Scaling behavior of the contact process in networks with long-range connections

We present simulation results for the contact process on regular cubic networks that are composed of a one-dimensional lattice and a set of long edges with unbounded length. Networks with different sets of long edges are considered that are characterized by different shortest-path dimensions and random-walk dimensions. We provide numerical evidence that an absorbing phase transition occurs at some finite value of the infection rate and the corresponding dynamical critical exponents depend on the underlying network. Furthermore, the time-dependent quantities exhibit log-periodic oscillations in agreement with the discrete scale invariance of the networks. In case of spreading from an initial active seed, the critical exponents are found to depend on the location of the initial seed and break the hyperscaling law of the directed percolation universality class due to the inhomogeneity of the networks. However, if the cluster-spreading quantities are averaged over initial sites, the hyperscaling law is restored.