Accurate prediction of noise propagation in biological networks is key to understanding faithful signal propagation in gene networks as well as for designing noise-tolerant synthetic gene circuits. Knowledge on how biological fluctuations propagate up the development ladder of biological systems is currently lacking. Similarly, little research effort has been devoted to the analysis of error propagation in biological networks. To capture and characterize error evolution, this paper considers a Boolean network (BN) model representation of a biological network such that nodes on the graph represent diverse biological entities, e.g., proteins, genes, messenger-RNAs, etc. In addition, the network edges capture the interactions between nodes. By conducting a density evolution analysis on the graphical model based on node functionalities, a recursive closed-form expression for error propagation is derived. Subsequently, the recursive equation allows us to obtain a necessary condition to guarantee noise-error elimination in dynamic discrete gene networks. Our analytical formulations provide a step toward achieving optimal network parameters for resilience against variability or noise in biology