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 lambda(c) of the control parameter lambda. Since the coordination
number k of the vertices of the Apollonian network is cumulatively
distributed according to a power-law P(k) alpha 1/k(eta-1), with
exponent eta similar or equal to 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 beta/nu(perpendicular to), gamma/nu(perpendicular to),
nu(perpendicular to), beta. 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
eta larger than 3.
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