Sampling methods for absorbing-state phase transitions on complex networks

Phase transitions into absorbing states, configurations from which the system cannot escape, are a current topic on the frontier of the non-equilibrium statistical mechanics. Despite the fact that there are still problems being investigated on regular topologies, phase transitions on complex networks have been subject of increasing interest to the scientific community due to the fact that such networks describe a wide variety of systems, relevant both in the technological and intellectual aspects. Considering the dynamical nature and huge size of real complex networks, the statistical physics approach has proved to be suitable since its association with the graph theory permits a characterization of emergent macroscopic phenomena in terms of the time evolution of basic elements composing the system. Since numerically we deal with finite systems necessarily, and finite-size effects are particularly stronger on power-law complex networks than on its regular lattice counterparts, a finite-size scaling (FSS) analysis is required. In this work, we perform simulations for the susceptible-infected-susceptible (SIS) model and the contact process (CP) on power-law networks with a degree distribution $P(k) \sim k^{-\gamma}$, using three different sampling techniques: the reflecting boundary condition (RB), the coupled vanishing external field (EF) and the quasi-stationary (QS) simulation methods. We show that the three methods are equivalent for the CP, successfully capturing its critical behavior on networks. The SIS on power-law networks presents multiple transitions and for this model the three sampling techniques successfully characterize the transition associated to an endemic phase (diverging lifespan), but provide different results for transitions involving localized states. In this regime, the RB and EF method do not capture, for example, a transition associated with the activation of the most connected vertex of the network, as predicted by the quenched mean-field theory (QMF).
Acknowledgement: FAPEMIG