Critical properties of the SIS model dynamics on the Apollonian network

Abstract

We present an analysis of the classical SIS (susceptible–infected–susceptible) model on the Apollonian network which is scale free and displays the small word effect. Numerical simulations show a continuous absorbing-state phase transition at a finite critical value λc of the control parameter λ. Since the coordination number k of the vertices of the Apollonian network is cumulatively distributed according to a power-law P(k) ∝ 1/kη−1, with exponent η ≃ 2.585, finite size effects are large and the infinite network limit cannot be reached in practice. Consequently, our study requires the application of finite size scaling theory, allowing us to characterize the transition by a set of critical exponents β/ν⊥, γ/ν⊥, ν⊥, β. We found that the phase transition belongs to the mean-field directed percolation universality class in regular lattices but, very peculiarly, is associated with a short-range distribution whose power-law distribution of k is defined by an exponent η larger than 3.