  We present a degree-based theoretical framework to study the
susceptible-infected-susceptible (SIS) dynamics on time-varying (rewired)
configuration model networks. Using this framework, we provide a detailed
analysis of the stationary state that covers, for a given structure, every
dynamic regimes easily tuned by the rewiring rate. This analysis is suitable
for the characterization of the phase transition and leads to three main
contributions. (i) We obtain a self-consistent expression for the
absorbing-state threshold, able to capture both collective and hub activation.
(ii) We recover the predictions of a number of existing approaches as limiting
cases of our analysis, providing thereby a unifying point of view for the SIS
dynamics on random networks. (iii) We reinterpret the concept of hub-dominated
phase transition. Within our framework, it appears as a heterogeneous critical
phenomenon : observables for different degree classes have a different scaling
with the infection rate. This leads to the successive activation of the degree
classes beyond the epidemic threshold.
