Circadian rhythms are ubiquitous and are widely observed in plants, animals, fungi and cyanobacteria. They are believed to control many processes of varying scales, right from cell regeneration to sleeping and feeding patterns of animals. Although there are many different network architectures that can produce circadian oscillations, it is interesting to know if there are any common design principles underlying all these oscillators. Further, how does performance vary between these different architectures? A similar study for identifying networks undergoing perfect adaptation was done by Ma et al (2009; Cell. 138, 760–773). In a similar vein, we here carry out a thorough unbiased search of all the topologies of 2-node and 3-node networks to determine the minimum requirements needed to achieve circadian behaviour and also the key topological and dynamic features of these networks. The search space of topologies and parameters grows exponentially due to combinatorial explosion, leading to a computationally challenging problem. However, these searches are independent and hence are easily parallelisable, particularly across Graphics Processing Units (GPUs), which are capable of efficiently running highly parallel programs and outperform CPUs in terms of raw computing power. We hence decided to parallelize our search using the GPU; our parallelized simulations of the 2-node network and 3-node networks on the GPU and obtained speed-ups of up to 150x. We determined the topologies that showed circadian oscillations and also evaluated the key topological and parametric constraints needed to sustain these oscillations. We will also illustrate the robustness of the favourable topologies.