Generic absorbing transition in coevolution dynamics

We study a coevolution voter model on a network that evolves according to the state of the nodes. In a single update, a link between opposite-state nodes is rewired with probability $p$, while with probability $1-p$ one of the nodes takes its neighbor's state. A mean-field approximation reveals an absorbing transition from an active to a frozen phase at a critical value $p_c=\frac{\mu-2}{\mu-1}$ that only depends on the average degree $\mu$ of the network. The approach to the final state is characterized by a time scale that diverges at the critical point as $\tau \sim |p_c-p|^{-1}$. We find that the active and frozen phases correspond to a connected and a fragmented network respectively. We show that the transition in finite-size systems can be seen as the sudden change in the trajectory of an equivalent random walk at the critical rewiring rate $p_c$, highlighting the fact that the mechanism behind the transition is a competition between the rates at which the network and the state of the nodes evolve.