  The contact process and the slightly different
susceptible-infected-susceptible model are studied on long-range connected
networks in the presence of random transition rates by means of a strong
disorder renormalization group method and Monte Carlo simulations. We focus on
the case where the connection probability decays with the distance $l$ as
$p(l)\simeq\beta l^{-2}$ in one dimension. Here, the graph dimension of the
network can be continuously tuned with $\beta$. The critical behavior of the
models is found to be described by an infinite randomness fixed point which
manifests itself in logarithmic dynamical scaling. Estimates of the complete
set of the critical exponents, which are found to vary with the graph
dimension, are provided by different methods. According to the results, the
additional disorder of transition rates does not alter the infinite randomness
critical behavior induced by the disordered topology of the underlying random
network. This finding opens up the possibility of the application of an
alternative tool, the strong disorder renormalization group method to dynamical
processes on topologically disordered structures. As the random
transverse-field Ising model falls into the same universality class as the
random contact process, the results can be directly transferred to that model
defined on the same networks.
