  We investigate the dynamics of network minority games on Kauffman's NK
networks (Kauffman nets), growing directed networks (GDNets), as well as
growing directed networks with a small fraction of link reversals (GDRNets). We
show that the dynamics and the associated phase structure of the game depend
crucially on the structure of the underlying network. The dynamics on GDNets is
very stable for all values of the connection number $K$, in contrast to the
dynamics on Kauffman's NK networks, which becomes chaotic when $K>K_c=2$. The
dynamics of GDRNets, on the other hand, is near critical. Under a simple
evolutionary scheme, the network system with a "near" critical dynamics evolves
to a high level of global coordination among its agents. In particular, the
performance of the system is close to the optimum for the GDRNets; this
suggests that criticality leads to the best performance. For Kauffman nets with
$K>3$, the evolutionary scheme has no effect on the dynamics (it remains
chaotic) and the performance of the MG resembles that of a random choice game
(RCG).

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