Coherence in scale-free networks of chaotic maps

We study fully synchronized states in scale-free networks of chaotic logistic maps as a function of both dynamical and topological parameters. Three different network topologies are considered: (i) a random scale-free topology, (ii) a deterministic pseudofractal scale-free network, and (iii) an Apollonian network. For the random scale-free topology we find a coupling strength threshold beyond which full synchronization is attained. This threshold scales as $k^{-\mu }$, where $k$ is the outgoing connectivity and $\mu$ depends on the local nonlinearity. For deterministic scale-free networks coherence is observed only when the coupling strength is proportional to the neighbor connectivity. We show that the transition to coherence is of first order and study the role of the most connected nodes in the collective dynamics of oscillators in scale-free networks.