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) random scale-free topology, (ii) deterministic pseudo-fractal scale-free network, and (iii) 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.