Stochastic spreading processes on a network model based on regular   graphs
  The dynamic behaviour of stochastic spreading processes on a network model based on k-regular graphs is investigated. The contact process and the susceptible-infected-susceptible model for the spread of epidemics are considered as prototype stochastic spreading processes. We study these on a network consisting of a mixture of 2- and 3-fold oordinated randomly-connected nodes of concentration p and 1-p, respectively, with p varying between 0 and 1. Varying the parameter p from p=0 (3-regular graph of infinite dimension) to p=1 (2-regular graph - 1D chain) allows us to investigate their behaviour under such structural changes. Both processes are expected to exhibit mean-field features for p=0 and features typical of the directed percolation universality class for p=1. The analysis is undertaken by means of Monte Carlo simulations and the application of mean-field theory. The quasi-stationary simulation method is used to obtain the phase diagram for the processes in this environment along with critical exponents. Predictions for critical exponents obtained from mean-field theory are found to agree with simulation results over a large range of values for p up to a value of p=0.95, where the system is found to sharply cross over to the one-dimensional case. Estimates of critical thresholds given by mean-field theory are found to underestimate the corresponding critical rates obtained numerically for all values of p. 
S. V. Fallert, S. N. Taraskin