Claims of “brain criticality” suggest that the generation of activity on all scales in a critical state (e.g. neuronal avalanches) may underlie complex, adaptive cognition. However the relationship of the powerlaw distributed network observables behind these claims to brain functional output is not known. Here we use a novel approach to this problem by considering functional output directly: the networks generated by Drosophila melanogaster during pre-copulatory courtship. This courtship body language, treated as a symbolic dynamics, has been shown to exhibit Context-Free and Context-Sensitive grammars in the Chomsky hierarchy; here we show that the underlying networks corresponding to such grammars deviate from scale-free structure. We provide a simple network growth model, which matches the degree distributions of the non-scale-free networks by breaking the preferential attachment paradigm with a second internal linking process. From these observations, we suggest that the higher level cognition associated with such grammatical structure may not be compatible with a critical state.