  Discrete gene regulatory networks (GRNs) play a vital role in the study of
robustness and modularity. A common method of evaluating the robustness of GRNs
is to measure their ability to regulate a set of perturbed gene activation
patterns back to their unperturbed forms. Usually, perturbations are obtained
by collecting random samples produced by a predefined distribution of gene
activation patterns. This sampling method introduces stochasticity, in turn
inducing dynamicity. This dynamicity is imposed on top of an already complex
fitness landscape. So where sampling is used, it is important to understand
which effects arise from the structure of the fitness landscape, and which
arise from the dynamicity imposed on it. Stochasticity of the fitness function
also causes difficulties in reproducibility and in post-experimental analyses.
  We develop a deterministic distributional fitness evaluation by considering
the complete distribution of gene activity patterns, so as to avoid
stochasticity in fitness assessment. This fitness evaluation facilitates
repeatability. Its determinism permits us to ascertain theoretical bounds on
the fitness, and thus to identify whether the algorithm has reached a global
optimum. It enables us to differentiate the effects of the problem domain from
those of the noisy fitness evaluation, and thus to resolve two remaining
anomalies in the behaviour of the problem domain
of~\citet{espinosa2010specialization}. We also reveal some properties of
solution GRNs that lead them to be robust and modular, leading to a deeper
understanding of the nature of the problem domain. We conclude by discussing
potential directions toward simulating and understanding the emergence of
modularity in larger, more complex domains, which is key both to generating
more useful modular solutions, and to understanding the ubiquity of modularity
in biological systems.
