  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.
