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Rapid advance in additive manufacturing techniques promises that, in the near future, the fabrication of functional cellular structures will be achieved with desired cellular microstructures tailored to specific application in mind. In this perspective, it is essential to develop a detailed understanding of the relationship between mechanical response and cellular microstructure. The present study reports on the results of a series of computational experiments that explore the effect cellular topology and microstructural irregularity (or non-periodicity) on overall mechanical response of cellular solids. Compressive response of various 2D topologies such as honeycombs, stochastic Voronoi foams as well as tetragonal and triangular lattice structures have been investigated as functions of quantitative irregularity parameters.