  We study the absorbing phase transition for the model of epidemic spreading,
Susceptible- Infected- Refractory (SIR), on one dimensional small world
networks. This model has been found to be in the universality class of the
dynamical percolation class, the mean field class corresponding to this model
is d = 6. The one dimensional case is special case of this class in which the
percolation threshold goes to one (boundary value) in the thermodynamic limit.
This behavior resembles slightly the behavior of one dimensional Ising and XY
models where the critical thresholds for both models go to zero temperature
(boundary value) in the thermodynamic limit. By analytical arguments and
numerical simulations we demonstrate that, increasing the connectivity (2k) of
this model on regular one dimensional lattice does not alter the criticality of
this model. Whereas we find that, this model crosses from a one dimensional
structure to mean field type for any finite value of the rewiring probability
(p), in manner is similarly to what happened in the Ising and XY models on
small world networks. In additional to that, we calculate the critical
exponents and the full critical phase space of this model on small world
network. We also introduce the crossover scaling function of this model from
one dimensional behavior to mean field behavior. Furthermore we reveal the
similarity between this model, and the Ising and XY models on the small world
networks.
