Spreading processes are ubiquitous in natural and artificial systems, being disease contagion and rumor spreading the most important of these processes due to their practical relevance. Most of the literature focuses on continuous-time Markov chains (CTMC), for which exponential inter-event times are imposed. Despite its mathematical elegance and soundness for the analysis of critical behavior, its computational cost is often prohibitive. The discrete-time counterpart of CTMC is a Cellular-automaton (CA) approach that belongs to the class of discrete-time Markov chains (DTMC) and in which all events happen in the same time-window. As its computational cost is much lower CA models allow computationally studying and simulating a wider range of real and artificial systems. Interestingly, DTMC models also distinguish between fully reactive (RP) and contact processes (CP). The critical properties of them are different, with the RP having a vanishing critical point in scale-free networks and the CP showing a non-zero finite critical point, regardless of the underlying structure. Here we focus on the analytical and computational aspects of Cellular-automaton models, extending the computational algorithms and the models themselves. Specifically, we extend Monte Carlo methods, making them more suitable for the analysis of CA models and including all the transitions that are plausible for both disease contagion and rumor propagation processes. We round off our work by showing how the proposed framework can be applied to the study of spreading processes occurring on social networks. As a consequence of our work, we expect to reduce the gap in our understanding of CTMC and CA models, allowing other researchers to further apply this formalism in different applications.
