Relaxation time as a tool to characterize phase transitions in dynamic processes on complex networks

The criticality of non-equilibrium processes in regular lattices can be investigated using the generalization of susceptibility of equilibrium systems which is intrinsically associated to the spatial and temporal correlations. However, many complex networks are highly heterogeneous and requires alternative definitions of susceptibility to capture different properties of system, generating an ambiguity in the susceptibility definition. Moreover, spatial correlations in complex networks becomes meaningless due to small-world property but temporal correlations are well defined and emerges as an alternative or complementary tool to characterize phase transitions in complex networks. We analyzed the integrated correlation time for two basic reaction-diffusion processes related to the epidemic spreading and observed that this characteristic time provides the correct critical point and exponents for the contact process (CP), a model that exhibits an absorbing state phase transition, on regular lattices in 1, 2 and 3 dimensions. The method was also able to identify the epidemic threshold of CP model on synthetic scale-free networks with a power-law degree distribution $P(k)\sim k^{-\gamma}$. Also, we applied the method in a double random regular network where epidemic processes undergoes two phase transitions and both were identified. Currently we are investigating the susceptible-infected-susceptible (SIS) model where multiple transitions are also observed in finite-size networks.
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We acknowledge the financial support of FAPEMIG.