Many real-world systems are connected together, in natural and man-made networks. A complex-valued laser network can simulate the working mechanism of human brain. However, amplitude control of a complex-valued laser network is seldom studied. In this paper, a ring network of complex-valued Lorenz laser systems is investigated. The ring network exhibits complex dynamics including hyper-chaos, quasi-periodic orbits, and coexisting hyper-chaos. Three kinds of single-parameter oriented amplitude controls are realized with varying or unvarying Lyapunov exponents in the ring network. Meanwhile, a simple knob can realize the amplitude rescaling of hyper-chaotic signals, which reduces the cost of circuit implementation. Moreover, a criterion of chaotic complete synchronization among all the nodes is established for a network with strong coupling. For relatively weak coupling, quasi-periodic complete synchronization is found, and the performance of chaotic synchronization is studied in terms of the cross-correlation coefficient. It is moreover revealed that the improvement and trend of synchronization performance are robust to the parity of the number of nodes for the same-scale laser networks.
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