Networks play a vital role in the development of predictive models of physical, biological, and social collective phenomena. A quite remarkable feature of many of these networks is that they are believed to be approximately scale free: the fraction of nodes with k incident links (the degree) follows a power law p(k)∝k−λ for sufficiently large degree k. The value of the exponent λ as well as deviations from power law scaling provide invaluable information on the mechanisms underlying the formation of the network such as small degree saturation, variations in the local fitness to compete for links, and high degree cutoffs owing to the finite size of the network. Indeed real networks are not infinitely large and the largest degree of any network 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 empirical 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. Notably this is the case of biological protein interaction networks, technological computer and hyperlink networks and informational citation and lexical networks. Marked deviations appear in other examples, especially infrastructure and transportation networks, but also social, affiliation and annotation networks. Strikingly, the values of the scaling exponents are not independent but satisfy an approximate exponential relationship.
Scale-free networks revealed from finite-size scaling
Matteo Serafino, Giulio Cimini, Amos Maritan, Samir Suweis, Jayanth R. Banavar, Guido Caldarelli