For populations with small effective population size, a genomic relationship matrix constructed for a large number of animals is singular. The APY algorithm exploits the reduced dimensionality of the matrix for lower computing cost, by splitting animals into core and noncore, and using recursion to predict noncore animals from core animals. Typically, the core animals are randomly selected, and their number is approximately equal to the number of eigenvalues explaining 98% variance in the matrix. While correlations in GEBV obtained when using 2 random cores is >0.99, some animals rerank. The purpose of this study was find the extent and origins of reranking using simulated and field data sets across species, and propose methods to reduce the reranking. In general, changes in GEBV obtained with 2 random cores are small but for some animals can be as large as 0.5 additive SD but predictivity and genetic merit of top 20 or 100 animals are nearly identical. Large changes occur nearly always for lower accuracy animals. Animals with low residual in the recursion do not show large changes. The changes are smaller for core than noncore animals. Some changes originate from blending. The changes can be minimized by increasing the number of core animals, and by treating important animals as core. Keeping same core animals over time reduces the changes for existing animals. GEBV generated with the APY algorithm exhibit some variations with the choice of core animals without affecting accuracy of selection.