recently. Computer scientists, theoretical physicists, mathematicians, and others use them as basis for their studies. At least partly due to the different mind-sets of these disciplines, these random graph models have not always been correctly applied to questions in, e.g., peer-to-peer computing. This paper tries to shed some light on common misunderstandings in the study of small-world peer-to-peer networks. It shows that, contrary to some recent publications, Gnutella can indeed be described by a model with power-law degree distribution. To further distinguish the proposed model from other random graph models, this paper also applies two mathematical concepts, dimension and curvature, to the study of random graphs. These concepts help to understand the distribution of node distances in small-world networks. It thus becomes clear that the observed deficit in the number of reachable nodes in Gnutella-like networks is quite natural and no sign of any wrong or undesirable effect like, e.g., network partitioning.