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Discrete gene regulatory networks (GRNs) play a vital role in the study ofrobustness and modularity. A common method of evaluating the robustness of GRNsis to measure their ability to regulate a set of perturbed gene activationpatterns back to their unperturbed forms. Usually, perturbations are obtainedby collecting random samples produced by a predefined distribution of geneactivation patterns. This sampling method introduces stochasticity, in turninducing dynamicity. This dynamicity is imposed on top of an already complexfitness landscape. So where sampling is used, it is important to understandwhich effects arise from the structure of the fitness landscape, and whicharise from the dynamicity imposed on it. Stochasticity of the fitness functionalso causes difficulties in reproducibility and in post-experimental analyses.  We develop a deterministic distributional fitness evaluation by consideringthe complete distribution of gene activity patterns, so as to avoidstochasticity in fitness assessment. This fitness evaluation facilitatesrepeatability. Its determinism permits us to ascertain theoretical bounds onthe fitness, and thus to identify whether the algorithm has reached a globaloptimum. It enables us to differentiate the effects of the problem domain fromthose of the noisy fitness evaluation, and thus to resolve two remaininganomalies in the behaviour of the problem domainof~\citet{espinosa2010specialization}. We also reveal some properties ofsolution GRNs that lead them to be robust and modular, leading to a deeperunderstanding of the nature of the problem domain. We conclude by discussingpotential directions toward simulating and understanding the emergence ofmodularity in larger, more complex domains, which is key both to generatingmore useful modular solutions, and to understanding the ubiquity of modularityin biological systems.

Resolving Anomalies in the Behaviour of a Modularity Inducing Problem Domain with Distributional Fitness Evaluation