Scale-free networks revealed from finite-size scaling

Network theory is a powerful tool to develop predictive models of physical, biological and social collective phenomena. A remarkable feature of many networks observed in nature is that they are approximately scale free: the fraction of nodes with k incident links (the degree) follows a power law for sufficiently large k. The value of the power law exponent as well as deviations from such scaling behavior provide invaluable information on the mechanisms underlying the formation of the network. Importantly, real networks are not infinitely large and the largest degree cannot be larger than the number of nodes. Finite size scaling is a useful tool for analyzing deviations from power law behavior in the vicinity of a critical point in a physical system arising due to a finite correlation length. Here we show that despite the essential differences between networks and critical phenomena, finite size scaling provides a powerful framework for analyzing self-similarity and the scale free nature of real networks. We analyze about two hundred naturally occurring networks with distinct dynamical origins, and find that a large number of these follow the finite size scaling hypothesis without any self-tuning.