Connection between scale-free networks and nonextensive statistical mechanics

Abstract

The degree distribution of the so-called scale-free networks exhibits, quite often, the form p(k) ∝1/(k0+k)γ (with γ>0 and k0>0), in the limit of large networks. It happens that this form precisely coincides with the q-exponential p(k) ∝exp q(-k/κ) (q ≥1 and κ>0), with γ=1/(q-1) and k0=κ/(q-1). It optimises the nonadditive entropy \(S_q= \frac{k_B}{q-1}\left\{ 1-\sum_{k=1}^\infty [p(k)]^q \right\}\) with mathematically the same constraints that yield the stationary (or quasi-stationary) distribution in nonextensive statistical mechanics. In other words, the most ubiquitous form of the degree distribution of scale-free networks is a realisation of the hypothesis involved within the q-generalisation of Boltzmann-Gibbs statistical mechanics. In addition to this, we show that growth is not a necessary condition for having scale-free networks, in contrast with a widely spread belief.